Set Theory and Its Applications: Annual Boise Extravaganza in Set Theory, Boise, Idaho, 1995-2010

CONTEMPORARYMATHEMATICS

American Mathematical Society

533

Set Theory and Its Applications

Annual Boise Extravaganzain Set TheoryBoise, Idaho1995–2010

L. BabinkostovaA. E. Caicedo

S. GeschkeM. Scheepers

Editors

conm-533-scheepers2-cov.indd 1 10/5/10 2:10 PM


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American Mathematical SocietyProvidence, Rhode Island

CONTEMPORARYMATHEMATICS

533


Annual Boise Extravaganza in Set Theory Boise, Idaho 1995–2010

L. Babinkostova A. E. Caicedo

S. Geschke M. Scheepers

Editors

Editorial Board

Dennis DeTurck, managing editor

George Andrews Abel Klein Martin J. Strauss

2000 Mathematics Subject Classification. Primary 03C55, 03E15, 03E17, 03E35, 03E60,46L05, 54A20, 54A25, 54D20, 91A44.

Library of Congress Cataloging-in-Publication Data

Boise Extravaganza in Set Theory Conference.Set theory and its applications : annual Boise Extravaganza in Set Theory, 1995–2010, Boise,

Idaho / L. Babinkostova . . . [et al.], editors.p. cm. — (Contemporary mathematics ; v. 533)

Includes bibliographical references.ISBN 978-0-8218-4812-8 (alk. paper)1. Set theory—Congresses. I. Babinkostova, L. (Liljana) II. Title.

QA248.B66 2011511.3′22—dc22

2010030559

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10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11

Contents

Preface vii

Past Invited Speakers at BEST ix

Some Positive Results in the Context of Universal ModelsMirna Dzamonja 1

A Survey of D-spacesGary Gruenhage 13

Combinatorics of Filters and IdealsMichael Hrusak 29

More Structural Consequences of ADRichard Ketchersid 71

αi-selection Principles and GamesLjubisa D.R. Kocinac 107

Jensen’s Diamond Principle and Its RelativesAssaf Rinot 125

Paracompactness and Normality in Box Products: Old and NewJudith Roitman 157

Some Problems and Techniques in Set-Theoretic TopologyFranklin D. Tall 183

Menger’s and Hurewicz’s Problems: Solutions from “The Book” andRefinements

Boaz Tsaban 211

A Trichotomy Theorem in Natural Models of AD+

Andres E. Caicedo and Richard Ketchersid 227

The Coinitialities of Efimov SpacesStefan Geschke 259

Uniforming n-place Functions on Well Founded TreesEsther Gruenhut and Saharon Shelah 267

A Classical Proof of the Kanovei-Zapletal CanonizationBenjamin D. Miller 281

Lords of the IterationAndrzej Ros�lanowski and Saharon Shelah 287

v


Preface

The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small,locally funded conference dedicated to Set Theory and its Applications. A numberof years after its inception BEST started being funded by the National ScienceFoundation. Without this funding it would not have been possible to maintain theconference.

The conference remained relatively small with many opportunities for its par-ticipants to meet informally. We like to think that during these years BEST hasmade it possible for the numerous set theorists who have participated in it to ab-sorb, besides the new developments featured in the conference talks, also part of thefolklore and traditions of the field of set theory and its relatives. An explicit effortwas made to bring together role models from various career stages as well as thenew generation to support some notion of continuity in the field. A list of these in-vited speakers follows this preface. The interested reader can find more informationabout the past BEST conferences at http://math.boisestate.edu/∼best.

This volume has a similar purpose. In it the reader will find valuable papersranging from surveys that make available knowledge that has been around for sev-eral decades as unpublished lore, to hybrid survey-research papers, to pure researchpapers. Readers can be assured of the authority of each paper since each has beencarefully refereed. The reader will also find that the subjects treated in these pa-pers range over several of the historically strongly represented areas of set theoryand its relatives. Rather than expound the virtues of each paper individually here,we invite the reader to learn from the authors.

Bringing to publication such a collection of papers is not possible without thegenerous dedication of authors and referees and the services of a publisher. Wewould like to thank all authors and referees for their selfless contributions to thisvolume. And we particularly would like to thank the publisher, ContemporaryMathematics, and Christine Thivierge, for the guidance they provided during thisprocess.

Liljana Babinkostova,Andres E. Caicedo,Stefan GeschkeandMarion Scheepers.

vii


Past Invited Speakers at BEST

D. Aspero (BEST 13)Bristol University (Great Britain)

L. Babinkostova (BEST 12)University of St Cyril and Methodius(Macedonia)

J.E. Baumgartner (BEST 2)Dartmouth College

A.R. Blass (BEST 2, BEST 6)University of Michigan (Ann Arbor)

J. Brendle (BEST 5)Dartmouth College

D.K. Burke (BEST 13)Miami University (Ohio)

A.E. Caicedo (BEST 16)University of California (Los Angeles)

T.J. Carlson (BEST 16)Ohio State University (Columbus)

J.D. Clemens (BEST 13)Pennsylvania State university

J. Cummings (BEST 10)Carnegie Mellon University

C.A. DiPrisco (BEST 10)IVIC (Venezuela)

N. Dobrinen (BEST 15)Kurt Godel Research Center ofmathematical Logic (Austria)

A. Dow (BEST 5)York University (Canada)

M. Dzamonja (BEST 7)University of Wisconsin (Madison)

M. Dzamonja (BEST 14)University of East Anglia (GreatBritain)

T. Eisworth (BEST 11)University of Northern Iowa

I. Farah (BEST 8)York University (Canada)

M.D. Foreman (BEST 3)University of California (Irvine)

D.H. Fremlin (BEST 6)University of Essex (Great Britain)

S. Fuchino (BEST 7)Kitami Institute of Technology (Japan)

F. Galvin (BEST 3)University of Kansas (Lawrence)

M. Groszek (BEST 16)Dartmouth College

G. Gruenhage (BEST 12)Auburn University

A. Hajnal (BEST 3)Hungarian Academy of Science(Hungary)

J.D. Hamkins (BEST 10)City University of New York (StatenIsland)

K.P. Hart (BEST 13)Technical University Delft(Netherlands)

G. Hjorth (BEST 14)University of California (Los Angeles)

M. Hrusak (BEST 15)UNAM-Morelia (Mexico)

ix

x PARTICIPANTS

T. Ishiu(BEST 17)Miami University (Ohio)

S.C. Jackson (BEST 18)University of North Texas

I. Juhasz (BEST 15)Alfred Renyi Institute of Mathematics(Hungary)

W. Just (BEST 1)Ohio University (Athens)

A. Kanamori (BEST 6)Boston University

B. Kastermans (BEST 17)University of Wisconsin (Madison)

A.S. Kechris (BEST 3)California Institute of Technology

Lj.D.R. Kocinac (BEST 18)University of Nis (Serbia)

M. Kojman (BEST 17)Ben Gurion University (Israel)

P. Koszmider (BEST 9)Universidade de Sao Paulo (Brazil)

K. Kunen (BEST 9)University of Wisconsin (Madison)

C. Laflamme (BEST 2)University of Calgary (Canada)

J.A. Larson (BEST 5)University of Florida (Gainesville)

P.B. Larson (BEST 14)Miami University (Ohio)

D.A. Martin (BEST 4)University of California (Los Angeles)

R.D. Mauldin (BEST 4)University of North Texas (Denton)

H. Mildenberger (BEST 11)University of Vienna (Austria)

A.W. Miller (BEST 1, BEST 12)University of Wisconsin (Madison)

E.C. Milner (BEST 4)University of Calgary (Canada)

W.J. Mitchell (BEST 14)University of Florida (Gainesville)

J.T. Moore (BEST 19)Cornell University

J. Pawlikowski (BEST 8)Wroclaw University (Poland)

E. Pol (BEST 17)University of Warsaw (Poland)

D. Raghavan (BEST 19)University of Toronto (Canada)

A. Rinot (BEST 18)Tel Aviv University (Israel)

J. Roitman (BEST 6)University of Kansas (Lawrence)

A. Ros�lanowski (BEST 11)University of Nebraska, Omaha

M.E. Rudin (BEST 9)University of Wisconsin (Madison)

G. Sargsyan (BEST 18)University of California (Berkeley)

S. Solecki (BEST 8)University of Indiana (Bloomington)

S. Shelah (BEST 5)Hebrew University of Jerusalem (Israel)

J.R. Steel (BEST 8)University of California (Berkeley)

J. Steprans (BEST 1)York University (Canada)

P.J. Szeptycki (BEST 9)Ohio University (Athens)

F.D. Tall (BEST 19)University of Toronto (Canada)

S. Thomas (BEST 10)Rutgers University

S. Todorcevic (BEST 1, BEST 7)University of Toronto (Canada)

T. Usuba (BEST 19)University of Bonn (Germany)

PARTICIPANTS xi

B. Velickovic (BEST 15)Equipe de Logique Mathematique,Universite de Paris 7 (France)

W.A.R. Weiss (BEST 4)University of Toronto (Canada)

W.H. Woodin (BEST 7)University of California (Berkeley)

J. Zapletal (BEST 12)University of Florida (Gainesvilla)

M. Zeman (BEST 11)University of California (Irvine)


Some positive results in the context of universal models

Mirna Dzamonja

Abstract. Let (K,≤) be a quasi-ordered set or a class, which we think of as aclass of models. A universal family in K is a dominating family in (K,≤), andif there is such a family of size one then we call its single element a universalmodel in K. We survey some important instances of the existence of smalluniversal families and universal models in various classes and point out theinfluence of the axioms of set theory on the existence of such objects. Then wepresent some of the known methods of constructing small universal familiesand universal models and discuss their limitations, pointing out some of theremaining open questions.

1. Introduction

Let (K,≤) be a a quasi-ordered set or a class, which we think of as a class ofmodels. In the context that interests us this may be the class of models of a givencardinality of some first order theory ordered by elementary embedding or the classof models of a given cardinality of some abstract elementary class quasi-orderedby the inherited order. We may also consider classes whose membership is notdetermined by cardinality but by some other cardinal invariant such as topologicalweight. A universal family in K is a dominating family in (K,≤), and if there issuch a family of size one then we call its single element a universal model in K. Thesmallest size of a universal family is called the universality number of (K,≤).

Immediate examples of universal models are the the rationals considered as alinear order, which embed every countable linear order, or [0, 1]κ which contains aclosed copy of every compact space of weight κ, or the random graph which embedsevery countable graph. There are many other examples in just about every branchof mathematics. The purpose of this article is to discuss general methods whichcan be used to demonstrate the existence of universal models in various specificcontexts. In this presentation we concentrate on countable first order theories.The article does not deal with the related subject of methods that can be usedto demonstrate that a certain theory does not have a small universal family at acertain cardinal; we can refer the reader to the survey article [2] for a descriptionof some such ideas.

1991 Mathematics Subject Classification. Primary 03E35, 03C55.Key words and phrases. universal models, universal families.The author thanks Mittag-Leffler Institute for their support in September 2009 and EPSRC

for their support through grant EP/G068720.

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Contemporary MathematicsVolume 533, 2010

c©2010 American Mathematical Society

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1

2 MIRNA DZAMONJA

The article is organised as follows: it first presents some classical results frommodel theory that apply in ZFC or under the GCH-like assumptions. This isthe content of §2, which is divided to subsections relating to saturated and specialmodels and countable universal models. Section 3 moves into the realm where CHis violated and considers the possible existence of universal models in forcing exten-sions where CH fails, concentrating on ℵ1. This section is divided into subsectionsdealing with graphs, triangle-free graphs, linear orders and amenability at ℵ1.

Throughout κ stands for an infinite cardinal. An unattributed T stands fora theory, which means a complete first order theory with infinite models. Forsimplicity in this presentation we restrict ourselves to the case of countable theories.A type for us is any consistent set of sentences, and a complete type is a maximalconsistent set of sentences. By a universal model of T of size κ we mean a modelin which every other model of T of size κ embeds elementarily1.

2. Some classical results

We present some results on the existence of certain kinds of universal modelsfor complete first order theories, again restricting to the case of countable theories.Results presented in this section mostly come from 1960s.

2.1. Saturated and special models.

Definition 2.1. A model M of a theory T is said to be κ-saturated if forevery A ⊆ M of size < κ, the expansion (M,a)a∈A realises every type Γ(x) of theexpanded language which is consistent with the complete theory Th(M,a)a∈A. Mis said to be saturated if it is |M |-saturated.

A generalisation of Cantor’s proof that the rationals are a unique countabledense linear order with no first or last elements, gives us that saturated models areuniversal. See Theorem 2.4 for a detailed statement. The basic theorem about theexistence of saturated models at uncountable cardinals is the following

Lemma 2.2. (Vaught, [10]) Suppose that N is a model T of size ≤ 2κ. Thenthere is a κ+-saturated extension M of N of size 2κ.

Proof. Note that |[N ]κ| = 2κ, and for every A ∈ [N ]κ the language LA =L∪{ca}a∈A has size κ, therefore the total number of relevant types is 2κ. Introducefor each such type Σ a new symbol cΣ. We can form the set of sentences Γ consistingof the elementary diagram of N along with Σ(cΣ) for all relevant Σ. This is a finitelysatisfiable set of sentences, hence it has a model, so it has a model of size 2κ. LetM be the reduction of this model to the original language. �2.2 �

Provided that we assume some cardinal arithmetic this now gives us the exis-tence of saturated models in successor cardinals:

Theorem 2.3. (Vaught, [10]) Suppose that κ satisfies 2κ = κ+. Then there isa saturated model M of T of size κ+.

Proof. Let N be any model of T of size 2κ. By induction on α < 2κ we choosemodels Nα of T so that

• N0 = N , Nα ≺ Nα+1, Nδ =⋃

α<δ Nα for δ limit > 0,

1In this context, because of the compactness theorem, we could equivalently require thatevery model of infinite size ≤ κ embeds into the κ-universal model.

22

SOME POSITIVE RESULTS IN THE CONTEXT OF UNIVERSAL MODELS 3

• For every A ⊆ Nα of size κ and type Γ(x) consistent with the completetheory of (Nα, a)a∈A, Γ(x) is realised in Nα+1.

To do the construction at the successor stage α+1 we simply apply Lemma 2.2 tothe model Nα. At the end let M =

⋃α<κ+ Nα. �2.3 �

Saturation is not necessary for universality. A weaker notion that still sufficesis that of a special model: a model M of size κ is a special model if it is the unionof an elementary chain 〈Mλ : λ < λ∗〉 such that each λ is a cardinal and Mλ isλ+-saturated. By definition, saturated models are special. The opposite is nottrue. Relationship between saturation, speciality and universality is given by thefollowing:

Theorem 2.4. Every saturated model is special and every special model isuniversal.

Proof. The first sentence follows by definition. Suppose now that M is theunion of a specializing chain 〈Mi : i < i∗〉 where Mi+1 is κi-saturated for somecardinals κi increasing to κ, which is the size of M . We may without loss ofgenerality assume that this chain is continuous. Let N be a model of T of size κ,enumerated as {xα : α < κ}. By induction on α we choose yα ∈ M such thatxα → yα is an elementary embedding. We choose yαs in blocks of κi for i < i∗,that is the induction is on i, so that α < κi =⇒ yα ∈ Mi.

Suppose that 〈yα : α < κi〉 have been chosen. Choose yα ∈ Mi+1 for α ∈[κi, κi+1) by induction on α. Suppose that α < κi+1 and yβ for β < α have beenchosen. We use a modification of Cantor’s idea from the proof of the uniqueness ofthe rationals: let Γ(x) be the type of xα in (N, xβ)β<α. Therefore Γ is a type ofsize < κi+1 in (Mi+1, yβ)β<α. By the saturation of Mi+1 we can find yα ∈ Mi+1

which realises this type and the induction continues. �2.4 �Theorem 2.5. (Morley and Vaught, [10]) Suppose that κ = 2<κ is uncount-

able2. Then there is a special model of T of size κ.

Proof. If κ = λ+ then by the assumption 2λ = κ and hence by Theorem 2.3there is a saturated model of T of size κ. Suppose then that κ is a limit cardinal.Our assumptions allow us to choose an increasing sequence 〈κi : i < i∗〉 of infinitecardinals with limit κ, and such that 2κi = κi+1. Then we build an elementary chain〈Mi : i < i∗〉 by starting with any model M0 of T of size κ0, and applying Lemma2.2 at successor stages to get a model Mi+1 of size κi+1 which is κi

+-saturated.Letting M =

⋃i<i∗ Mi we obtain a special model as required. �2.5 �

Conclusion 2.6. Every countable first order theory T has a universal modelof size κ for every κ > ℵ0 satisfying 2<κ = κ.

Results presented above can be found in Chapter V of [1]. We also quote aselection of theorems which show that some assumptions on the kind of theoriesand on cardinal arithmetic are necessary for this conclusion.

Theorem 2.7. (1) (Hausdorff, [5]) There exists a saturated linear order of sizeκ > ℵ0 iff κ = κ<κ.

(2) (Shelah, see [7] for a proof) Suppose that V is a model of GCH in whichκ is a regular cardinal, and let G be V -generic for the Cohen forcing which adds

2The role of uncountability here is that we need κ to be larger than the size of T .

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4 MIRNA DZAMONJA

κ++ Cohen subsets to V . Then in V [G], in power κ+, there is no universal graph,linear order, or generally model of a complete first order T which is unstable in κ.

(3) (Kojman-Shelah, [7]) Suppose κ is a regular cardinal and there is somecardinal λ such that λ+ < κ < 2λ. Then there is no universal linear order ofcardinality κ.

Note that if T is stable in κ then it follows by an argument analogous to thatof Theorem 2.8 below that T has a saturated model of size κ.

2.2. Countable universal models. Countable universal models are abun-dant in mathematics. For example, in addition to the examples we mentioned inthe Introduction there are examples in topology: any countable dense subset ofthe Urysohn space is countably universal in the class of metric spaces with almostisometric embedding. In model theory however, the theory of countable modelsis different from the theory of the uncountable ones - as is most strikingly wit-nessed by the well developed classification theory using the number of pairwisenon-isomorphic models [11], versus the still unresolved Vaught conjecture aboutcountable models. In the theory of universal models it is similarly the case thatcountable models have a special role. Specifically, the methods presented in the restof this paper tend to apply only to uncountable models. However, this is not thecase with the concept of saturated models, which makes perfect sense in the case ofκ = ℵ0, see Definition 2.1. In fact, the concept of saturation was first introduced inthe countable case, by Vaught. Exactly the same proof as in the uncountable caseapplies to show that saturated models are universal. For example, a well knownexample of a countable universal model is the linear order of the rationals, and therationals are an example of a saturated model. In the case of countable modelsthere is a syntactic characterisation of the existence of saturated models, due toVaught in [17]. It also appears as Theorem 2.3.7 in [1].

Theorem 2.8. (Vaught, [17]) T has a countable saturated model iff for everyn < ω, T has only countably many complete types in n-variables.

Proof. In the forward direction, let M be a countable saturated model of T .Then for every n and complete type Γ in n variables, Γ is realised by an n-tuple inM . Since there are only countably many such tuples and none of them can realisemore than one complete type, the conclusion follows.

In the other direction, we shall expand T to a maximal consistent theory T ∗ inan expanded language, such that any countable model of T ∗ gives us a saturatedmodel of T . Let us form the language L∗ by adding countably many new constantsymbols {cn : n < ω} to L. Let {ϕn : n < ω} be an enumeration of all sentences inL∗. For every m-element subset Y of {cn : n < ω}, the complete types Γ(x) of Tin LY are in one-to-one correspondence with the complete types Σ(x0, . . . xm−1, x)of T in L, and hence there are only countably many of them by the assumption.We let {Γn(x) : n < ω} enumerate all such types Γ(x), where Y ranges over allfinite subsets of {cn : n < ω}.

By induction on n < ω we choose an increasing sequence Tn of consistenttheories with T0 = T , such that

: (1) each Tn contains only finitely many symbols from L∗ \ L,: (2) either ϕn ∈ Tn+1 or ¬ϕn ∈ Tn+1,: (3) if ϕn ∈ Tn+1 and ϕn = (∃x)ψ(x), then ψ(c) ∈ Tn+1 for some c ∈ L∗\L,

44


: (4) if Γn(x) is consistent with Tn, then Γn(c) ⊆ Tn+1 for some c ∈ L∗ \ L.The induction is straightforward and we obtain that T ∗ =

⋃n<ω Tn is a maximal

consistent theory in L∗. Let M ′ be a countable model of it, therefore by (3),M ′ = (M,an)n<ω where M = {an : n < ω}. M is clearly a model of T and (4)guarantees that it is saturated. �2.8 �

Of course, saturation is not a neccessary condition for universality: ω +Q is auniversal countable linear order but is not a saturated model. We shall not developthis topic further in this article and from now on we shall only consider uncountablemodels.

3. Changing the cardinal arithmetic

When we leave the realm of GCH and its remnants we are more or less leftwith universes which we construct with forcing and where instances of GCH areviolated by the construction of the extension. Theorem 2.7(2) shows that if weare not careful about how we do this, we shall end up basically with no universalmodels of any sort. Theorem 2.7(3) shows that for certain theories such as linearorders, no matter how careful we are, if we violate GCH sufficiently (including

making 2κ = 2κ+

), the universality number at κ+ will jump to the largest possible

value of 2κ+

. We shall see below that for certain other theories, for example theoryof graphs, it is possible to violate GCH as much as we like and still keep theuniversality number low. This indicates that the ability of having a small universalnumber in ‘reasonable’ forcing extensions in which the relevant instances of GCHare violated is a property of the theory itself, which is not possessed by all theories.In fact, in a series of papers, e.g. [7], [8], [6], [15], [16], [3] the thesis claiming theconnection between the complexity of a theory and its amenability to the existenceof universal models, has been pursued. In [4] we introduced the following definition,which formalised these notions:

Definition 3.1. We say that a theory T is amenable iff whenever λ is anuncountable cardinal satisfying λ<λ = λ and 2λ = λ+, while θ satisfies cf(θ) >λ+, there is a λ+-cc (< λ)-closed forcing notion that forces 2λ to be θ and theuniversality number of T at λ+ to be smaller than θ.

Localising this definition at a particular λ we define what is meant by theoriesthat are amenable at λ.

Connected to this definition there is a somewhat technical definition of highnon-amenability (see Definition 0.3 of [4]). We shall not quote the definition butstate only that high non-amenability of T implies that T is not amenable in the senseof Definition 3.1, and that the theory of dense linear order with no endpoints is aprototypical example of a highly non-amenable T . We should also comment that theamenability/high non-amenability is envisioned as a dividing in the classificationtheory of unstable theories. The exact syntactic properties that correspond to thisline have not been found yet, but is known that the property SOP4 implies high non-amenability and simplicity implies amenability. We shall not go into these model-theoretic considerations at this point but refer the reader to the articles mentionedabove. Here we shall simply be concerned to give examples and techniques whichapply to amenable theories. For simplicity in presentation we shall concentrate onthe case of amenability at λ = ℵ1 but we warn the reader that there are somecaveats in looking only at this case-we shall indicate them below.

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6 MIRNA DZAMONJA

3.1. Graphs. The goal of this section is to present a consistency result show-ing that it is possible to force over a model of GCH to obtain a model with a largecontinuum and a small universal family of graphs at ℵ1. In fact, we shall demon-strate that the theory of graphs is amenable at ℵ1, in the sense of Definition 3.1.The earliest theorem to this extent is in Shelah’s [13], but unfortunately there is anerror, which was noted and corrected by Shelah in [14]. The same year Mekler [9]found a different proof using, unlike [14], a ccc forcing. We shall present Mekler’sproof. Let T in this section stand for the model completion of the theory of graphs,so T is a complete model complete theory and we shall show that it is consistentthat CH fails and there is a universal model of T of size ℵ1.

To start with let V be a model of GCH and let us fix a sequence 〈Aα : α < ω1〉of subsets of ω1 with pairwise intersections countable. Let χ be a large enoughcardinal and we shall work with with elementary substructures of 〈H(χ),∈, . . .).The forcing will be an iteration 〈Pα, Q

˜β : α ≤ ω2, β < ω2〉 with finite supports

such that each Pα is ccc. The first coordinate will have a special role: it will adda generic graph G∗ on ω1 by finite conditions. Note that G∗ is a model of T . Wedefine the other coordinates by induction on α.

For α ≥ 1, if Pα has been defined and is ccc, suppose that a bookkeeping givesus a name τ

˜α, which is a Pα-name of a model of T . We first define Q

˜α. In V we

fix a continuous sequence Nα = 〈Nαi : i < ω1〉 of countable elementary submodels

of 〈H(χ),∈, . . .〉 with Nαi ∈ Nα

i+1 and 〈Pβ, Q˜

β : β < α〉, Pα ∈ Nα0 (so α ∈ Nα

0 ).For η < ω1 let i(η) denote the first i such that η ∈ Nα

i . Then Qα consists of pairsq = (Xq, fq) such that X = Xq is a finite subset of ω1 and f = fq is finite functionfrom ω1 to Aα whose range is disjoint from X and which satisfies

η ∈ dom(f) =⇒ f(η) ∈ Nαi(η)+1 \Nα

i(η).

The idea is that Qα adds an embedding of τα into G∗. To say this precisely, wedefine Pα+1 to consists of elements in Pα ∗ Q

˜α such that p � α decides the τ

˜α

structure of dom(fp(α)), ran(fp(α)) ⊆ p(0) and fp(α) is an embedding of the τ˜α

structure of dom(fp(α)) into p(0).Easy density arguments show that the P = Pω2

adds an embedding of each ταinto G∗ and that if the forcing is indeed ccc, we can do the bookkeeping so to coverall models of T of size ℵ1 in the extension. The main point of the proof is to showthat the forcing has ccc. The proof uses a certain amalgamation property that ispresent in the theory of graphs, but not for example in the theory of triangle-freegraphs or the theory of linear orders. We shall describe this amalgamation propertyat the point of the proof where we use it.

We need to pass to a dense set of conditions. Say that for 1 ≤ α ≤ ω2, p ∈ Pα

is complete if for every β < α and i < ω1

: (i) p � β∩Nβi is a condition and it determines the τβ

˜-structure of dom(fp(β))

and: (ii) if r ∈ Nβ

i extends p � β ∩Nβi and 1 ≤ γ < β then ran(fr(γ)) ∩ p(0) ⊆

ran(fp(γ)).

It can be proved that the set of complete conditions in Pα is dense, for all α. Boththis and the fact that the forcing is ccc are essentially implied by the followingLemma 3.2, which is the main point of the argument.

Let N = 〈Nα : α < ω2〉. For the Lemma we need to note that if p ∈ Pα is acomplete condition and N ≺ (H(χ),∈,N , . . .) countable with α ∈ N , then p ∩ N

66


is a condition. The proof of this is by induction on α, the main case being the case

α = β + 1. We have by (i) that p � β ∩ Nβ0 determines the τ

˜β structure of fp(β),

and we also have that Nβ0 ∈ N so p � β ∩N , which is a condition by the induction

hypothesis is stronger than p � β∩Nβ0 and hence it too determines the τ

˜β structure

of fp(β). We also need to note that the range of fp(β) is contained in N ∩ω1, whichfollows as fp(β) is finite.

Lemma 3.2. For any α ≤ ω2, if p ∈ Pα is a complete condition, N ≺ (H(χ),∈,N , . . .) countable with α ∈ N and r ∈ Pα ∩ N extends p ∩ N , then p ∪ r can beextended to a condition in Pα which extends both p and r3.

Proof. We shall present the main part of the proof, from which it can be seenwhere the amalgamation condition is being used. The final part of the proof willonly be indicated. Let us assume α ≥ 1, as otherwise the conclusion is trivial.

We shall need to refer to a following observation about elementary submodels.

Claim 3.3. For any γ ≤ ω2, γ ⊆⋃

i<ω1Nγ

i .

Proof of the Claim. Let β < γ. Since γ ∈ Nγ0 there is f ∈ Nγ

0 which maps ω1

onto γ. There is i < ω1 such that f−1(β) ∈ Nγi and then β ∈ Nγ

i . �3.3

The construction of the desired condition is by induction on k < ω, where theinduction will stop after some finite number of stages. At each stage we define

Nk ≺ (H(χ),∈, . . .) countable, rk ∈ Pα, ak ⊆ dom(p) \ {0}, (skβ)β∈ak, γk

where each skβ is a finite graph on a subset of ω1. For each k > 0, γk is a specialelement of ak, called the leading ordinal, with γ0 = ∞. The elements of ak arecalled the active ordinals. We denote δk = Nk∩ω1. The following are our inductivehypotheses:

• If β ∈ ak then skβ is a subgraph of rk(0) ∪ ran(fp(β)) (which in itself is a

graph), and the universe of skβ ∩ rk(0) is exactly skβ ∩ δk,

• if k > 0 then there is i < ω1 such that Nk = Nγk

i ,• rk ∈ Nk ∩ Pγk

and it extends p � γk ∩Nk and rm � γk for all m < k,• if β ≤ γk is active, then rk � β determines the τ

˜β structure of dom(fp(β))∪⋃

m≤k dom(frm(β)) ∩ δk and rk � β forces that fp(β) ∪⋃

m≤k frm(β) � δk is

an isomorphism of this structure with skβ,

• if β′ �= β′′ are both in ak and satisfy Aβ′∩Aβ′′ � δk, then skβ′∩skβ′′ ⊆ p(0).

We say that for β ∈ ak, the elements of dom(fp(β)) that have been used at the stagek are those in dom(fp(β)) ∩ δm, where m = max{l ≤ k : γl ≥ β} (since γ0 = ∞,

this maximum is well defined). We denote the set of such ordinals by ukβ.

The induction is not difficult to do: we let N0 = N , r0 = r and a0 = dom(p) \{0} ∩N . For β ∈ a0 let s0β = ran(r(β)) with the structure induced by r(0)- this iswell defined as p∩N ≤ r. If we have defined all the relevant objects at the stage k,do the following if possible: choose a minimal γ = γk+1 ∈ ak such that there is anunused element of dom(fp(γ)). Let i be the minimal such that there is an element

of dom(fp(γk+1) \ uk

γk+1in Nγ

i and satisfying Nk ∩ γk+1 ⊆ Nγk+1

i (note that such i

exists by Claim 3.3). Let Nk+1 = Nγk+1

i and ak+1 = ak∪ [dom(p)∩Nk+1∩γk+1]. If

3By p∪ r we mean the function q whose domain is dom(p)∪ dom(r), with q(0) = p(0)∪ r(0)and q(β) = (Xp(β) ∪Xr(β), fp(β) ∪ fr(β)) for β > 0 in dom(q).

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8 MIRNA DZAMONJA

no such choice is possible, stop the induction. Note that m ≤ k =⇒ γm ≤ γk andthat the induction must be stopped at some stage k∗ as all ordinals that appear asunused at any stage come from the finite set

⋃γ∈dom(p)\{0} dom(fp(γ)) (and once

an ordinal is used it remains so). Let γ∗ = γk∗ .We shall now proceed to define a condition q in Nγ∗ ∩ γ∗. We shall need the

following claim.

Claim 3.4. For all k ≤ k∗,⋃

m≤k Nm ∩ γk ⊆ Nk.

Proof of the Claim. The proof is by induction on k. If k = 0, the situation isclear. For the case of k + 1, we have by the induction hypothesis that for m ≤ k,Nm ∩ γk ⊆ Nk. Since γk+1 ≤ γk, certainly Nm ∩ γk+1 ⊆ Nk and hence it suffices toshow that Nk ∩ γk+1 ⊆ Nk+1. This conclusion follows by the choice of Nk+1. �3.4

Suppose that k ≤ k∗ and m ≤ k. Then rm ∈ Nm so all the finitely manyordinals involved in the construction of rm � γk are in Nm ∩ γk, and hence in Nk

by Claim 3.4. Therefore rm � γk ∈ Nk.At this point of the proof we shall get to use the Amalgamation Condition

(∗), which we define in terms of a partially ordered class (K,≤) of structures:(K,≤) satisfies a.c. (∗) if for any elements A,B, {Ci : i ∈ I} of K satisfying

• any two of these structures agree on their intersection• whenever i �= j then either Ci ∩ Cj ⊆ A or Ci ∩ Cj ⊆ B,

there is a structure in K which extends all A,B, {Ci : i ∈ I}.Obviously, the class of graphs with the induced subgraph relation satisfies the

a.c. (∗) since we can simply take a union of all the graphs involved.4

Suppose now that β′ �= β′′ ∈ ak∗ . By the induction hypothesis either Aβ′ ∩Aβ′′ � δk∗ , in which case sk

∗β′ ∩ sk

∗β′′ ⊆ p(0) or Aβ′ ∩ Aβ′′ ⊆ δk∗ and so sk

∗β′ ∩ sk

∗β′′ ⊆

rk∗(0). This says that, according to the a.c. (∗) property, we can define a graphthat extends p(0)∩δk∗ ∪rk∗(0)∪

⋃β∈ak∗ sk

∗β ∩δk∗ . Let s be such a graph. We define

q′, a candidate for a condition in Pα. Let q′(0) = s and for β ∈

⋃k≤k∗ dom(rk)∪ak∗

let q′(β) = p(β) ∩ δk∗ ∪⋃

k≤k∗ rk(β). Note that q′ as a function is an element of

Nk∗ . We now claim that q′ is a condition in Pα. By induction on 1 ≤ β ≤ α weprove that q′ � β ∈ Pβ. For β = 1 this follows from the choice of s and for β limitthis is clear. Suppose that β = γ + 1 and γ is in dom(q′). If γ /∈ dom(p) thenγ ∈

⋃k≤k∗ dom(rk) and q′(γ) =

⋃k≤k∗ rk(γ). This is well defined by the choice

of rk’s. If γ ∈ dom(p) \⋃

k≤k∗ dom(rk), then q′(γ) = p(γ) ∩ δk∗ , so q′ � β is a

condition because p satisfies (i) in the definition of completeness. Finally it mayhappen that γ ∈ dom(p)∩

⋃k≤k∗ dom(rk). In particular we have β ∈ ak∗ . Let k be

maximal such that γ ≤ γk. We claim that dom(fp(γ)) ∩ δk∗ = dom(fp(γ)) ∩ δm. Ifthis were not to be the case then at the stage k of the induction there would be anunused ordinal in dom(fp(γ)), so γ = γk+1, a contradiction. Having established theexistence of q′ we now need to keep going to extend to a condition which extendseach rk. This is done in stages using the assumptions of completeness, for whichwe refer the reader to the original article. �3.2 �

We now indicate how Lemma 3.2 implies that the forcing is ccc. We assumethat we have proved that the set of complete conditions is dense, so we only work

4In [9], Lemma 1, it is shown that a better known P(3−) amalgamation property impliesa.c. (∗).

88


with them. So assume that A is a maximal antichain of complete conditions andlet N be a rich enough countable elementary submodel with A ∈ N . We claimA = A ∩N , so A is countable. For this we have to show that A ∩N is a maximalantichain. Let p be any complete condition. By elementarity there is r ∈ N ∩ Awhich extends p∩N . By Lemma 3.2 there is a common extension of p and r. Thisproves the statement we claimed.

We remark that the above result, as it involves ccc forcing iteration, can beeasily generalised to cardinals of the form λ+ for some λ satisfying λ<λ = λ, asMekler does in Theorem 8 of [9]. Therefore the theory of random graphs is amenablein the sense of Definition 3.1.

3.2. Triangle-free Graphs. Having established that the theory of graphsis amenable the next natural question to ask to what extent the amalgamationcondition (∗) was necessary for this conclusion. A very good example to consideris that of the model completion of the theory of triangle-free graphs. In the senseof model-theoretic complexity, this theory provides a prototypical example of anon-simple ’simple enough’ theory, but also because this theory fails the a.c (∗).Namely

Example 3.5. Let Cl be a graph consisting of the edge (cl, cl+1/mod3), for l < 3,let A = C0 and B = C1. These structures satisfy the premises of the condition a.c(∗) but any graph extending C0, C1 and C2 contains the triangle {c0, c1, c2}.

Therefore methods of [9] do not apply to triangle-free graphs. Dzamonja andShelah proved in [3] that the theory of the model completion of triangle-free graphsis nevertheless amenable. We explain the main ideas of this proof.

Let us first concentrate on ℵ1. In the final extension we shall have 2ℵ0 = 2ℵ1 =ℵ3 and the universality number of the class of the triangle free graphs of size ℵ1

will be ℵ2. The values ℵ2 and ℵ3 are rather flexible, but we concentrate on thesefor concreteness. We consider the class K all finite triangle-free graphs G whoseuniverse is a subset of ω1 and that have the property that for every non-zero limitδ < ω1, G � δ is a ‘reflection’ of G in the sense that for every pair a0, a1 ∈ G � δ,if there is c with alRc for l < 2, then there is such c in G � δ. We can orderthis class by the relation of being an induced subgraph. Let us call this partiallyordered class the approximation family. A petition on the approximation family isa directed subset of it of size ℵ1. Notice that if we are given a petition Γ then itsunion is a graph of size ℵ1, and by directedness, this graph is triangle-free. Wecall this graph MΓ. Notice that by reenumerating, every triangle free graph ofsize ℵ1 is isomorphic to one whose family of finite subgraphs forms a petition onthe approximation family. We shall be interested in quorumed petitions, which arethose petitions that contain an isomorphic copy of every finite triangle-free graphon ω1 through an isomorphism which moves every ordinal α to an ordinal β offinite distance with α. The existence of such quorumed petitions is not somethingwe prove in ZFC, but we force them.

Let us now describe the forcing. We start with a model of GCH an in itwe denote by T the tree ω1>ω1. We first add ℵ3 many Cohen subsets of ω1 bycountable conditions. Notice that this introduces ℵ3 many branches to T . Callthe resulting universe V . We follow this with an iteration 〈Pα, Q

˜α : α < ω2 in

which each step Q˜

α is itself an iteration of ω3 steps, of ccc forcing, which we shallcall a block. In each block we shall have a preliminary forcing and an iteration of

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10 MIRNA DZAMONJA

length ω3. We first describe the iteration. It will have two kinds of coordinates.In each coordinate of the first kind. of it we are given a name for a petition Γ

õn

the approximation family and we force to embed MΓ˜into MΓ∗

˜for some quorumed

petition Γ∗˜. By remarks above, for our final universality result it will be sufficient

to ensure that there are ℵ2 triangle-free graphs on ω1 which embed all MΓ∗ forquorumed petitions Γ∗. By bookkeeping, we shall at each stage α < ω2 of the mainiteration assure by Q

˜α of forcing that we have dealt with all quorumed petitions

in V Pα . In fact we shall assure that there is in V Pα+1 a single triangle-free graphG∗

α on ω1 which embeds all MΓ∗ for quorumed petitions Γ∗ in V Pα . This is wherethe preliminary forcing of the block α comes in: in it we introduce a system Sα ofmembers of the approximation family indexed by the nodes in T in a such a waythat each branch through T gives a petition in the approximation family. It is easyto see that the union of this system is a triangle-free graph on ω1, which will beour G∗

α. In the second kind of coordinates in the block α we shall be embeddinga quorumed petition H

˜given by the bookkeeping into the subsystem of Sα given

by the elements indexed by the nodes on some branch of T . This assures that MH

embeds into G∗α.

The main point of the proof is to make sure that each individual forcing in ablock is ccc, and in assuring so in the second kind of coordinates we get to use anamalgamation property possessed by the elements of the approximation family K:

Suppose that M0, N0,M1,M2, N1, N2 and M are in K such that

• M0 = M1 ∩M2 and M1 is isomorphic with M2 by an isomorphism whichis identity on M0,

• N0 = N1 ∩N2 and N1 is isomorphic with N2 by an isomorphism which isidentity on N0,

• each Mi is an induced subgraph of the corresponding Ni and the universeof Ml consists of even ordinals in the universe of Nl,

• there are limit ordinals δ0 < δ1 < δ2 such that Nl ⊆ δl for each l < 3,• the universe ofM is contained in the even ordinals andM1,M2 are inducedsubgraphs of M .

Then there is N ∈ K whose induced subgraphs include M , N1 and N2.This property is called workability in [3]. Notice that checking that it is true

really uses the definition of the approximation family, not only the properties ofthe class of triangle-free graphs.

The proof in the case λ+ for λ = λ<λ in place of ℵ1 has to deal with a strongversion of λ+-cc needed in order to iterate, which introduces additional complica-tions which we shall not describe here. The structure of the proof as describedabove uses a tree of models rather than a linearly organised structure like in [9].The price we have to pay is that the final universe does not have one universalmodel, rather just a small universal family. The following question is still open:

Question 3.6. Is it consistent to have a universal triangle-free graph on ℵ1

and not CH?

3.3. Linear orders. The next step to consider in our increasing level of com-plexity of theories is a theory that does not have a workable approximation family.Such a theory is the theory of dense linear orders. Namely, the main result ofKojman-Shelah [7] is that this theory is highly non-amenable, and the proof wepresented for triangle-free graphs cannot be adopted to this case- as it would show

1010


that the theory is amenable. As an exercise, the reader may check that natural def-initions of approximation families for this class will fail to be workable. However,in [12] Shelah proved that it is consistent that there is a universal linear order ofsize ℵ1 in a model where CH fails 5. In the terminology of the rest of this paper,this shows that the theory of dense linear orders is amenable at ℵ1. Note that themethod of the proof uses oracle-proper forcing, which is a technique limited to ℵ1.Again by high non-amenability we conclude that the analogue of [12] cannot beobtained at cardinals larger than ℵ1, and at the same time that amenability is notimplied by amenability at ℵ1.

3.4. Amenability at ℵ1. Our presentation so far leaves open the question ifevery ‘reasonable’ theory T is amenable at ℵ1. Namely, let D(T ) denote the set ofcomplete types over the empty set in finitely many variables. It is known that ifthis set is uncountable then it has to have the cardinality of the continuum, andit is easy to see that every type in D(T ) must be realised in the universal model.Therefore if D(T ) is uncountable there cannot be a universal model of T in ℵ1 ifCH fails. It remains to ask what happens if D(T ) is countable, namely if it ispossible that every T with D(T ) countable is amenable at ℵ1. A negative answerto this is given in §1 of [7], where there is an example of a theory with countableD(T ) which has a universal model at ℵ1 iff CH holds.

Conclusion. In conclusion, we have presented the methods that are currentlyavailable for making the universality number at ℵ1 small while failing CH. Thesemethods come with amalgamation-type requirements on the theory in question andwe have discussed the prototypical examples of theories that satisfy or not theseamalgamation properties. We have discussed the division amenability/high non-amenability defined in terms of the ability of a theory to have a small universalitynumber in circumstances where the relevant instances of GCH are violated. We endby mentioning that there is a programme of characterising this division in syntacticterms i.e. in terms that do not discuss models of a theory but properties of its typesand formulae. The present state of this programme is described in the Introductionto [4].

References

[1] C. Chang and H.J. Keisler, Model Theory (3. edition), Studies in Logic, vol. 73, North-

Holland, (Amsterdam-New York-Oxford-Tokyo), 650 pp.,(1990).[2] M. Dzamonja, Club guessing and the universal models, Notre Dame Journal for Formal Logic,

vol. 46, No. 3, pp. 283-300, (2005).[3] M. Dzamonja and S. Shelah, On the existence of universals, Archive for Mathematical Logic,

vol. 43, pp. 901-936, (2004).[4] M. Dzamonja and S. Shelah, On properties of theories which preclude the existence of uni-

versal models, Annals of Pure and Applied Logic, vol. 139, no. 1-3, pp. 280-302, (2006).[5] F. Hausdorff, Grundzuge einer Theorie der geordneten Mengenlehre, Mathematische An-

nalen, vol. 65, pp. 435-505. (In German) (1908).[6] M. Kojman, Representing embeddability as set inclusion, Journal of the London Mathematical

Society (2nd series), vol 58, no. 185, Part 2, pp. 257-270 (1998).[7] M. Kojman and S. Shelah, Non-existence of Universal Orders in Many Cardinals, Journal

of Symbolic Logic, vol. 57, pp. 875-891, (1992).

5Unfortunately, the published proof of this very important result is very sketchy and theauthor of this article would not claim that she understands it completely. Shelah has kindlyexplained some of the major missing and inaccurate points in a recent conversation and promisedto make a written version available.

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[8] M. Kojman and S. Shelah, The universality spectrum of stable unsuperstable theories, Annalsof Pure and Applied Logic, vol. 58, pp. 57-72, (1992).

[9] A. H. Mekler, Universal structures in power ℵ1, Journal of Symbolic Logic vol. 55, no.2, pp.466–477, (1990).

[10] M. Morley and R. Vaught, Homogeneous universal models, Math. Scand., vol. 11, pp. 37-57,(1962).

[11] S. Shelah, Classification Theory, revised ed., Studies in Logic, vol. 73, North-Holland

(Amsterdam-New York-Oxford-Tokyo), 705 pp., (1990).[12] S. Shelah, Independence results, Journal of Symbolic Logic, vol. 45, pp. 563–573, (1980).[13] S. Shelah, On universal graphs without instances of CH, Annals of Pure and Applied Logic,

vol. 26, pp. 75–87, (1984).[14] S. Shelah, Universal graphs without instances of GCH: revisited, Israel Journal of Mathe-

matics, vol. 70, no. 1, pp. 69–81, (1990).[15] S. Shelah, The Universality Spectrum: Consistency for more classes in Combinatorics, Paul

Erdos is Eighty, Bolyai Society Mathematical Studies vol. 1, pp. 403-420, (1993). (proceedingsof the Meeting in honor of P. Erdos, Keszthely, Hungary 7. 1993, an improved version availableat http://www.math.rutgers.edu/ shelarch).

[16] S. Shelah, Toward classifying unstable theories, Annals of Pure and Applied Logic vol. 80,pp. 229-255, (1996).

[17] R. Vaught, Denumerable models of complete theories, in lnfinitistic Methods (Pergamon,London) pp. 303-321, (1961).

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

E-mail address: [email protected]

1212

Contemporary Mathematics

A survey of D-spaces

Gary Gruenhage

Abstract. A space X is a D-space if whenever one is given a neighborhoodN(x) of x for each x ∈ X, then there is a closed discrete subset D of X suchthat {N(x) : x ∈ D} covers X. It is a decades-old open question whethersome of the standard covering properties, such as Lindelof or paracompact,imply D . This remains unsettled, yet there has been a considerable amount

of interesting recent work on D-spaces. We give a survey of this work.

1. Introduction

A neighborhood assignment for a space (X, τ ) is a function N : X → τ withx ∈ N(x) for every x ∈ X. X is said to be a D-space if for every neighborhoodassignment N , one can find a closed discrete D ⊂ X such that N(D) = {N(x) :x ∈ D} covers X, i.e., X =

⋃x∈D N(x) =

⋃N(D)

The notion of a D-space seems to have had its origins in an exchange of lettersbetween E.K. van Douwen and E. Michael in the mid-1970’s,1 but the first paper onD-spaces is a 1979 paper of van Douwen and W. Pfeffer [25]. The D property is akind of covering property; it is easily seen that compact spaces and also σ-compactspaces are D-spaces, and that any countably compact D-space is compact.

Part of the fascination with D-spaces is that, aside from these easy facts, verylittle else is known about the relationship between the D property and many of thestandard covering properties. For example, it is not known if a very strong coveringproperty such as hereditarily Lindelof implies D, and yet for all we know it couldbe that a very weak covering property such as submetacompact or submetalindelofimplies D!

While these questions about covering properties remain unsettled, there nev-ertheless has been quite a lot of interesting recent work on D-spaces. The purposeof this article is give a survey of this work. In Section 2, we describe some impor-tant examples of spaces which are not D, in Section 3 we discuss what is known(and not) about the relationship of covering properties to D-spaces, in Section 4

2010 Mathematics Subject Classification. Primary 54D20.1Michael sent van Douwen a letter with a proof that semistratifiable spaces are D, and van

Douwen replied with an alternate proof. Michael kindly provided me with a copy of van Douwen’sletter to him, which is dated June 6, 1975, and another letter, with undetermined date, in whichvan Douwen proves that strong Σ-spaces are D. These results, however, did not appear in theliterature until 1991 [16] and 2002 [13], respectively.

c©0000 (copyright holder)

1



13



13

2 GARY GRUENHAGE

we list various results which say that spaces which have certain generalized metricproperties or base properties are D-spaces, Sections 5, 6, and 7 are about unions,products, and mappings of D-spaces, respectively, and finally Section 8 is on someother notions closely related to D-spaces.

All spaces are assumed to be regular and T1.

2. Examples

To help get a feeling for the property, we start by discussing ways to recognizespaces that aren’t D and give a couple of important examples. We mentioned in theintroduction that countably compact D-spaces must be compact. This illustratesa more general fact. Recall that the extent, e(X) of a space X is the supremumof the cardinalities of its closed discrete subsets, and the Lindelof degree, L(X), isthe least cardinal κ such that every open cover of X has a subcover of cardinality≤ κ. Note that e(X) ≤ L(X) for any X. It’s easy to see that if X is a D-space,and U is an open cover with no subcover of cardinality < κ, then there must be aclosed discrete subset of X of size κ; hence e(X) = L(X). Since closed subspacesof D-spaces are D [16], we have the following folklore result:

Proposition 2.1. If X is aD-space, then e(Y ) = L(Y ) for any closed subspaceY .

The result about countably compact D-spaces being compact is a corollary.Actually, a little bit stronger conclusion holds: if there is an open cover with nosubcover of cardinality less than L(Y ), then the supremum in the definition of e(Y )is attained, i.e., there is a closed discrete subset of cardinality e(Y ) = L(Y ). Thisgives the following fact: a stationary subset S of a regular uncountable cardinal κis not D.

A useful example to know, because it illustrates some of the limits of whatproperties could imply D, is an old example due to van Douwen and H. Wicke,which they denoted by Γ.

Example 2.2. [26] There is a space Γ which is non-Lindelof and has count-able extent, and is Hausdorff, locally compact, locally countable, separable, first-countable, submetrizable, σ-discrete, and realcompact.

This space is not D because e(X) = ω < L(X).

For a time, all known examples of non-D-spaces failed to be D because theconclusion of 2.1 failed. This was noted by W. Fleissner, essentially bringing upthe question (see [39]): IsX aD-space iff e(Y ) = L(Y ) for every closed subspace Y ?R. Buzyakova [14] asked a related question: Is X hereditarily D iff e(Y ) = L(Y ) forevery Y ⊂ X? Fleissner’s question was first answered consistently in the negativeby T. Ishiu, who obtained counterexamples both by forcing [38] and then from ♦∗

[39].

Example 2.3. It is consistent that there is a Hausdorff locally countable locallycompact non-D-space topology on ω1 such that e(Y ) = L(Y ) for every closedsubspace Y .

Ishiu’s space in [38] is constructed so that any uncountable closed set Y containsan uncountable closed discrete set, hence e(Y ) = L(Y ). The space is made not a D-space by constructing a neighborhood assignment N such that both D and ∪N(D)are nonstationary for any closed discrete set D.

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A SURVEY OF D-SPACES 3

Ishiu [39] showed that spaces like his do not exist under MA+ ¬CH:

Theorem 2.4. (MA + ¬CH) Let X be a locally compact locally countablespace of size < 2ω. Then either there is a closed subset F such that e(F ) < L(F )or X is a D-space.

Indeed, this follows rather easily from Balogh’s result [9] that under MA +¬CH, any locally compact, locally countable space of cardinality < 2ω is eitherσ-closed discrete, or contains a perfect preimage of the ordinal space ω1.

Ishiu’s examples are obtained from certain club guessing sequences. He alsoshowed that under PFA, no example similarly defined from a club guessing sequence(locally compact or not) can be a counterexample to Fleissner’s question (see [39],Proposition 5.5).

Later, P. Nyikos found a ZFC example of a non-D-space in which e(Y ) = LY )for every subspace Y , closed or not, thus answering Buzyakova’s question. Nyikos’sspace is also locally countable and locally compact. Recall that the interval topologyon a tree T is the topology whose base is the collection of intervals of the form(s, t] = {u ∈ T : s < u ≤ t}.

Example 2.5. [47] Let S be a stationary co-stationary set in ω1, and let Tbe the tree of all closed-in-ω1 subsets of S, ordered by end-extension. Give T theinterval topology. Then T is not D, yet e(Y ) = l(Y ) for every subspace Y .

The neighborhood assignment witnessing not D is the obvious one: N(x) ={y : y ≤ x}. One observes that any closed discrete D such that N(D) covers Xmust be cofinal in T , that a closed discrete cofinal set can be written as a countableunion of antichains, and then uses the known fact that T is Baire (in the sense of thetopology generated by {t+ : t ∈ T}, where t+ = {s : s ≥ t}) to get a contradiction.

Nyikos shows that T contains no Aronszajn subtrees, and from that arguesthat any uncountable subset of T contains an antichain of the same cardinality. Itfollows that e(Y ) = L(Y ) for any subspace Y .

3. Covering properties

Why isn’t it easy to prove that Lindelof spaces (for example) are D? Given aneighborhood assignment N , it is natural to consider a countable cover {N(xn) :n ∈ ω} of X. If it could be arranged that xn ∈

⋃i<n N(xi) for each n, then

D = {xn : n ∈ ω} would be closed discrete and N(D) would cover X. Butarranging this is the problem. What about paracompact spaces? One can considera locally finite open refinement U of {N(x) : x ∈ X}. If for each U ∈ U , one couldfind a point x(U) ∈ U such that U ⊂ N(x(U)), then the x(U)’s would be closeddiscrete and witness the D property. But it may be that the only N(x)’s whichcontain U are for x not in U .

In their article in Open Problems in Topology II, M. Hrusak and J.T. Moore[36] list twenty central problems in set theoretic topology; the question whetheror not Lindelof spaces are D is Problem 15 on this list, and is attributed to vanDouwen. T. Eisworth’s survey article [28] in the same volume lists several relatedquestions. In fact, it is not known if any of the following covering properties, evenif you add “hereditarily”, imply D:

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4 GARY GRUENHAGE

Lindelof, paracompact, ultraparacompact, strongly paracompact, metacom-pact, metalindelof, subparacompact, submetacompact2, submetalindelof, paralin-delof, screenable, σ-metacompact.

Since the non-D-space Γ mentioned in the previous section is σ-discrete, itfollows that weakly submetacompact does not imply D. But the following questionis open:

Question 3.1.

(1) Let X be a countably metacompact space. Must X be a D-space if it isweakly submetacompact? σ-metrizable? [3];

(2) Is there a ZFC example of a normal weakly submetacompact non-D-space?[47]

There is a consistent example answering 3.1(2): de Caux’s σ-discrete Dowkerspace [21] is not D because e(X) = ω < L(X). I don’t know if Balogh’s Dowkerspace [10] is or could be made not D.

There is a covering property implied by D: irreducibility. Recall that a spaceX is irreducible if every open cover U has a minimal open refinement V , i.e., eachV ∈ V contains a point not in any other V ′ ∈ V . Observe that an open coverU has a minimal open refinement iff there is a closed discrete set D and functionθ : D → U such that d ∈ θ(d) for each d ∈ D and {θ(d) : d ∈ D} is a cover. It isobvious from this that D-spaces are irreducible.

The D property is closed hereditary [16], but irreducibility is not. E.g., 2ω1 \{p}, where p ∈ 2ω1 , is irreducible but not D [7] since it contains a closed copy ofω1. But the following question is unsettled:

Question 3.2. Is X a D-space iff every closed subspace of X is irreducible?3

This question is essentially due to A.V. Arhangel’skii, who with Buzyakovaintroduced a property called aD [4], and later [3] proved that for T1-spaces aD isequivalent to every closed subspace being irreducible, and asked if aD is equivalentto D. Since submetalindelof implies irreducible [44], a positive answer to thisquestion would settle almost all of the questions about the relationship of standardcovering properties to the D property.

Since X irreducible implies e(X) = L(X), Question 3.2 could be considereda refinement of the question of Fleissner that was answered in the negative byexamples of Ishiu and Nyikos (see Section 2). Nyikos’s example is not irreducible,so doesn’t answer 3.2, but I don’t know if Ishiu’s is irreducible or not.

We mentioned above the question of Arhangel’skii whether countably metacom-pact, weakly submetacompact spaces are D. He also asked about aD; this questionis (for T1-spaces) equivalent to the question whether such spaces are irreducible,which was asked by J.C. Smith [62] in 1976 and apparently is still open.

2A space X is submetacompact (submetalindelof) if for each open cover U , there is a sequenceUn, n ∈ ω, of open refinements of U covering X, such that, for any x ∈ X, there is some n ∈ ωsuch that {U ∈ Un : x ∈ U} is finite (countable). Submetacompact (submetalindelof) spaces arealso called θ-refinable(δθ-refinable). Weakly submetacompact or weakly θ-refinable is defined thesame way as submetacompact except that the Un’s need not be covers. For definitions of the othercovering properties in the list or elsewhere in this article, see [29] or [19].

3Very recently Daniel Soukup [63] gave a consistent negative answer to this question byconstructing, assuming ♦∗, a space which is not D (or even linearly D –see Section 8) but everyclosed subspace is irreducible.

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See Section 8 on D relatives for more about aD and irreducibility.Most of the questions about covering properties implying D have a positive an-

swer for spaces that are either scattered or locally D (i.e., every point has a closedneighborhood which is D). That paracompact locally D spaces are D is straight-forward, and that paracompact scattered spaces are D is an easy induction of theCantor-Bendixson height. But the same result for weaker covering properties seemsto require a more subtle argument. L.-Xue Peng’s proof of this for submetacompactspaces is one illustration of the effective use of topological games in this area.

His result follows from a more general result concerning the following gamedue to Telgarsky [67]. Let K be a closed hereditary class of spaces, and X a space.We define the game G(K, X). There are two players, I and II. Player I begins bychoosing a nonempty closed subset A0 of X such that A0 ∈ K. II responds bychoosing a closed set B0 ⊂ X \ A0. At round n > 0, I chooses a nonempty closedsubset An of Bn−1 such that An ∈ K, and II responds by choosing a closed setBn ⊂ Bn−1 \ An. We say I wins the game if

⋂n∈ω Bn = ∅; otherwise II wins.

The space X is said to be K-like if Player I has a winning strategy in this game.Trivially, K is contained in the class of K-like spaces.

Now, let D be the class of D-spaces. Peng showed that D-like spaces are in factD:

Theorem 3.3. [49] Every D-like space is a D-space.

Proof. The argument shows in fact that X is D as long as II has no winningstrategy in G(D, X). Let N be a neighborhood assignment on X. If An is I’s playin round n, let II choose a closed discrete subset Dn of An such that N(Dn) coversAn, and then let II play Bn = Bn−1 \ ∪N(Dn). This defines a strategy for II.Since the strategy is not winning, there is a play A0, B0, A1, B1, . . . of the gamesuch that

⋂n∈ω Bn = ∅. Noting that Bn = X \ ∪N(D0 ∪D1 ∪ ... ∪Dn) and that

Dn+1 ⊂ An+1 ⊂ Bn, it is easy to check that if D =⋃

n∈ω Dn, then D is closeddiscrete and N(D) covers X. �

A space X is said to be K-scattered if every closed subset F of X contains apoint p with a closed neighborhood (relative to F ) in K. Peng proved:

Theorem 3.4. [50] Suppose K is a class of spaces which is closed under closedsubspaces and topological sums, and every K-like space is in K. Then every sub-metacompact K-scattered space is in K.

Note that the class D of all D-spaces satisfies the hypotheses of the theorem, soany submetacompact D-scattered space is D. Any scattered space and any locallyD-space is D-scattered. Hence the following corollary holds:

Corollary 3.5. If X is submetacompact and either scattered or locally D,then X is a D-space.4

But I don’t know the answer to the following:

Question 3.6.

(1) Are (sub)metalindelof scattered spaces D?(2) Does Question 3.2 have a positive answer in the class of scattered spaces?

4See the paragraph following Corollary 5.4 for discussion of a direct proof (without usinggames) of this result.

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Another nice illustration of game theory in this area involves the Menger prop-erty, which says that, given a countable sequence Un, n ∈ ω, of open covers of X,one can select for each n a finite subcollection Vn of Un such that

⋃n∈ω Vn is a

cover. Obviously, σ-compact spaces are Menger. The following is a recent result ofL. Aurichi. We give the proof, which uses a nontrivial game characterization of theMenger property due to W. Hurewicz.

Theorem 3.7. [6] Menger spaces are D-spaces.

Proof. Suppose X is Menger, and let N be a neighborhood assignment on X.We play a game in which Player I chooses, in each round n, an open cover Un closedunder finite unions, and Player II responds by choosing Un ∈ Un. Player II wins if{Un : n ∈ ω} covers X. Hurewicz [37] proved that X is Menger iff Player I has nowinning strategy in this game.

Now, let Player I’s first play be {∪{N(x) : x ∈ F} : F ∈ [X]<ω}. If Player IIresponds with ∪{N(x) : x ∈ F0}, let I then play

{∪{N(x) : x ∈ F0 ∪ F} : F ∈ [X]<ω, F ∩ ∪{N(x) : x ∈ F0} = ∅}.Then similarly, if II’s reply is ∪{N(x) : x ∈ F0 ∪ F1}, I plays

{∪{N(x) : x ∈ F0 ∪ F1 ∪ F} : F ∈ [X]<ω, F ∩ ∪{N(x) : x ∈ F0 ∪ F1} = ∅},and so on. This defines a strategy for Player I. X is Menger, so this can’t be awinning strategy. Therefore there is some play of the game with I using this strategysuch that, if F0, F1, . . . code the plays of II, then ∪{N(x) : x ∈ F0 ∪ F1 ∪ ... ∪ Fn :n ∈ ω} covers X. Let D =

⋃n∈ω Fn. Then N(D) covers X. Since for each n, we

have Fn ∩ ∪{N(x) : x ∈ F0 ∪ F1 ∪ ... ∪ Fn−1} = ∅, it is easy to check that D is aclosed discrete subset of X. Hence X is a D-space. �

Menger implying D gives that certain other Lindelof spaces are D. It wasproved in [7] that if a Lindelof space X can be covered by fewer than cov(M)-many compact sets, where M is the ideal of meager subsets of the real line, then Xis D; in fact, as was pointed out in [6], under these assumptions X is Menger. Letus note that this result implies that, e.g., there are no absolute examples of Lindelofnon-D-spaces of cardinality ℵ1. But I don’t know the answer to the following:

Question 3.8. Is it consistent that every paracompact space of cardinality ℵ1

is a D-space?

In the definition of the Menger property, if we require that each Vn has car-dinality 1, then we obtain the stronger Rothberger property. Scheepers and Tall[61] show that adding ℵ1 Cohen reals to a model makes any ground model Lindelofspace Rothberger (hence D), and that Moore’s ZFC L-space [46] is Rothberger.

A space X is said to be productively Lindelof if its product with any Lindelofspace is Lindelof. Alster [5] showed that under the Continuum Hypothesis(CH),every productively Lindelof space of weight ≤ ℵ1 has the following property, calledAlster in [8] and [64]:

• If G is a cover of X by Gδ-sets and G contains a finite subcover of eachcompact set, then G has a countable subcover.

Every space which is Alster is Menger [8] and hence D. Alster asked if everyproductively Lindelof space is Alster, but this remains unsolved. However, Tall [65]recently proved that under CH, every productively Lindelof space is Menger andhence D. It is not known if every productively Lindelof space is D in ZFC.

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A space X is indestructibly Lindelof (resp., indestructibly productively Lindelof)if X remains Lindelof (resp., productively Lindelof) in every countably closed forc-ing extension. Rothberger spaces, as well as Lindelof spaces which are scatteredor P -spaces, are indestructibly Lindelof. It is not known if every indestructiblyLindelof space is D [8], but Tall [66] has shown that indestructibly productivelyLindelof spaces are Menger and hence D.

We conclude this section with an interesting forcing result on D-spaces due toAurichi, L. Junqueira, and P. Larson.

Theorem 3.9. [7] Let T be a tree of height ω.

(1) If X is a D-space, then X remains D after forcing with T ;(2) If X is a Lindelof space, and every countable subset of T can be refined

to an antichain5, then X becomes D after forcing with T .

4. Generalized metrizable spaces and base properties

One can often show that spaces with an additional structure such as a baseproperty or generalized metric property are D-spaces.

Theorem 4.1. The following are D-spaces:

(1) semistratifiable spaces (hence Moore, semimetric, stratifiable, andσ-spaces) [16](see also [30]);

(2) subspaces of symmetrizable spaces [20];(3) strong Σ-spaces (hence paracompact p-spaces) [13];6

(4) protometrizable spaces (hence nonarchimedean spaces) [17];(5) spaces having a point-countable base [4];(6) spaces having a point-countable weak base [51], [20];(7) sequential spaces with a point-countableW -system [20] or point-countable

k-network [52];(8) spaces with an ω-uniform base [6];(9) base-base paracompact spaces (hence totally paracompact spaces) [60](see

also [59]).

Some of these properties are defined in [31]; see the references given abovefor the definitions of others. We give a rough idea of why these spaces are D.In the first two cases, and the third with some extra refinements, the proof thatmetrizable spaces are D basically works. Given a neighborhood assignment N , thespace naturally divides into countably many pieces Xn, where the neighborhoodsin each piece are “large” in some sense. E.g., for symmetrizable spaces, put apoint x in Xn if N(x) contains a ball of radius 1/2n; for a σ-space, put x in Xn ifthere is some member F in the nth discrete collection of a σ-discrete network withx ∈ F ⊂ N(x). Then well-order the space so that the points of X0 come first, thenX1, etc. Let x0 be the least point and if xα has been defined for all β < α, let xα

be the least point of X \ ∪{N(xβ) : β < α}. Continue until the N(xα)’s cover X.If D is the set of xα’s, then D is closed discrete. So X is a D-space.

Protometrizable spaces and nonarchimedean spaces have a base B such that,given any cover C by members of B, the largest members of C form a locally finite

5A subset S of a tree T can be refined to an antichain if for each s ∈ S one can choose a(s)above s in the tree such that {a(s) : s ∈ S} is an antichain.

6In [41], S. Lin shows that spaces having what he calls a σ-cushioned (mod k) pair-networkare D, which generalizes (1) and (3) and implies that Σ#-spaces are D.

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family. Thus, given a neighborhood assignment N whose range may be assumedto be included in B, the points corresponding to the largest members of the rangewitness the D property. Somewhat similar are the proofs for totally paracompact,which means that every base contains a locally finite subcover, and base-base para-compact, which means that there is a base B such that every subset C of B whichis a base has a locally finite subcover. Given N , apply the defining property to thecollection C of all members B of B such that x ∈ B ⊂ N(x) for some x ∈ X.

The proof for point-countable base uses a different method. One defines bya transfinite induction countable closed discrete sets Dα as follows. Let B be apoint-countable base for X, and N a neighborhood assignment whose range wemay assume to be included in B.

Now supposeDβ has been defined for all β < α. Let Fα = X\∪N(⋃

β<α Dβ). If

Fα = ∅, the induction stops. Otherwise, let M be a countable elementary submodelof H(θ) for sufficiently large θ, such that M contains X, B, Fα, ... Let ≺ well-orderM in type ω. Choose xα,0 ∈ Fα ∩ M . Suppose xα,i has been defined for eachi < n. If there no point x ∈ Fα \∪i<nN(xα,i) such that N(x)∩{xα,i}i<n = ∅, thenDα = {xα,i : i < n}. On the other hand, if there is such a point, let xα,n be sucha point in M chosen so that N(xα, n) is ≺-least. If the induction continues for alln ∈ ω, then Dα = {xα,n : n ∈ ω}.

One may check that not only is Dα closed discrete, but it is also “sticky” inthe sense that x ∈ Fα and N(x) ∩Dα = ∅ implies x ∈ ∪N(Dα). Note that by theway Dα was defined, it is automatically relatively discrete, and if it is also sticky,it is closed discrete. Stickyness was introduced by Fleissner and A. Stanley [30],and the idea was extended in [32], as a way to simplify certain D arguments. Onealso uses stickyness to show that the union of the Dα’s is closed discrete as well.So then, to complete the argument, simply continue the inductive construction ofthe Dβ ’s until Fα as defined above is empty, and let D =

⋃β<α Dβ .

In [32], I used the idea of stickyness to argue that in many cases to show thata space is a D-space, one only has to produce a single nonempty closed discretesticky subset D of an arb itrary nonempty closed subspace F . The advantage ofthis is that one can ignore the induction on α and concentrate on constructing asingle closed discrete D ⊂ F , which can be countable (even finite); note that youdon’t need N(D) to cover F .

The proofs that “sequential with a point-countable W -system”, “sequentialwith a point-countable k-network”, and “ω-uniform base” imply D are suitablymodified versions of the argument for point-countable base.

H. Guo and H.J.K. Junnila [34] proved that t-metrizable spaces are hereditarilyD-spaces. Here, a space (X, τ ) is t-metrizable if there exists a weaker metrizabletopology π on X and an assignment H �→ JH from [X]<ω to [X]<ω such thatclτ (A) ⊂ clπ(

⋃H∈[A]<ω JH). Since Cp(K) is t-metrizable for compact K, a corollary

is the earlier result of Buzykova [14] that Cp(K) is hereditarily D for any compactK. I improved this to:

Theorem 4.2. [32] If X is a Lindelof Σ-space, then Cp(X) is hereditarily aD-space.

The above result can be viewed as an explanation for the classical result of D.P.Baturov that Lindelof degree equals extent for subspaces of these Cp(X)’s.

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Subsequently, V. Tkachuk [69] defined the class of monotonically monolithicspaces, which includes spaces Cp(X) for X Lindelof Σ, as well as spaces witha point-countable base, and proved, using Buzyakova’s argument as the model,that monotonically monolithic spaces are hereditarily D. And recently Peng [58]defined weakly monotonically monolithic spaces, a class which besides includingmonotonically monolithic spaces is also more general than some of the classes inthe list in Theorem 4.1 (e.g., 4.1(6)(7)), and proved they are D-spaces.

From Buzyakova’s result about Cp(K) it follows that Eberlein compacta areD-spaces. I generalized this to:

Theorem 4.3. [32] Corson compacta are hereditarily D.

Corson compact spaces are monolithic but it is not known if they must bemonotonically monolithic [69]. Corson compacta are also Frechet, hence sequential.I mention this because the following question of I. Juhasz and Z. Szentmiklossy isunsolved:

Question 4.4. [40] Is every compact hereditarily D-space sequential?

Possibly relevant to this question is the result of M. Tkacenko [68] that acompact space which is the union of countably many left-separated spaces (whichare well-known to be D) is sequential.

A result van Douwen proved long before his death in 1987, but which didn’tappear in print until 1997 [24], is that paracompact GO-spaces (i.e., subspaces oflinearly ordered spaces) are D-spaces. Since a GO-space is paracompact iff it doesnot contain a closed subset homeomorphic to a stationary subset of an uncountableregular cardinal, and such subsets are not D (see Section 2), it follows that a GO-space is a D-space iff it is paracompact.

The class of monotonically normal spaces( [35]; see also [31]) is a commongeneralization of both metrizable and GO-spaces, and like GO-spaces, they areparacompact iff they do not contain a closed subspace homeomorphic to a station-ary subset of an uncountable regular cardinal [11]. Thus monotonically normalD-spaces are paracompact, which suggests the following question of Borges andWehrly:

Question 4.5. [16] Is every paracompact monotonically normal space a D-space?

A space (X, τ ) is quasimetrizable if there is a function g : ω × X → τ suchthat (i) {g(n, x) : n ∈ ω} is a base at x. and (ii) y ∈ g(n, x) ⇒ g(n + 1, y) ⊂g(n, x). If in (ii) we put “ y ∈ g(n, x) ⇒ g(n, y) ⊂ g(n, x)” instead, then X issaid to be nonarchimedean quasimetrizable. Equivalently, X is nonarchimedeanquasimetrizable iff X has a σ-interior-preserving base, where a collection U of opensets is interior preserving if for each x ∈ X,

⋂{U ∈ U : x ∈ U} is open.

Question 4.6. Are (nonarchimedean) quasimetrizable spaces D-spaces?

Another class of spaces relevant to generalized metrizable spaces are the so-called “butterfly spaces”. Let (M, τ ) be a metrizable space. If τ ′ is a finer topologyon M such that each point p ∈ M has a neighborhood base consisting of sets Bsuch that B \ {p} is open in the metric topology τ , then (M, τ ′) is called a butterflyspace over (M, τ ). Many examples in the area of generalized metrizable spacesare butterfly topologies over separable metric spaces (e.g., the tangent disc space,bow-tie space, even the Sorgenfrey line). I don’t know the answer to the following:

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Question 4.7. Is every butterfly space over a separable metrizable space aD-space?

It is not difficult to show that every such butterfly space is subparacompact,so a negative answer would be most interesting.

5. Unions

The van Douwen-Wicke space Γ is σ-discrete but not D, so the D property isnot preserved by countable unions. But the following remains open:

Question 5.1. [3] Is the union of two (equivalently, finitely many) D-spacesalways a D-space?

The answer is positive in some special cases.

Theorem 5.2. If X =⋃

i≤nXi, then X is D if each Xi:

(a) is metrizable [4];(b) is strong Σ, Moore, DC-like, or regular subparacompact C-scattered [53](see

also [72] for strong Σ and [43] for C-scattered);(c) has a point-countable base [2].

Recall that a space X is C-scattered if every nonempty closed subspace F hasa point with a compact neighborhood (relative to F ). A space is DC-like if I hasa winning strategy in the game G(DC, X), where DC is the class of all topologicalsums of compact spaces. (See Section 3 for the definition of G(K, X).) The classof DC-like spaces includes all subparacompact C-scattered spaces and all spaceswhich admit a σ-closure preserving cover by compact sets.

The result about finite unions of Moore spaces gives a positive answer to aquestion in [2].

There are also some positive results about infinite unions. Borges and Wehrly[16] show that a countable infinite union of closed D-subspaces is D. The followingresult of Guo and Junnila is more general:

Theorem 5.3. [34] Suppose X =⋃

α<λ Xα, where each Xα is D, and for eachβ < λ,

⋃α<β Xα is closed. Then X is a D-space.

From this, they derive the following corollary.

Corollary 5.4. If X has σ-closure-preserving cover by closed D subspaces,then X is a D-space.

Then they use this to obtain a direct proof (without using games) of Peng’sresult (see Corollary 3.5) that submetacompact locally D spaces are D; note that itthen follows by a straightforward induction on height that submetacompact scat-tered spaces are D.

Arhangel’skii [3] asked whether a countably compact space which is a union ofcountably many D subspaces must be compact. Juhasz and Szentmiklossy gave apositive answer.

Theorem 5.5. [40](see also [52]) If X is countably compact and a countableunion of D subspaces, then X is compact.

See Section 8 on D relatives for some generalizations of this result.Finally, we mention that J.C. Martinez and L. Soukup [43] showed that a

countable or locally finite union of Lindelof C-scattered spaces is D.

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6. Products

The fact that semistratifiable spaces and strong Σ-spaces are D-spaces andare countably productive as well shows that in many special cases, finite and evencountable products of D-spaces are D. Also, Borges and Wehrly [16] showed thatthe product of a D-space with a compact space isD. But in general, the D propertyis not finitely productive.

Example 6.1. [1] There is a Lindelof D-space X and separable metric M suchthat X ×M not D.

Outline of the construction. Van Douwen [23] shows that one can put a finerlocally compact locally countable topology with countable extent on any subset ofthe real line R of cardinality continuum. Do this on a Bernstein set B, and letX = R with points of B having van Douwen’s neighborhoods and points of R \ Bhaving their usual Euclidean neighborhoods. One proves that this space is Lindelofand D using the fact that any open superset of R \ B contains all but countablymany points of R. Now X ×B, where B has its usual topology, contains the closedcopy {(x, x) : x ∈ B} of B with van Douwen’s topology, which is not D as it is notLindelof but has countable extent. �

Assuming CH, Borges and Wehrly [18] show that there exists a first countableregular hereditarily Lindelof space Y such that Y 2 is perfectly normal and hereditar-ily separable but not a D-space. Tall [?] obtains consistent examples of Rothbergerspaces (for the definition, see the discussion after Theorem 3.9) whose squares arenot D; the key idea here is to take known examples of two Lindelof spaces whoseproduct is not Lindelof but has countable extent, and note that adding ω1-manyCohen reals makes ground model Lindelof spaces Rothberger [61] but preservescountable extent.

An interesting specific example is the Sorgenfrey line and its countable powers.In the first published paper on D-spaces, van Douwen and Pfeffer [25] prove thatthe Sorgenfrey line and its finite powers are D-spaces. Later, answering one of theirquestions, P. de Caux [22] proved that every finite power of the Sorgenfrey line ishereditarily D. But the following is still open.

Question 6.2. [22] Let S be the Sorgenfrey line. Is Sω a D-space? Is ithereditarily D?

Since Sω is hereditarily subparacompact, and it is not known if subparacompactimplies D, it would be particularly interesting if the answer to the question is “no”.

The D-property in products of scattered spaces and some more general spaceshave been investigated. In the previous section, we discussed unions of DC-like andC-scattered spaces. We recall here that any countable product of (sub)paracompactDC-like spaces is (sub)paracompact.

Theorem 6.3. [55] A countable product of paracompact DC-like (in particular,C-scattered) spaces is a D-space.

Fleissner and Stanley [30] showed that if each Xα is scattered of height 1,then the box product �α<κXα is a D-space. They also showed that a subset of afinite product of ordinals is D iff it is metacompact iff it does not contain a closedsubset homeomorphic to stationary subset of regular uncountable cardinal. Thebox product result was improved by Peng [55] to scattered of height ≤ k, wherek ∈ ω.

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7. Mappings

There are only a small number of results on mappings of D-spaces. Borges andWehrly proved the following useful result:

Theorem 7.1. [16] If X is a D-space, then so is every closed image of X andevery perfect pre-image of X.

Note that fact that the product of a D-space with a compact space is D is acorollary to the result on perfect pre-images.

As was pointed out in [20], quotient or even open images of D-spaces neednot be D. Any first countable space is the open image of a metrizable space, butof course there are first-countable spaces which are not D-spaces. That spaceswith a point-countable base are D (Theorem 4.1(5)) implies that open s-imagesof metrizable spaces are D (where “s-image” means that fibers are separable). D.Burke extended this with the following result:

Theorem 7.2. [20] The quotient s-image of a space with a point-countablebase is a D-space.

This also follows from Theorem 4.1(7).Of course, continuous images of D-spaces need not be D, but the following

question is open:

Question 7.3. (Arhangel’skii, see [13]) Let X be a Lindelof D-space. Is everycontinuous image of X a D-space?

8. D relatives

8.1. Property aD. Arhangel’skii and Buzyakova [4] introduced a weakeningof the D property called aD. A space X is aD if for each closed F ⊂ X and foreach open cover U of X, there is a locally finite A ⊂ F and φ : A → U with a ∈ φ(a)and F ⊂ ∪φ(A).

It is obvious that D-spaces are aD. In a later paper, Arhangel’skii showed that(for T1-spaces) the aD property has an equivalence in terms of irreducibility (seeSection 3 for the definition of irreducible):

Theorem 8.1. [3] A T1-space X is aD iff every closed subspace of X is irre-ducible.

So questions about aD become questions about irreducibility. Arhangel’skiiasked in [3] if every Tychonoff aD space is D. By the previous theorem, this isequivalent to Question 3.2.

J. Mashburn [44] proves (for T1-spaces) that submetalindelof spaces are irre-ducible (also weakly δθ-refinable, which we shall not define here). Z. Yu and Z.Yun [72] improve this by showing that any finite union of submetalindelof spacesis irreducible.

We mentioned Arhangel’skii’s question whether the union of two D-spaces is D.The following question of his is also open and seems more likely to have a positiveanswer:

Question 8.2. [3] If X is the union of two D subspaces, must X be aD (i.e.,irreducible)?

2424


8.2. Dually discrete. One can obtain generalizations of D-spaces by weak-ening the requirement in the definition that D be closed discrete. If P is a propertyof topological spaces, J. van Mill, Tkachuk, and Wilson [45] call a space X duallyP if for every neighborhood assignment N , there is a subspace D of X having prop-erty P such that N(D) covers X. In particular, a space X is dually discrete if forany neighborhood assignment N , there is a relatively discrete D with ∪N(D) = X.

Dually discrete is a much weaker property thanD, as evidenced by the followingresult.

Theorem 8.3.

(1) [54] Every GO-space is dually discrete;(2) [27] Every tree with the interval topology is dually discrete.

Theorem 8.3(1) answers a question in [15] and 8.3(2) a question in [47].Surprisingly, the same questions about whether or not standard covering prop-

erties imply D are also unsolved for this weaker version. In particular:

Question 8.4. [1] Are hereditarily Lindelof spaces dually discrete?

Since GO-spaces are dually discrete, so is any ordinal space. Answering aquestion in [1], Peng [56] showed that any finite product of ordinals is alwaysdually discrete. But as with the D-property, in general dually discrete is not finitelyproductive, at least consistently. In [1] it is noted that if in Example 6.1 above,the van Douwen line is replaced by the Kunen line (which exists only under CH),then we have a D-space X and a separable metrizable M such that X ×M is notdually discrete. The Kunen line, being hereditarily separable and not Lindelof, isclearly not dually discrete. One would have a ZFC example if the van Douwen linewere (or could be made to be) not dually discrete, but I don’t know if that is thecase or not.

In [15], the authors show that under ♦, Rκ is not dually discrete for κ > ω,and ask if this is true in ZFC.

An easy argument shows that every space X is dually scattered. There ishowever a ZFC example of a scattered space X which is not dually discrete [15]:take a right-separated subset of type κ+ in a space whose hereditary density is κand hereditary Lindelof degree is greater than κ (such spaces were constructed inZFC by Todorcevic [70]).

Here are some more questions from [15]:

Question 8.5. Do any of the following imply dually discrete:

(1) Monotonically normal;(2) σ-(relatively) discrete;(3) dually metrizable?

8.3. Linearly D and transitively D. Putting conditions on the neighbor-hood assignment yields generalizations of the D property. Guo and Junnila de-fine X to be linearly D if for every neighborhood assignment N whose range{N(x) : x ∈ X} is a linearly ordered (by ⊆) collection of sets, then there is aclosed discrete set D such that X = ∪N(D).

Theorem 8.6. [33]

(1) X is linearly Lindelof iff X is linearly D and e(X) = ω;(2) Submetalindelof spaces are linearly D;

2525

14 GARY GRUENHAGE

(3) Finite unions of linearly D spaces are linearly D;(4) Countably compact spaces which are the union of countably many linearly

D subspaces are compact.

In [54], Peng had a different definition of linearly D; his property is now called“transitively D”. A space X is transitively D if for every neighborhood assignmentN such that y ∈ N(x) implies N(y) ⊂ N(x), there is a closed discrete set D suchthat X = ∪N(D).

Peng proved the following:

Theorem 8.7. [57]

(1) Every transitively D space is linearly D;(2) Metalindelof spaces are transitively D;7

(3) Finite unions of transitively D spaces are transitively D.

It follows from Theorem 8.6(1) that any linearly Lindelof non-Lindelof space islinearly D but not D. But the following is unsettled:

Question 8.8. [57] Is there a transitively D space which is not D?

I don’t know if any of the known examples of linearly Lindelof non-Lindelofspaces8 are transitively D, and I don’t know if Ishiu’s examples (Example 2.3)are transitively D. Nyikos’s space (Example 2.5) is not transitively D since theneighborhood assignment witnessing not D is transitive; however, D. Soukup [63]noted that it is linearly D. Van Douwen-Wicke’s space Γ (Example 2.2) is not evenlinearly D since e(Γ) = ω but Γ is not linearly Lindelof.

Acknowledgement. The author is grateful to the following mathematiciansfor valuable comments on earlier drafts of this article: Leandro Aurichi, RaushanBuzyakova, Bill Fleissner, Tetsuya Ishiu, Heikki Junnila, Dave Lutzer, Ernie Michael,Peter Nyikos, Liang-Xue Peng, Frank Tall, and Vladimir Tkachuk. He also thanksErnie for providing him with copies of some of his correspondence with Eric vanDouwen.

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2163–2170.[3] A.V. Arhangel’skii, D-spaces and covering properties, Topology Appl. 146-147(2005), 437-449.[4] A.V.Arhangel’skii and R. Buzyakova, Addition theorems and D-spaces, Comment. Mat. Univ.

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35 (2010),73-82.[8] L. Aurichi and F. Tall, Lindelof spaces which are indestructible, productive, or D, preprint.[9] Z. Balogh, Locally nice spaces and Martin’s Axiom, Comment. Math. Univ. Carolin. 24 (1983)

63–87.[10] Z. Balogh, A small Dowker space in ZFC, Proc. Amer. Math. Soc. 124(1996), 2555–2560.

7I recently proved that submetalindelof spaces are transitively D.8See the references in [48] for various examples of linearly Lindelof non-Lindelof spaces.

2626


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ducible nor realcompact, Topology Proceedings I(1976), 67–77.[22] P. de Caux, Yet another property of the Sorgenfrey plane, Topology Proc. 6(1981), 31-43.[23] E.K. van Douwen, A technique for constructing honest locally compact submetrizable exam-

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(1977), no. 1, 141–152.[27] A. Dow and R. Mendoza, unpublished note.[28] T. Eisworth, On D-spaces, in: Open Problems in Topology II, E. Pearl, ed., Elsevier, Ams-

terdam, 2007, 129-134.

[29] R. Engelking, General Topology, Heldermanan Verlag, Berlin, 1989.[30] W. Fleissner and A. Stanley, D-spaces, Topology Appl. 114(2001), 261-271.[31] G. Gruenhage, Generalized metric spaces, in the Handbook of Set-theoretic Topology, K.

Kunen and J.E. Vaughan, eds., North-Holland, Amsterdam, 1984, 423-501.[32] G. Gruenhage, A note on D-spaces, Topology and Appl. 153(2006), 2229–2240.[33] H. Guo and H.J.K. Junnila, On spaces which are linearly D, Topology Appl. 157(2010),

102-107.[34] H. Guo and H.J.K. Junnila, On D-spaces and their unions, preprint.[35] R. Heath, D. Lutzer, and P. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc.

178 (1973), 481–493.[36] M. Hrusak and J.T. Moore, Twenty problems in set theoretic topology, in: Open Problems in

Topology II, E. Pearl, ed., Elsevier, Amsterdam, 2007, 111-114.[37] W. Hurewicz, Uber eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24(1926),

401-421.[38] T. Ishiu, More on perfectly normal non-realcompact spaces, Topology Appl. 153 (2006), 1476–

1499.[39] T. Ishiu, A non-D-space with large extent, Topology Appl. 155 (2008), 1256–1263.[40] I. Juhasz and Z. Szentmiklossy, Two improvements of Tkachenko’s addition theorem, Com-

ment. Mat. Univ. Car. 46(2005), 705-710.[41] S. Lin, A note on D-spaces, Comment. Math. Univ. Carolin. 47 (2006), 313–316.[42] D. Lutzer, Ordered topological spaces, in: Surveys in General Topology, G.M. Reed, ed.,

Academic Press, 1980, pp. 247-295.[43] J.C. Martinez and L. Soukup, The D-property in unions of scattered spaces, Topology Appl.

156(2009), 3086-3090.[44] J.D. Mashburn, A note on irreducibility and weak covering properties, Topology Proc. 9

(1984), no. 2, 339–352.[45] J. van Mill, V. Tkachuk, and R. Wilson, Classes defined by stars and neighborhood assign-

ments, Topology Appl. 154(2007), 2127-2134.

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16 GARY GRUENHAGE

[46] J.T. Moore, A solution to the L space problem, J. Amer. Math. Soc. 19 (2006), 717–736.[47] P. Nyikos, D-spaces, trees, and an answer to a problem of Buzyakova, Topology Proc., to

appear.[48] E. Pearl, Linearly Lindelof problems, in: Open Problems in Topology II, E. Pearl, ed., Else-

vier, Amsterdam, 2007, 225-231.[49] L.-Xue Peng, On some sufficiencies of D-spaces, Journal Beijing Inst. Tech., Vol. 16, (1996),

229-233.

[50] L.-Xue Peng, About DK-like spaces and some applications, Topology Appl. 135(2004), 73-85.[51] L.-Xue Peng, The D-property of some Lindelof spaces and related conclusions , Topology

Appl. 154(2007), 469-475.[52] L.-Xue Peng, A note on D-spaces and infinite unions, Topology Appl. 154 (2007), 2223–2227.[53] L.-Xue Peng, On finite unions of certain D-spaces , Topology Appl. 155(2008), 522-526.[54] L.-Xue Peng, On linear neighborhood assignments and dually discrete spaces , Topology Appl.

155(2008), 1867-1874.[55] L.-Xue Peng, On products of certain D-spaces , Houston J. Math. 34(2008), 165–179.

[56] L.-Xue Peng, Finite unions of weak θ-refinable spaces and products of ordinals, Topologyand Appl. 156(2009), 1679-1683.

[57] L.-Xue Peng, On spaces which are D, linearly D and transitively D, Topology Appl.157(2010), 378-384.

[58] L.-Xue Peng, On weakly monotonically monolithic spaces, Comment. Math. Univ. Carolin.,to appear.

[59] S. Popvassilev, Base-cover paracompactness, Proc. Amer. Math. Soc. 132(2004), 3121–3130.[60] J.E. Porter, Generalizations of totally paracompact spaces, Dissertation, Auburn University,

2000.[61] M. Scheepers and F. Tall, Lindelof indestructibility, topological games and selection princi-

ples, Fund. Math., to appear.[62] J.C. Smith, Irreducible spaces and property b1, Topology Proc. 5 (1980), 187–200.[63] D. Soukup, Properties D and aD are different, preprint.[64] F. Tall, Lindelof spaces which are Menger, Hurewicz, Alster, productive, or D, Topology

Appl., to appear.[65] F. Tall, Productively Lindelof spaces may all be D, preprint.[66] F. Tall, A note on productively Lindelof spaces, preprint.

[67] R. Telgarsky, Spaces defined by topological games, Fund. Math. 88(1975),193-223.[68] M. Tkacenko, O bikompaktah predstavimyh...I and II, Comment. Math. Univ. Carolinae

20(1979), 361-379 and 381-395.[69] V.V. Tkachuk, Monolithic spaces and D-spaces revisited, Topology and Appl. 156(2009),

840-846.[70] S. Todorcevic, Partition problems in topology, Contemporary Mathematics, 84. American

Mathematical Society, Providence, RI, 1989.[71] J.M. Worrell and H.H. Wicke, Characterizations of developable topological spaces, Canad. J.

Math. 17 1965 820–830.[72] Z. Yu and Z. Yun, D-spaces, aD-spaces, and finite unions, Topology Appl. 156(2009), 1459-

1462.

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36830


2828


Combinatorics of filters and ideals

Michael Hrusak

Abstract. We study the combinatorial aspects of filters and ideals on count-able sets, concentrating on Borel ideals and their interaction with non-definableones. The basic tools for this study are cardinal invariants naturally associatedto ideals (filters) and the Katetov and Tukey orders.

Introduction

This paper is part survey and part research announcement. It contains noproofs, though in some cases short hints at proofs are given. It deals with combi-natorial aspects of filters and ideals on countable sets. The focus is on definable(Borel, analytic, . . . ) ideals and filters. The reason for this is twofold. On onehand, in a great number of cases there are “critical” ideals with respect to a givencombinatorial property, which are definable. In most cases these critical ideals areeven Borel of a low Borel complexity. On the other hand, definable filters andideals allow for fewer “pathologies” and the study of these can take advantage ofdescriptive set-theoretic methods, such as Borel determinacy, as well as forcing andcombinatorial methods combined with an absoluteness argument.

We are also interested in the interaction between definable and non-definableideals (filters). One of the first results linking properties of non-definable filtersto definable ones is A. Mathias’ characterization of selective ultrafilters as exactlythose ultrafilters which intersect every tall analytic ideal [72]. We will show howdefinable ideals can be used to naturally classify non-definable ones such as maximalideals (or, dually, ultrafilters) and maximal almost disjoint families.

The principal tools for our considerations are cardinal invariants of the con-tinuum and closely related partial orders on ideals (filters). Many such orderingshave been successfully used in the literature. We will closely examine two of them,the Katetov order and the Tukey order, as they apparently reflect combinatorial

2000 Mathematics Subject Classification. Primary and Secondary: 03E15, 03E17, 03E05,03E35.

Key words and phrases. Filter, ideal, cardinal invariants of the continuum, Katetov order,Tukey order.

The research was partially supported by PAPIIT grant IN101608 and CONACYT grant80355.


1



29



29

2 MICHAEL HRUSAK

properties of ideals better than the more rigid Rudin-Keisler order and Borel re-ducibility.

The paper is organized as follows. The first section contains basic definitionsof ideals, filters and their combinatorial properties as well as the definitions of therelevant cardinal invariants.

The second section deals with the problem of destructibility of ideals by forc-ing. It is largely based on [45] where a connection is established between forcingsof the type Borel(X)/I, where X is a Polish space and I is a σ-ideal of Borelsubsets of X, and forcings of the type P(ω)/I, where I is a definable ideal onω. In particular, it is shown that for a large class of definable forcings there is adefinable ideal naturally associated to the forcing critical (in the Katetov order) forideal destructibility by the forcing. The second part of section two deals with theMathias-Prikry and Laver-Prikry type forcings and is based mostly on [50]. Wegive a combinatorial characterization of Martin’s number for these forcing notionsand briefly outline a rather general scheme for analyzing preservation propertiesfor these forcing notions. In particular, we characterize for which ideals the corre-sponding Mathias-Prikry forcing adds a dominating real, and state sufficient andnecessary criteria for preservation of ω-hitting families.

In the third section we present a list of Borel ideals critical for various combi-natorial properties and calculate their cardinal invariants. These calculations aresometimes routine and sometimes nontrivial. The details of many of these can befound in [42, 48, 78].

The short fourth section is included mostly as a further motivation for studyof the Katetov order on Borel ideals. Here it is shown how Borel ideals natu-rally classify non-definable objects such as ultrafilters and maximal almost disjointfamilies.

The fifth section is devoted to basic structural analysis of the Katetov order onBorel ideals. First we present a theorem of D. Meza showing that the structure ofthe order is quite complex and we briefly discuss the (open) problem of the existenceof (locally) minimal tall Borel ideals and its connection to Ramsey type propertiesof Borel ideals. Finally, we present two dichotomies for Borel ideals and analyticP-ideals, respectively. This section is based on [46, 49, 48, 78].

The Tukey order, cofinal types and cofinalities of analytic ideals are consideredin section 6. We review basic theory of the Tukey order on analytic ideals asdeveloped by Todorcevic, Louveau-Velickovic and Fremlin in [99, 102, 70, 37] andintroduce a new class of fragmented Fσ ideals. We present a dichotomy theorem forthe fragmented Fσ ideals and prove some consistency results concerning cofinalitiesof Borel ideals ([51]).

In section 7 we propose a Wadge-like order on Borel ideals based on a naturalgame associated to a pair of Borel ideals (see [47]).

The last section (section 8) treats the quotient Boolean algebra P(ω)/I, fordefinable ideals I. We very briefly review the extensive body of work on rigid-ity phenomena and gap structure of the quotients done by Farah [25, 28, 29],Todorcevic [99, 100, 102] and Kanovei-Reeken [57, 58]. We also mention someisolated results on cardinal invariants of the quotients [1, 96, 94, 41, 31].

We have included a rather large number of open problems. They are scatteredthroughout the text.

3030

COMBINATORICS OF FILTERS AND IDEALS 3

Acknowledgments. The paper was written during my stay at the Universitatde Barcelona, January-August 2009. I wish to express my gratitude to the Gener-alitat de Catalunya for supporting my stay with the Research fellowship for visitingprofessors no. 00032. I also wish to thank to Prof. Joan Bagaria for inviting meand letting me present large parts of this work at his seminar, as well as to himand the other members of the seminar for listening to my talks and for making mystay so enjoyable.

The research presented here depends heavily (explicitly or implicitly) on previ-ous work of many mathematicians. Several of our determinacy arguments are basedon games considered by C. Laflamme [67, 66]. We take advantage of the works ofS. Solecki [89, 87, 88], I. Farah [25, 27, 28], S. Todorcevic [98, 101, 99, 100],A. Louveau and B. Velickovic [70], and D. Fremlin [36, 37] on analytic P-idealsand Tukey order, J. Brendle’s and S. Shelah’s work on cardinal invariants andultrafilters [16, 15, 18], and J. Zapletal’s work on definable forcing [109].

Large parts of this text are based on the PhD thesis Ideals and filters on count-able sets written by D. Meza under my supervision. Included are also results ofjoint work with B. Balcar, J. Brendle, F. Hernandez, D. Meza, H. Minami, D. Ro-jas, E. Thummel and J. Zapletal, some of them published, some of them in the finalstages of preparation.

Finally, I want to thank the anonymous referee and O. Zindulka for a verycareful reading of the manuscript and for patiently pointing out many factual andgrammatical errors and typos. I also want to thank the referee for pointing me tothe papers [10, 22] and [69].

1. Preliminaries and definitions

1.1. Ideals and filters. A family I ⊂ P(X) of subsets of a given set X is anideal on X if

(1) for A,B ∈ I, A ∪B ∈ I,(2) for A,B ⊂ X, A ⊂ B and B ∈ I implies A ∈ I and(3) X �∈ I.

In this paper we assume that all ideals on X contain all finite subsets of X. Dualis the notion of a filter on X, i.e. F ⊂ P(X) is a filter on X if

(1) for F,G ∈ F , F ∩G ∈ F ,(2) for F,G ⊂ X, F ⊂ G and F ∈ F implies G ∈ F and(3) ∅ �∈ F .

Given an ideal I onX we denote by I∗ the dual filter, consisting of complementsof the sets in I. Similarly, if F is a filter on X, F∗ denotes the dual ideal. We sayan ideal I on X is tall1 if for each Y ∈ [X]ω there exists I ∈ I such that I ∩ Yis infinite. Given an ideal I on a set X, we denote by I+ the family of I-positivesets, i.e. subsets of X which are not in I. If I is an ideal on X and Y ∈ I+, wedenote by I � Y the ideal {I ∩ Y : I ∈ I} on Y .

We will consider mostly ideals and filters on countable sets. In that case, wetypically pretend that they are, in fact, ideals or filters on ω.

We consider P(ω) equipped with the natural topology induced by identifyingeach subset of ω with its characteristic function, where 2ω is given the product

1Many authors prefer the term dense, which is probably more descriptive.

3131

4 MICHAEL HRUSAK

cov (N ) �� non (M) �� cof (M) �� cof (N )

b

��

�� d

��

add (N )

��

�� add (M) ��

��

cov (M) ��

��

non (N )

��

Cichon’s Diagram

topology. We call an ideal I a Borel (analytic, co-analytic,. . . ) ideal on ω if I isan ideal on ω and I is Borel (analytic, co-analytic,. . . ) in this topology. The sameapplies to filters.

An extensively studied class of ideals is the class of analytic P-ideals. An idealI on ω is a P-ideal if for any sequence Xn ∈ I, n ∈ ω, there is an X ∈ I suchthat Xn ⊆∗ X for all n ∈ ω, i.e. X \Xn is finite for all n ∈ ω. An ideal I on ω iscountably tall (or ω-hitting) [24] if for any sequence Xn ∈ [ω]ω, n ∈ ω, there is anX ∈ I such that |Xn ∩X| = ℵ0 for all n ∈ ω.

Let I be an ideal on ω. We say that I is a P+-ideal if for every decreasingsequence {Xn : n < ω} of I-positive sets there is an I-positive set X such thatX ⊆∗ Xn, for all n < ω. We say that I is a Q-ideal if for every partition {Fn : n <ω} of ω into finite sets there is an I-positive set Y ⊆ ω such that |Y ∩ Fn| ≤ 1,for all n < ω. We say that I is a Q+-ideal if its restriction to every positive setis a Q-ideal, i.e. if for every I-positive set X and every partition {Fn : n < ω} ofX into finite sets there is an I-positive set Y ⊆ X such that |Y ∩ Fn| ≤ 1, for alln < ω.

1.2. Cardinal invariants. Given an ideal I on a set X, the following arestandard cardinal invariants associated with I :

add (I) = min {|A| : A ⊆ I ∧⋃A /∈ I} ,

cov (I) = min{|A| : A ⊆ I ∧⋃A = X},

cof (I) = min {|A| : A ⊆ I ∧ (∀I ∈ I) (∃A ∈ A) (I ⊆ A)} ,non (I) = min{|Y | : Y ⊆ X ∧ Y /∈ I}.

We denote by M the ideal of meager subsets of R and by N the ideal ofLebesgue null subsets of R (or 2ω). For f, g ∈ ωω, we consider the order by eventualdominance f ≤∗ g if f(n) ≤ g(n) for all but finitely many n < ω. A family F ⊆ ωω

is bounded if there is h ∈ ωω such that f ≤∗ h for all f ∈ F ; and we say F isdominating if for any g ∈ ωω there is f ∈ F such that g ≤∗ f . The correspondingcardinal invariants are the minimal cardinality b of an unbounded family, and d,the minimal cardinality of a dominating family. The provable inequalities betweenthe cardinal invariants of M and N are summarized in Cichon’s diagram2.

For more on cardinal invariants in general and the Cichon’s diagram in partic-ular consult [5] and [11].

2As usual, the arrows in the diagram point from the smaller to the larger cardinal.

3232


non∗ (I)

��

ℵ0�� add∗ (I)

��

��cof∗ (I) �� 2ℵ0

cov∗ (I)

��

When we deal with ideals on countable sets, the only one of these cardinalinvariants giving any information is the cofinality, as all the others are less than orequal to ℵ0.

Definition 1.1 ([42]). Let I be a tall ideal on ω. Define the following cardinalsassociated with I :

add∗ (I) = min {|A| : A ⊆ I ∧ (∀X ∈ I) (∃A ∈ A) (A �∗ X)} ,cov∗ (I) = min{|A| : A ⊆ I ∧ (∀X ∈ [ω]ℵ0) (∃A ∈ A) (|A ∩X| = ℵ0)},cof∗ (I) = min {|A| : A ⊆ I ∧ (∀I ∈ I) (∃A ∈ A) (I ⊆∗ A)} ,non∗ (I) = min{|A| : A ⊆ [ω]

ℵ0 ∧ (∀I ∈ I) (∃A ∈ A) (|A ∩ I| < ℵ0)}.

The cof∗ is, of course, equal to cof for any uncountably generated ideal.3 Ourchoice of names is somewhat justified by the following: For every tall ideal I on ω,there is a natural ideal of Borel subsets of P (ω) associated with I defined as

I ={X ⊆ P (ω) : (∃I ∈ I) (X ⊆I)

},

where I = {X ⊆ ω : |X ∩ I| = ℵ0}. One can easily check that I ⊆∗ J if and

only if I ⊆ J . Hence, J ⊆ P (ω) is a P -ideal if and only if J is a σ-ideal. Then

add(I) = add∗(I), cov(I) = cov∗(I), non(I) = non∗(I) and cof(I) = cof∗(I).The inequalities holding among these cardinals are summarized in the above

diagram.It follows directly from the definition that cov∗ (I) ≥ p for any tall ideal I.

Also, add∗ (I) ≥ ℵ1 if and only if I is a P-ideal, and non∗ (I) ≥ ℵ1 if and only if Iis ω-hitting.

1.3. Orders on ideals on ω. We consider four (pre)orders on ideals on ωand discuss their impact on cardinal invariants of the ideals. Let I and J be idealson ω.

• (Katetov order) I ≤K J if there is a function f : ω → ω such thatf−1[I] ∈ J , for all I ∈ I.

• (Katetov-Blass order) I ≤KB J if there is a finite-to-one function f :ω → ω such that f−1[I] ∈ J , for all I ∈ I.

3Some of these cardinals have been originally introduced in the dual language of filters.Brendle and Shelah in [18] introduced cardinal invariants p(F) and πp(F) associated with an(ultra)filter F . For tall ideal I, add∗ (I) = p(I∗) , cov∗ (I) = πp(I∗ ), non∗ (I) = πχ (I∗) andcof∗ (I) = cof (I) = χ (I∗) .

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• (Rudin-Keisler order) I ≤RK J if there is a function f : ω → ω such thatA ∈ I if and only if f−1[I] ∈ J .

• (Tukey order) I ≤T J if there is a function f : I → J such that for every⊆-bounded set X ⊆ J , f−1[X] is ⊆-bounded in I.

We will say I and J are Katetov-equivalent if I ≤K J and J ≤K I. Analo-gously are defined Katetov-Blass, Rudin-Keisler and Tukey-equivalences.

There is a close relationship between the cardinal invariants of ideals and cor-responding orders. The Rudin-Blass order is the strongest; obviously, I ≤RB Jimplies I ≤RK J , and I ≤RK J implies both I ≤K J and I ≤T J .

Theorem 1.2. Let I and J be ideals on ω.

(1) If I ≤K J then cov∗(J ) ≤ cov∗(I).(2) If I ≤KB J then non∗(I) ≤ non∗(J ).(3) If I ≤T J then cof(I) ≤ cof(J ) and add∗(J ) ≤ add∗(I).

Shoenfield’s absoluteness entails that the Katetov order among Borel ideals isabsolute. When dealing with Borel (or analytic) ideals in several models of settheory, we do not consider the same set, which is unlikely to be either an ideal orBorel, but rather the ideal with the same Borel code. It should also be mentionedthat the it does not matter which code we take, as codes which give the same Borelset in one model give the same Borel set in any other model containing the codes.

Proposition 1.3. If I and J are Borel ideals on countable sets then the rela-tion I ≤K J is absolute for models M ⊆ N such that ωN

1 ⊆ M and I,J ∈ M

The same is, of course, also true for the Rudin-Keisler order, but not necessarilyfor the Tukey order. T. Matrai [73] has recently described two analytic ideals whichare Tukey equivalent if and only if CH holds. As of now there are no Borel examples.On the other hand, Solecki and Todorcevic [91] showed that among analytic P-idealsthe Tukey order reduces to a Borel function and therefore is also absolute.

1.4. Ideals and submeasures. There is an extremely close and useful con-nection between Fσ ideals and analytic P-ideals, and lower semicontinuous submea-sures.

Definition 1.4. A submeasure on a set X is a function ϕ : P(X) → [0,∞]satisfying:

• ϕ(∅) = 0,• If A ⊆ B then ϕ(A) ≤ ϕ(B) and• ϕ(A ∪B) ≤ ϕ(A) + ϕ(B).

To avoid trivialities, we also require that

• ϕ(F ) < ∞ for all finite subsets of X.

If ϕ is a submeasure on ω and satisfies:

• ϕ(A) = limn→∞ ϕ(A ∩ n)

then ϕ is called a lower semicontinuous submeasure, abbreviated by lscsm. To eachlscsm ϕ on ω naturally correspond the following two ideals:

• Fin(ϕ) = {A ⊆ ω : ϕ(A) < ∞} and• Exh(ϕ) = {A ⊆ ω : limn→∞ ϕ(A \ n) = 0}.

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It is immediate from the definition that Exh(ϕ) ⊆ Fin(ϕ), Fin(ϕ) is an Fσ

ideal and Exh(ϕ) is an Fσδ P-ideal. The following fundamental theorems of Mazurand Solecki are key to the study of both Fσ-ideals and analytic P-ideals.

Theorem 1.5 (Mazur [76]). Let I be an ideal on ω. Then I is an Fσ ideal ifand only if there is a lscsm ϕ such that I = Fin(ϕ).

Theorem 1.6 (Solecki [87, 88]). Let I be an ideal on ω. Then:

• I is an analytic P-ideal if and only if there is a lscsm ϕ such that I =Exh(ϕ).

• I is an Fσ P-ideal if and only if there is a lscsm ϕ such that I = Exh(ϕ) =Fin(ϕ).

In particular, all analytic P-ideals are Fσδ.

1.5. MAD families and ultrafilters. Given two infinite subsets A,B of ωwe say A and B are almost disjoint if A∩B is finite. A family A of infinite subsetsof ω is an almost disjoint family if A and B are almost disjoint for any A,B distinctelements of A. A MAD family is an infinite maximal almost disjoint family, i.e. analmost disjoint family such that for every infinite set X ⊆ ω there is an A ∈ A suchthat A ∩ X is infinite. Given an almost disjoint family A we denote by I(A) theideal generated by A. Note that I(A) is a tall ideal if and only if A is a MAD family.In [72], A. Mathias proved that ideals generated by MAD families are meager butnot analytic.

Every filter can be extended to a maximal filter (ultrafilter) by the Kuratowski-Zorn lemma. We only consider free ultrafilters, i.e. ultrafilters consisting of infinitesets. Ultrafilters have been thoroughly studied by both set-theorists and topologists.The most important classes of ultrafilters are: selective ultrafilters, P-points, Q-points, rapid ultrafilters and nowhere dense ultrafilters. An ultrafilter U on ω is:

• selective if for every partition {In : n ∈ ω} of ω into sets not in U there isU ∈ U such that |U ∩ In| = 1 for every n ∈ ω.

• a P-point if for every partition {In : n ∈ ω} of ω into sets not in U thereis U ∈ U such that |U ∩ In| is finite for every n ∈ ω.

• a Q-point if for every partition {In : n ∈ ω} of ω into finite sets there isU ∈ U such that |U ∩ In| = 1 for every n ∈ ω.

• rapid if the family of increasing enumerations of elements of U is domi-nating.

• nowhere dense (or a nwd-ultrafilter) if for every map f : ω → R there is aU ∈ U such that f [U ] is a nowhere dense subset of R.

It is well known that an ultrafilter U is selective if and only if it is both aP-point and a Q-point. Also every Q-point is rapid and every P-point is nwd [5].

2. Destructibility of ideals by forcing

Our interest in the Katetov order stems from the study of destructibility ofideals by forcing.

Definition 2.1. Given an ideal I and a forcing notion P, we say that P destroysI if there is a P-name X for an infinite subset of ω such that

�P “I ∩ X is finite for every I ∈ I”.

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Destroying an ideal (which really means destroying tallness of the ideal) is,in the dual language of filters, called also diagonalizing or zapping a filter. Thegeneral question, central in combinatorial set theory of the reals, is the following:

Question 2.2. When does a given forcing destroy a given ideal?

Many open problems boil down to instances of this question: The consistencyof p < t, the question of Roitman as to whether the existence of a dominatingfamily of size ℵ1 implies the existence of a MAD family of size ℵ1, . . .

2.1. Trace ideals. It turns out that there is a deep connection between theproper forcings of the type PI of I-positive Borel subsets of a Polish space X,ordered by inclusion, where I is a σ-ideal on X, studied by Zapletal in [109],and definable ideals on countable sets and their corresponding quotient Booleanalgebras. PI is a non-separative partial order whose separative quotient is theσ-algebra Borel(X)/I. Zapletal [109] has given the following characterization ofproperness of these forcing notions:

PI is proper if and only if for every countable elementary submodel M of alarge enough H(θ) and every condition B ∈ M ∩ PI the set C = {x ∈ B : x isM -generic} is I-positive.

Another important property of forcings of the type PI is the Continuous Read-ing of Names (CRN).

Definition 2.3 (Zapletal [109]). If PI is a proper forcing then it has the CRNif for every Borel function f : B → 2ω with an I-positive Borel domain B there isan I-positive Borel set C ⊆ B such that f � C is continuous.

Many of the common proper forcing notions, such as Cohen, random, Sacks,Miller, Laver, . . . can be naturally presented as forcings of the form PI with theCRN. In particular, every proper ωω-bounding poset PI has the continuous readingof names, and if the ideal I is σ-generated by closed sets then the forcing PI is properand it has the continuous reading of names (see [45] and [109]).

With Zapletal [45] we have studied the relationship between the forcings oftype PI and quotients P(ω)/I, where I is an ideal on ω. The link between theseclasses of posets is provided by the following definition.

Definition 2.4 (Brendle [19]). Given a σ-ideal I on ωω, its trace ideal tr(I)is an ideal on ω<ω defined by a ∈ tr(I) if and only if {r : ∃∞n ∈ ω (r � n ∈ a)} ∈ I.

Of course, if the σ-ideal I is reasonably definable, so is the ideal tr(I).

Theorem 2.5 ([45]). Let I be a σ-ideal on ωω. If PI is a proper forcing withCRN then P(ω<ω)/tr(I) is a proper forcing as well and it is naturally isomorphicto a two-step iteration of PI followed by an ℵ0-distributive forcing.

In some cases, we have been able to identify the ℵ0-distributive “tail” forcingas P(ω)/fin of the PI -extension; however, we do not know what it is in many othercases, such as in the case of Cohen and random forcings.

Proposition 2.6 ([45]). Let I be a σ-ideal on ωω σ-generated by a σ-compactfamily of closed sets. Then the forcing PI is proper and ωω-bounding4, andP(ω<ω)/tr(I) = PI ∗ P (ω)/fin.

4Recall that a forcing is ωω-bounding if it does not add unbounded reals.

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It turns out that the trace ideals are critical, in the Katetov order, with re-spect to PI-destructibility. The following theorem was discovered independently byKurilic [63] and Hrusak [43], for the special case of Cohen forcing, then extendedby Brendle and Yatabe [19] to a larger class of forcings and finally took the currentform in [45].

Theorem 2.7 ([45]). If PI is a proper forcing with CRN and I is an ideal onω then the following are equivalent:

(1) there is a B ∈ PI such that B � “the ideal I is destroyed”, and(2) there is a tr(I)-positive set a such that I ≤K tr(I) � a.

We say that an ideal I on ω is K-uniform if I � X ≤K I for every I-positive setX. A forcing of the form PI where I is a σ-ideal on ωω is continuously homogeneousif for every I-positive Borel set B there is a continuous function F : ωω → B suchthat F−1(A) ∈ I for all A ∈ I � B. It is easy to see that if PI is continuouslyhomogeneous, then tr(I) is K-uniform and hence the theorem takes a nicer form.

Theorem 2.8. Let PI be a proper forcing with CRN, which is continuouslyhomogeneous, and let J be an ideal on ω. Then the following conditions are equiv-alent:

(1) PI destroys J(2) J ≤K tr(I).

Many of the aforementioned forcings are indeed continuously homogeneous, e.g.Cohen, random, Miller, Sacks, . . .

There is a close relation between the covering number of the σ-ideal and thecov∗-number of the corresponding trace ideal.

Proposition 2.9 ([45]). Suppose that I is a σ-ideal on ωω generated by ana-lytic sets such that PI is a proper forcing with the CRN. Then

cov(I) ≤ cov∗(tr(I)) ≤ max{cov(I), d}.

The trace ideals associated to definable forcing notions are themselves definable,though they are rarely Borel. They are Borel, in fact Fσδ, for Cohen and randomforcing; however, for most other simple forcing notions they are already co-analytic(or worse). The only known Borel trace ideals come from c.c.c. forcings.

Question 2.10. Is there a non-c.c.c. forcing PI such that tr(I) is Borel?

We conjectured in [45] that if the trace ideal is analytic then it is even Borel.We also do not have an example of a Borel trace ideal of Borel complexity higherthan Fσδ.

2.2. Laflamme, Mathias-Prikry and Laver-Prikry type forcings. Stillconsidering the question 2.2, rather than fixing a forcing and investigating whichideals are being destroyed, one can fix an ideal and try to find a forcing withadditional “nice” properties destroying the ideal, for instance forcing not addingunbounded or dominating reals. Laflamme [65] has shown that

Theorem 2.11 (Laflamme [65]). Every Fσ ideal can be destroyed by a properωω-bounding forcing.

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A variant of Laflamme’s forcing can be easily described using Mazur’s charac-terization of Fσ ideals. Let I be an Fσ ideal, I = Fin (ϕ) for some lower semicon-tinuous submeasure ϕ by Theorem 1.5. Define the poset Pϕ as the set of all perfectfinitely branching trees T ⊆ ω<ω such that limt∈T ϕ (succT (t)) = ∞5, ordered byinclusion.

The forcing Pϕ destroys I, is ωω-bounding and adds a bounded eventuallydifferent real. There are several interesting problems concerning this forcing. Forinstance, it is not known whether it can add random reals or even just independentreals. Also, it would be interesting to characterize those submeasures ϕ such thatthe forcing Pϕ preserves outer Lebesgue measure.

Laflamme’s theorem cannot be extended even to Fσδ ideals. However, thefollowing seems to be an open problem:

Question 2.12. Can every Fσδ-ideal be destroyed by a proper forcing notadding a dominating real? What about the density zero ideal Z?

Two natural and commonly used forcing notions that destroy a given ideal Jare the Mathias-Prikry and Laver-Prikry forcings associated to J .

Definition 2.13. Let J be an ideal on ω.

The Mathias-Prikry forcing MJ associated to the ideal J is defined as the setof all pairs 〈t, a〉 where t ⊂ ω is a finite set, a ⊂ ω is a set in the ideal J , and〈u, b〉 ≤ 〈t, a〉 if t ⊂ u, a ⊂ b and a ∩ u \ t = 0.

We will refer to the union of the first coordinates of conditions in the genericfilter as the generic subset of ω, and denote it by agen.

The Laver-Prikry forcing LJ associated to the ideal J consists of perfect sub-trees T ⊆ ω<ω with stem sT such that for every t ∈ T with sT ⊆ t the setsuccT (t) ∈ J ∗, ordered by inclusion.

We denote by fgen the name for the generic function (the union of the stems

of the trees in the generic filter) and by agen the range of fgen.

In fact, both forcings do more than destroy the ideal J , they separate J fromJ +, i.e. agen is forced to be almost disjoint from all ground model sets in J andhave an infinite intersection with all J -positive ground model sets.

It is useful to introduce the corresponding cardinal invariant, the separatingnumber of an ideal J .

sep(J ) =min{|H|+ |K| : K ⊂ J ,H ⊂ J+ and

∀A ⊂ ω ((∃J ∈ K(|A ∩ J | = ω) or ∃H ∈ H(|A ∩H| < ω))}.

It is clear from the definition that add∗(J ) ≤ sep(J ) ≤ cov∗(J ), and thatsep(J ) = cov∗(J ) if J is a maximal ideal (i.e. if J ∗ is an ultrafilter).

Proposition 2.14. Let I and J be ideals on ω. If I ≤RK J then sep(J ) ≤sep(I).

5By succT (t) we denote the set {n ∈ ω : t�n ∈ T} and limt∈T ϕ (succT (t)) = ∞ means thatfor every N ∈ ω the set of those t ∈ T such that ϕ (succT (t)) < N is finite.

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Both the Mathias-Prikry and Laver-Prikry forcing notions are clearly c.c.c., infact, σ-centered. Also, LJ adds a dominating real (the generic function fgen isdominating).

The rank analysis of names introduced by Baumgartner and Dordal [7] forHechler forcing is the basic tool for analyzing forcing properties of the Laver-Prikrytype forcings (see e.g. [16, 15, 17]). There does not seem to be a direct analogueof this for the Mathias-Prikry type forcings; however, they can be analyzed bystudying the following ideal associated to an arbitrary ideal I. Let fin denote thefamily of all non-empty finite subsets of ω.

Definition 2.15. Given I an ideal on ω, let

I<ω = {A ⊆ fin : (∃I ∈ I)(∀a ∈ A) a ∩ I �= ∅}.

This ideal was probably first considered implicitly by Sirota [86] and explic-itly by Louveau [68] in the construction of an extremally disconnected topologicalgroup.

We are now going to characterize basic preservation properties of the forcingsMJ and LJ . The first property we consider is the property of not adding a Cohenreal.

Theorem 2.16 ([14]). Let J be an ideal on ω. Then

(1) MJ does not add a Cohen real if and only if J ∗ is a selective ultrafilter.(2) LJ does not add a Cohen real if and only if J ∗ is a nwd-ultrafilter.

Moreover, if J ∗ is a selective ultrafilter, then MJ and LJ are forcing equivalent.

Shelah and B�laszczyk have extended (2) to prove the following:

Theorem 2.17 ([12]). There is a σ-centered forcing that does not add Cohenreals if and only if there is a nowhere dense ultrafilter.

The behavior of any forcing notion P can be to a large extent described by itsMartin number

m(P) = min{κ : ¬MAκ(P)},i.e. m(P) is the minimal size of a collection of dense subsets of P such that no filteron P intersects them all.

In [18], Brendle and Shelah characterized the Martin numbers of the Mathias-Prikry and Laver-Prikry type forcings for ultrafilters as follows:

Theorem 2.18 ([18]). Let U be an ultrafilter. Then:

(1) m(MU∗) = cov∗(U∗) and(2) m(LU∗) = min{b, cov∗(U∗)}.

For arbitrary ideal the situation is similar, with three changes, first the coveringnumber has to be replaced by the separating number, in the case of the Mathias-Prikry forcing the ideal I<ω has to be considered, and the fact that the forcingadds Cohen reals has to be taken into account.

Theorem 2.19 ([50]). Let I be an ideal on ω which is not maximal. Then:

(1) m(MI) = min{sep(I<ω), cov(M)} and(2) m(LI) = min{sep(I), add(M)}.

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We have already seen that the forcing LI always adds a dominating real. Thequestion of when the forcing MI adds a dominating real was considered by Canjar[20] and Brendle [15]. Canjar [20] has, assuming d = c, constructed an ultrafilter Usuch that the forcing MU∗ does not add a dominating real and noticed that such anultrafilter necessarily has to be a P-point without rapid Rudin-Keisler predecessors.Brendle in [15] has (among other things) noticed that MI does not add dominatingreals for any Fσ ideal I.

Here we present a simple combinatorial characterization of not adding a dom-inating real by the Mathias-Prikry type forcings.

Theorem 2.20 ([50]). Let I be an ideal on ω. Then MI does not add adominating real if and only if the ideal I<ω is a P+-ideal.

Both Canjar’s and Brendle’s results follow as simple corollaries. It is not clear atthe moment whether for an ultrafilter U not adding dominating reals is equivalentto U being a P-point without rapid Rudin-Keisler predecessors. We also do notknow whether for a Borel ideal I, MI does not add a dominating real if and onlyif I is Fσ. A result in this direction is the following:

Theorem 2.21 ([50]). Let J be a Borel ideal on ω. Then the following areequivalent:

(1) J can be extended to an ideal I (not necessarily definable) such that MIdoes not add a dominating real

(2) J can be extended to an Fσ ideal.

The combinatorics of the Mathias-Prikry forcing and the ideal I<ω is closelyrelated to the problem of Malykhin (see [80, 39]) in general topology: Is therea separable non-metrizable Frechet topological group? To each ideal I on ω onecan naturally associate a group topology τI on the countable Boolean group [ω]<ω

with the symmetric difference as the group operation. The ideal I<ω is the idealof sets whose closure does not contain the neutral element ∅ in this topology. Theresulting group topology τI on [ω]<ω is Frechet iff every I<ω-positive set containsan infinite set in (I<ω)⊥6 and it is metrizable if and only if the ideal I is countablygenerated.

An ideal I is Frechet if I = I⊥⊥. In other words, τI is Frechet iff I<ω isa Frechet ideal. Gruenhage and Szeptycki in [39] asked the following instance ofMalykhin’s question:

Question 2.22 (Gruenhage-Szeptycki [39]). Is there an uncountably generatedideal I such that I<ω is Frechet?

The answer is known to be positive in various models of ZFC, see [39]. In [17]we have given the following partial negative answer:

Theorem 2.23 ([17]). It is consistent with ZFC that I<ω is not Frechet forany ℵ1-generated ideal I.

One of the crucial elements of our proof was preservation of ω-hitting familiesby the Laver-Prikry type forcing:

Theorem 2.24 ([17]). Let I be an ideal on ω. Then the following are equiva-lent:

6Recall that if I is an ideal on a set X then I⊥ = {J ⊆ X : (∀I ∈ I)|I ∩ J | < ω}.

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(1) LI preserves ω-hitting families,(2) ∀X ∈ I+ ∀J ≤K I � X (J is not ω-hitting).

Note that, in particular, LI preserves ω-hitting families if I is a Frechet ideal,but also in many other cases, such as for I being the ideal nwd of nowhere densesubsets of the rationals. A similar result also holds for the Mathias-Prikry forcing:

Theorem 2.25 ([50]). Let I be an ideal on ω. Then the following are equiva-lent:

(1) MI preserves ω-hitting families,(2) ∀X ∈ (I<ω)+ ∀J ≤K I<ω � X (J is not ω-hitting).

While there is an abundance of ideals for which the forcing LI preserves ω-hitting, there is no known ZFC example of an ideal I such that MI preserves ω-hitting. This can be seen as a variant of the question of Gruenhage and Szeptycki.

Question 2.26. Is there in ZFC an uncountably generated ideal I such thatMI preserves ω-hitting families?

The results concerning Borel ideals contained here depend on the study ofKatetov order contained in subsequent sections. We present them here as part ofthe motivation for the study of the Katetov order on Borel ideals.

3. Some critical Borel ideals and their cardinal invariants

3.1. The nowhere dense ideal nwd. The nowhere dense ideal nwd is theideal on the set of rational numbers Q whose elements are the nowhere dense subsetsof Q. nwd is an Fσδ ideal. It is naturally isomorphic to the trace ideal correspondingto the Cohen forcing. In particular, for an ideal I on ω,

I is Cohen-destructible if and only if I ≤K nwd.

It is a result of Keremedis [62] (see also [1]) that cov∗(nwd) = cov(M). Fremlin[37] proved that cof(nwd) = cof(M). Finally, any countable base for open sets ofQ is a witness for non∗(nwd) = ℵ0.

To highlight the close relationship between the Katetov order and the cov∗-number of an ideal, we mention the following

Proposition 3.1 ([48]). Let I be a Borel ideal on ω. Then I ≤K nwd if andonly if ZFC � cov∗(I) ≥ cov(M).

Proof. One implication follows directly from 1.2. To see the other, assumethat ZFC � cov∗(I) ≥ cov(M). Add c+-many Cohen reals. Then cov∗(I) >cov∗(I)V , so I is Cohen-destructible and hence I ≤K nwd. �

3.2. The eventually different ideals. The eventually different ideal is de-fined by

ED = {A ⊂ ω × ω : (∃m,n ∈ ω)(∀k > n) (|{l : 〈k, l〉 ∈ A}| ≤ m)}.It is easily seen that the ideal ED is not ω-hitting, so add∗(ED) = non∗(ED) =

ℵ0. Furthermore, cov∗(ED) = non(M) and cof∗(ED) = c (see [48]). The ideal EDis critical for selective ideals:

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14 MICHAEL HRUSAK

Proposition 3.2. Let I be an ideal on ω. Then ED ≤K I if and only if thereis a partition of ω into sets in I such that every selector is in I.

We also consider the ideal EDfin = ED � �, where � = {〈m,n〉 : n ≤ m}. Itis critical for Q-ideals, in much the same way:

Proposition 3.3 ([48]). Let I be an ideal on ω. Then EDfin ≤KB I if andonly if there is a partition of ω into finite sets such that every selector is in I.

Moreover, it is the KB-least ω-hitting ideal among definable ideals. We statethe theorem for Borel ideals, in order to keep the determinacy arguments simpleand intuitive. However, this and other similar theorems are typically true either foranalytic or co-analytic ideals in ZFC, and for ideals of higher complexities assumingdeterminacy at corresponding levels of the projective hierarchy.

Theorem 3.4 ([48]). If I is a Borel ideal on ω, then non∗(I) = ω or EDfin ≤KB

I.

Proof. For a Borel ideal I, consider the following two-player game: In stagek, Player I chooses a finite subset Fk of ω and then Player II chooses a naturalnumber nk �∈ Fk.

I F0 ∈ [ω]<ω F1 ∈ [ω]<ω . . .II n0 �∈ F0 n1 �∈ F1 . . .

Player I wins if {ni : i ∈ ω} ∈ I and Player II wins {ni : i ∈ ω} ∈ I+.Now, by Borel determinacy, the game is determined, so it suffices to note that:

(1) If Player I has a winning strategy then EDfin ≤KB I.(2) If Player II has a winning strategy, then non∗(I) = ω. �

In particular, non∗(I) = ω or non∗(EDfin) ≤ non∗(I) for all Borel ideals I. Italso follows that every ω-splitting7 Borel ideal contains a perfect ω-splitting subset,which is a special case of a theorem of Spinas [92].

The cov∗(EDfin) and non∗(EDfin) can be viewed as bounded versions of non(M)and cov(M), respectively, and extend in a natural way Cichon’s diagram.

Proposition 3.5 ([48]). The following hold:

(1) cov(M) = min{d, non∗(EDfin)} and(2) non(M) = max{b, cov∗(EDfin)}.

Let us also mention that the min and max in the proposition are sharp. In theRandom real model, i.e. a model obtained from a model of CH by adding at leastℵ2-many random reals, cov∗(EDfin) > d, and cov∗(EDfin) < add(M) holds in theHechler model. In a sense dually, non∗(EDfin) < b holds after adding ℵ1-manyrandom reals to a model of Martin’s Axiom. The fact that cof(M) < non∗(EDfin)holds after adding ℵ1-many Hechler reals to a model of MA, is an unpublished resultof J. Brendle.

7A family S of infinite subsets of ω is ω-splitting if for every countable collection of infinitesubsets of ω there is an element of S which splits all elements of the collection.

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3.3. Fubini products. Given two ideals I,J on ω, the Fubini product I ×Jis defined by

I × J = {A ⊆ ω × ω : {n : (A)n /∈ J } ∈ I}.It is easy to see that the Fubini product of Borel ideals is also a Borel ideal.

Actually, if I is a Σα ideal and J is a Σβ ideal then I × J is a Σβ+α ideal.

Here we violate our implicit agreement, that all ideals contain all finite subsetsof their underlying sets, this facilitates the definitions (and natural names) for thefollowing two ideals.

fin × ∅ can be thought of as the ideal generated by an infinite partition of ωinto infinite sets. It is countably generated, hence not tall.

∅ × fin can be viewed as an ideal I for which there is a partition of ω intoinfinitely many infinite sets {Pn : n < ω}, such that I ∈ I if and only if I ∩ Pn isfinite for all n < ω. It is not a tall ideal and consequently is Katetov equivalentwith fin. It is an Fσδ P -ideal, with cof(∅ × fin) = d and add∗(∅ × fin) = b.

fin × fin is an Fσδσ ideal. It is critical with respect to the following P-likeproperty:

Proposition 3.6. Given an ideal I on ω, I ≥K fin × fin if and only if thereis a partition {Qn : n < ω} of ω into infinite sets in I such that every A ⊆ ωsatisfying |A ∩Qn| < ℵ0 is in I. �

Its cardinal invariants are: add∗(fin×fin) = non∗(fin×fin) = ℵ0, cov∗(fin×fin) =

b and cof(fin× fin) = d.

Building on an earlier work of Solecki [89] and a natural game introduced byLaflamme [66], Laczkovich and Rec�law [64] proved the following dichotomy.

Theorem 3.7 ([64]). Let I be a Borel ideal. Then either

(1) I ≥K fin× fin, or(2) I and I∗ can be separated by an Fσ set, i.e there is an Fσ set X containing

I and disjoint from I∗.

In particular, no ideal Katetov-above fin× fin can be extended to an Fσδ idealas any Fσδ ideal can be separated from its dual according to a theorem of Solecki[89].

3.4. conv. An ideal closely related to fin× fin is the ideal conv, defined as theideal on Q ∩ [0, 1] generated by sequences in Q ∩ [0, 1] convergent in [0, 1]. convis an Fσδσ ideal. Every conv-positive set contains a positive subset X such thatconv � X is naturally isomorphic to fin × fin. The cardinal invariants of conv aretrivial: add∗(conv) = non∗(conv) = ℵ0 and cov∗(conv) = cof(conv) = c.

The following theorem characterizes those ideals which are Katetov above theideal conv.

Theorem 3.8 ([78]). For any ideal I on ω the following are equivalent

(1) I ≥K conv,(2) there is a countable family X ⊆ [ω]ω such that for every Y ∈ I+ there is

X ∈ X such that |X ∩ Y | = |Y \X| = ℵ0.

the ideal conv is also a lower bound for all trace ideals in the Katetov order:

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16 MICHAEL HRUSAK

Theorem 3.9 ([78]). Let I be an ideal on ω such that the quotient P(ω)/Iis a proper forcing adding a new real. Then there is an I-positive set X such thatI � X ≥K conv.

3.5. The random graph ideal R. Given a graph G on ω, one can define a(possibly improper) ideal IG as the ideal generated by the set of all the subsets ofω which are homogeneous (cliques and free sets) for G. We consider the ideal R onω generated by the homogeneous sets in the Rado graph (also called the randomgraph) E . The Rado graph is determined uniquely (up to an isomorphism) by thefollowing extension property:

Given a and b disjoint finite subsets of ω there is k < ω such that {{k, l} : l ∈a} ⊆ E and {{k, l} : l ∈ b} ∩ E = ∅.

It immediately follows that the Rado graph is universal, i.e. given a graph〈ω,G〉, there is a subset X ⊆ ω such that 〈ω,G〉 ∼= 〈X,E � X〉.

The ideal R is Fσ and, by Ramseys theorem, it is tall. Its cardinal invariantsare trivial: add∗(R) = non∗(R) = ℵ0, cov

∗(R) = cof(R) = c.

Consider the following Ramsey property of ideals:

Definition 3.10. Let I be an ideal on ω. We will say that I satisfies

ω −→ (I+)22

if for every coloring ϕ : [ω]2 → 2 there is an I-positive set X homogeneous withrespect to ϕ. We will say that I satisfies

I+ −→ (I+)22

if for every I-positive set X and every coloring ϕ : [X]2 → 2 there is an I-positivesubset Y of X homogeneous with respect to ϕ.

The ideal R is critical with respect to the property ω −→ (I+)22.

Proposition 3.11. Let I be an ideal on ω. Then,

ω −→ (I+)22 if and only if I �K R.

In particular, the following conditions are equivalent:

• I+ −→ (I+)22,• R �K I � X, for all X ∈ I+.

3.6. Solecki’s ideal S. Denote by Clop(2ω) the (countable) family of allclopen subsets of the Cantor set 2ω, and let λ denote the standard Haar measureon 2ω. Solecki’s ideal S [89] is the ideal on the countable set

Ω = {A ∈ Clop(2ω) : λ(A) =1

2},

generated by the subsets of Ω with non-empty intersection. Equivalently, a subbasefor S is the family of all subsets of Ω of the form:

Ix = {A ∈ Ω : x ∈ A}where x is an element of 2ω.

The ideal S is a tall Fσ ideal, whose cardinal invariants are: add∗(S) =non∗(S) = ℵ0, cov

∗(S) = non(N ) and cof(S) = c [48].

The ideal S is critical for ideals which fail to satisfy the Fubini property.

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Definition 3.12. We will say that an ideal I satisfies the Fubini property iffor any Borel subset A of ω× 2ω and any ε > 0, {n < ω : λ(An) > ε} ∈ I+ impliesλ∗({x ∈ 2ω : Ax ∈ I+}) ≥ ε, where λ∗ denotes the outer Lebesgue (Haar) measureon 2ω.

Of course, for ideals which are universally measurable, in particular, for Borelideals, the outer measure in the definition can be replaced by measure. Solecki [89]noticed that:

Theorem 3.13 (Solecki [89]). An ideal I fails to satisfy the Fubini property ifand only if there is an I-positive set X such that S ≤K I � X.

3.7. Summable ideals. Given f : N → R+ such that∑

n∈ω f (n) = ∞, thesummable ideal corresponding to f is the ideal

If =

{

A ⊆ ω :∑

n∈A

f (n) < ∞}

.

The ideal If is tall if and only if limn→∞ f (n) = 0. The lower semicontinuoussubmeasure on ω corresponding to If is: ϕf (A) =

∑n∈A f (n). By definition

If = Fin (ϕf ). So, summable ideals are Fσ. A typical example of a summable idealis the ideal

I 1n=

{

A ⊆ ω :∑

n∈A

1

n< ∞

}

.

3.8. Asymptotic density zero ideal. The ideal Z of subsets of ω of asymp-totic density zero is the ideal

Z ={A ⊆ ω : lim

n→∞|A∩n|

n = 0}.

Equivalently, A ∈ Z if and only if

limn→∞

∣∣A ∩ [2n, 2n+1)

∣∣

2n= 0.

Both the tall summable ideals and the density zero ideal Z are tall analyticP-ideals, hence are KB-above EDfin. Consequently, their covering numbers arebelow cov∗(EDfin) and, dually, their uniformity numbers are above non∗(EDfin).Also, the summable ideals are random-destructible; hence cov (N ) ≤ cov∗ (I) andnon∗ (I) ≤ non (N ) for every tall summable ideal I [42].

For the density zero ideal, there are some upper and lower bounds

min {b, cov (N )} ≤ cov∗ (Z) ≤ max {b, non (N )}

(see [42]) and many questions. It is not even known whether it can be destroyedby an ωω-bounding forcing.

For additivity and cofinality the results are optimal [37]: add∗ (I) = add (N )and cof∗ (I) = cof (N ), for every tall ideal I which is either summable or a densityideal (see [25]) for the definition of a density ideal).

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18 MICHAEL HRUSAK

4. Katetov order, ultrafilters and MAD families

The Katetov order was introduced by Miroslav Katetov in [59] together withan order that became known as the Rudin-Keisler order. On ultrafilters (or equiv-alently maximal ideals) the two orders coincide. Whereas the Rudin-Keisler orderhas been extensively studied, the Katetov order has been somewhat neglected. Oneof the primary objectives of this survey is to show that it is both useful and intrin-sically interesting, and deserves further study. For some early results on Katetovorder see [22].

4.1. Elementary facts about Katetov order. Some immediate propertiesof Katetov order are listed here. Let I and J be ideals on ω.

(1) I �K fin if and only if I is not tall.(2) If I ⊆ J then I ≤K J .(3) If X ∈ I+ then I ≤K I � X.(4) I ⊕ J ≤K I and I ⊕ J ≤K J .(5) I,J ≤K I × J .

Here I ⊕J denotes the disjoint sum of I and J . Properties (4) and (5) show thatKatetov order is both upward and downward directed. The following propositionlists some of the order-theoretic properties of the Katetov order.

Proposition 4.1 ([78, 44]). The following hold.

(1) Every family A of at most c ideals has a ≤K-lower bound.(2) The family of maximal ideals is cofinal in Katetov order.(3) Ideals generated by MAD families are coinitial among tall ideals in Katetov

order.

Theorem 4.2 ([78, 44]). Let I be a tall ideal on ω. Then

(1) there is a ≤K-antichain below I of cardinality c and(2) there is a ≤K-decreasing chain of length c+ below I.

4.2. Ultrafilters and Katetov order. In this section we study the criticalideals for well studied classes of ultrafilters: P-points, Q-points, selective ultrafiltersand rapid ultrafilters. We conclude this section with the study of S-ultrafilters, i.e.the ultrafilters which satisfy the Fubini property.

Theorem 4.3 (Mathias [72]). Let U be an ultrafilter on ω. Then U is selectiveif and only if U intersects every tall analytic ideal I.

Zapletal [110] has recently found a characterization of P-points similar in spirit.

Theorem 4.4 (Zapletal [110]). Let U be an ultrafilter on ω. Then U is a P-point if and only if for every Borel ideal I disjoint from U there is an Fσ-ideal Jdisjoint from U and containing I.

J. Baumgartner introduced the following definition in [6]. Let I be a family ofsubsets of a set X such that I contains all singletons and is closed under subsets.An ultrafilter U is an I-ultrafilter if for every function F : ω → X there is an A ∈ Usuch that F [A] ∈ I.

Proposition 4.5. Let I be an ideal on ω. Then an ultrafilter U on ω is anI-ultrafilter if and only if I �K U∗.

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Many standard combinatorial properties of ultrafilters are easily seen to becharacterized in this way by Borel ideals of a low complexity (see [35] for details).Let U be an ultrafilter and U∗ the dual ideal. Then

• U is selective iff ED �≤K U∗ iff R �≤K U∗,• U is a P-point iff fin× fin �≤K U∗ iff conv �≤K U∗,• U is a nowhere dense ultrafilter iff nwd �≤K U∗,• U is a Q-point iff EDfin �≤KB U∗,• U is rapid iff I �≤KB U∗ for any analytic P-ideal I.

Another, perhaps less standard property of ultrafilters was considered by M.Benedikt [8, 9]. Given an ultrafilter U on ω, and a sequence 〈An : n ∈ ω〉 of Borelsubsets of the Cantor space 2ω, the U-limit of the sequence 〈An : n ∈ ω〉 is the set

U- limAn = {x ∈ 2ω : {n ∈ ω : x ∈ An} ∈ U}.If 〈xn : n < ω〉 is a sequence of real numbers then l ∈ R is the U-limit of 〈xn : n < ω〉provided that {n < ω : |xn − l| < ε} ∈ U for all ε > 0.

Proposition 4.6. Let U be a free ultrafilter. Then the following conditions areequivalent:

(1) S �≤K U∗,(2) U∗ satisfies the Fubini property and(3) for any sequence 〈An : n < ω〉 of Borel subsets of 2ω,

if U- limλ(An) > 0 then U- limAn �= ∅.It is well known that for every one of the properties considered in this section

it is relatively consistent with ZFC that there are no ultrafilters satisfying it (see[5]).

Question 4.7. Is there a Borel ideal I such that in ZFC there is an ultrafilterU such that I �≤K U∗? What about the density zero ideal Z?

Concerning the density zero ideal, Gryzlov [40] showed that in ZFC there is anultrafilter U such that for any injective function f : ω → ω there is a U ∈ U suchthat f [U ] ∈ Z. Flaskova in [34] improved on Gryzlov’s result by showing that thedensity ideal can be replaced by the summable ideal I 1

n.

There is also the well-known problem:

Question 4.8. Is it consistent with ZFC that there are neither P-points norQ-points?

There are either P-points or Q-points in every model of c ≤ ℵ2 (see [5]).

4.3. MAD families and Katetov order. Just as we saw that the upwardKatetov-cones of definable ideals stratify and classify ultrafilters, the downwardcones do the same for MAD families. Given a (Borel) ideal I one could, dualizingBaumgartner’s definition, call a MAD family A I-MAD if I(A) �≤K I.

For the K-uniform trace ideals, of course, A is tr(I)-MAD if and only if it isPI -indestructible. It is easy to see that I(A) ≤K fin × fin, for every MAD familyA. There are many natural questions concerning MAD families and the Katetovorder. We mention three of them:

Question 4.9 ([44]). Is there (consistently) a MAD family ≤K-maximal amongMAD families?

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20 MICHAEL HRUSAK

Question 4.10 (Steprans [95]). Is there a Cohen-indestructible MAD familyin ZFC?

Question 4.11 ([52]). Is there a Sacks-indestructible MAD family in ZFC?

There is a published incorrect answer to the last question. In [43] I showedthat there is a ctbl-MAD family in ZFC, here ctbl denotes the ideal of the subsetsof the rationals with countable closure (in the reals). The mistake is that ctbl isnot (K-equivalent to) the trace ideal tr(ctbl), corresponding to the Sacks forcing.

5. Katetov order on Borel ideals

Motivated by the results of the previous sections we now turn our attention tothe study of the Katetov order restricted to Borel ideals.

5.1. Katetov order is complex. Of course, there are only c-many Borelideals, so, for instance, the fact that there are decreasing chains of length c+ in theKatetov order no longer holds when restricted to Borel ideals. However, Katetovorder even when restricted to Borel ideals is complex.

Theorem 5.1 (D. Meza [78]). There is an order embedding of P(ω)/fin intoBorel ideals ordered by the Katetov order. In fact, there is such an embedding intosummable ideals, in particular, Fσ P-ideals.

Proof. Fix a partition of ω into finite intervals 〈In : n < ω〉 such thatmin(In+1) = max(In) + 1, and a sequence 〈rn : n < ω〉 of real numbers in (0, 1]such that:

(1) |In| · rn ≥ |⋃

j<n Ij | and(2) |In| · rn+1 ≤ 2−n−1.

For each infinite subset A of ω, define a function fA : ω → (0, 1] such that forevery k < ω

fA(k) =

{rn if k ∈ In and n /∈ A,

rn+1 if k ∈ In and n ∈ A.

Then:

• For every infinite and co-infinite subset A of ω, IfA is a non-trivial tallsummable ideal.

• Let A,B ∈ [ω]ω. Then A ⊆∗ B if and only if IfA ≤K IfB . �

Note that this, in particular, shows that there are antichains of size c andboth increasing and decreasing chains of length b in the Katetov order restricted toBorel ideals. We do not know whether there are in ZFC increasing and/or decreasingchains of length c in the Katetov order on Borel ideals.

Question 5.2. Are there consistently two (or more) distinct covering numbersof summable ideals?

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5.2. Looking for (locally) Katetov minimal Borel ideals. We do notknow the answer to the following basic question:

Question 5.3. Is there a tall Borel ideal Katetov-minimal among tall Borelideals?

This is, of course, equivalent to asking whether the Katetov order restricted totall Borel ideals is c-downwards closed. We conjecture that the answer is negative.However:

Proposition 5.4. There is a tall projective ideal Katetov below all tall Borelideals.

Proof. Let U ⊆ P(ω)× 2ω be a universal analytic set. Let Y = {x ∈ 2ω : Ux

is a tall Borel ideal}. Then, Y is projective and by changing U on the coordinatesoutside Y we get a projective V ⊆ P(ω)× 2ω such that

(1) for every x ∈ 2ω the set V x is a tall Borel ideal, and(2) for every tall Borel ideal I there is an x ∈ 2ω such that I = V x.

Having fixed such V one can define an ideal J on 2<ω generated by antichainsof 2<ω and sets of the form {x � n : n ∈ I}, where x ∈ 2ω and I ∈ V x.

The ideal J is then tall and projective and such that for any tall Borel idealI there is a J -positive set X such that I is isomorphic (hence K-equivalent) toJ � X. �

A reasonable weaker question is

Question 5.5. Is there a Borel tall ideal J such that for every Borel tall idealI there is an I-positive set X such that J ≤K I � X?

We call such an ideal J locally minimal. There is a natural candidate, the idealR introduced in section 3. Recall that for a Borel ideal I there is an I-positive setX such that R ≤K I � X, if and only if I+ �−→ (I+)22. We have been able to give apositive answer to the question in a restricted class of Borel ideals (containing allFσ ideals).

Theorem 5.6 ([78]). Let I be a tall Borel ideal on ω such that P(ω)/I isproper. Then there is an I-positive set X such that I � X ≥K R.

Let us remark that there are even Fσ ideals which are not Katetov above R, soR is not K-minimal. We do not even know whether Fσ ideals are co-initial amongtall Borel ideals:

Question 5.7. Does every tall Borel ideal contain a tall Fσ subideal?

5.3. Ramsey and related properties. We will take a closer look at theRamsey property I+ −→ (I+)22 and the related P+ and Q+ properties here. Thefollowing is a well known fact, essentially, a reformulation of the standard proof ofRamsey’s theorem.

Proposition 5.8. If I is an ideal which is both P+ and Q+ then I+ −→ (I+)22.

It is easy to see that if I+ −→ (I+)22 then I has to be a Q+-ideal. On theother hand, the P+-property is not indispensable.

Claim 5.9. There is a non-P+-ideal satisfying I+ −→ (I+)22.

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22 MICHAEL HRUSAK

Proof. Fix for every n < ω a MAD familyAn such thatAn+1 =⋃

A∈AnAA

n+1,

where every AAn+1 is a MAD family in P(A). Let

I =⋂

n<ω

I(An).

Then I+ −→ (I+)22, but I is not P+. �Also, there are no tall Borel ideals satisfying the conditions above.

Proposition 5.10. There are no tall analytic ideals which are both P+ andQ+.

Proof. Let I be an analytic ideal on ω, and suppose that I is a P+ andQ+-ideal. Then, by the P+ condition I+ is a σ-closed forcing, hence, it does notadd new reals. Let G be an I+-generic ultrafilter. Then, in V [G], G is a selectiveultrafilter and IV [G] = I is an analytic ideal disjoint from G, contradicting Mathias’theorem 4.3 (in V [G]). �

We have already seen that the ideal EDfin is critical with respect to the Q+-property: I is a Q+-ideal if and only if I � X �KB EDfin for all I-positive sets X,and for Borel ideals if and only if I � X is not ω-hitting for all I-positive sets X.

The P+-property is a lot more slippery. Let I be an ideal on ω. We will saythat I is decomposable if there is an infinite partition {Xn : n < ω} of ω intoI-positive sets such that for every X ⊆ ω

X ∈ I if and only if (∀n < ω)(X ∩Xn ∈ I).We will say that I is indecomposable if it is not decomposable.

Proposition 5.11. Let I be an ideal. Then I is a P+-ideal if and only if I isindecomposable and fin× fin �K I � X, for all X ∈ I+.

There is a close relationship between the P+ property and Fσ ideals. Just andKrawczyk [56] were probably the first to notice that every Fσ ideal is P+.

Theorem 5.12 ([78]). Let I be a Borel ideal on ω. Then the following condi-tions are equivalent

1. there is an Fσ ideal J containing I,2. there is a P+-ideal K containing I.

Proof. 1 → 2 follows trivially from the above observation. Let us prove 2 → 1.Let G be an K+-generic ultrafilter. Since K is P+, K+ is a σ-closed forcing and itdoes not add new reals, sequences of real numbers and Borel sets. Then, in V [G], Gis a P-point, and by theorem 4.4, there is an Fσ ideal J containing I and disjointfrom G. Since K+ does not add new real numbers, the ideal J was already inV . �

Fσ ideals can actually be combinatorially characterized among Borel ideals bya slight strengthening of the P+- property.

Definition 5.13 (Laflamme and Leary [67]). Let X be a set of infinite subsetsof ω. A tree T ⊆ ([ω]<ω)<ω is an X -tree of finite sets if for each s ∈ T there is anXs ∈ X such that s a ∈ T for each a ∈ [Xs]

<ω.An ideal I on ω is a P+(tree)-ideal if every I+-tree of finite sets has a branch

whose union is in I+.

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Laflamme and Leary [67] have proved that an ideal I is not P+(tree) if andonly if Player I has a winning strategy for the following game G: In step n, PlayerI chooses an I-positive set Xn and Player II chooses a finite set Fn ⊆ Xn. PlayerII wins if

⋃n<ω Fn ∈ I+.

Lemma 5.14. Let I be a Borel ideal. Then, Player II has a winning strategyin the game G if and only if I is an Fσ ideal.

So we have proved the following theorem.

Theorem 5.15 ([78]). Let I be a Borel ideal. Then I is a P+(tree)-ideal ifand only if I is an Fσ ideal. �

Question 5.16. Is it true that, if I is a Borel ideal then either I ≥K conv orthere is an Fσ ideal J containing I?

An approximation to this conjecture is the following result.

Theorem 5.17 ([78]). Let I be a Borel ideal such that the forcing quotientP(ω)/I is proper. Then, either there is an I-positive set X such that conv ≤K I �X or there is an Fσ ideal J containing I.

A similar problem is to characterize those Borel ideals that can be extended toan Fσδ ideal:

Question 5.18. Is it true that, if I is a Borel ideal then either I ≥K fin× finor there is an Fσδ ideal J containing I?

Of course, the main open problem remains whether R is locally minimal:

Question 5.19. Is there a tall Borel ideal I such that I+ −→ (I+)22?

More results on Ramsey type properties of definable and non-definable idealswill appear in [49]. R. Filipow, N. Mrozek, I. Rec�law and P. Szuca also studiedRamsey type properties and related convergence properties in [33, 32]. Many oftheir results can be readily reformulated as results about Katetov order.

5.4. Category dichotomy. In this section we will prove the following struc-tural theorem for Borel ideals.

Theorem 5.20 (Category Dichotomy [46, 78]). Let I be a Borel ideal. Theneither I ≤K nwd or there is an I-positive set X such that I � X ≥K ED.

Proof. The proof goes through a determinacy argument for the followinggame G(I) associated to an ideal I: In step k, Player I chooses an element Ikof I and then Player II chooses an element nk of ω not in Ik. Player I wins if{nk : k < ω} ∈ I.

If for every I-positive set X Player II has a winning strategy in the gameG(I � X) then I ≤K nwd.

On the other hand, if there is an I-positive set Y such that Player I has awinning strategy for G(I � Y ) then there is an I-positive set X ⊆ Y such thatI � X ≥K ED. �

One can easily turn the Category Dichotomy into a trichotomy: For every Borelideal I either I ≤K nwd or there is an I-positive set X such that I � X ≥K fin×finor there is an I-positive set X such that I � X ≥KB EDfin.

Recall that cov∗(nwd) = cov(M) and cov∗(ED) = non(M).

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24 MICHAEL HRUSAK

Corollary 5.21. Let I be a K-uniform Borel ideal. Then cov∗(I) ≥ cov(M)or cov∗(I) ≤ non(M).

This is a particular case of a heuristically confirmed “rule of thumb” that forany simply definable (Borel) cardinal invariant j either ZFC � j ≤ non(M) orZFC � j ≥ cov(M). The only standard cardinal invariant I know that does notsatisfy this is the groupwise density number g [11], which is, of course, not Borel.

Question 5.22. Is it true, that for any Borel cardinal invariant j either ZFC �j ≤ non(M) or ZFC � j ≥ cov(M)?

5.5. Measure dichotomy. In this section we present a dichotomy for ana-lytic P-ideals similar in form to the Category dichotomy. It is somewhat analogousto Christensen’s result [21] linking the Fubini property to non-pathologicity of sub-measures on atomless Boolean algebras.

Theorem 5.23 (Measure Dichotomy [46, 78]). Let I be an analytic P-ideal.Then, either I ≤K Z or there is X ∈ I+ such that S ≤K I � X.

Recall that a submeasure ϕ on a set X is non-pathological if for every A ⊆ X

ϕ(A) = ϕ(A) =def sup{μ(A) : μ is a measure on X dominated by ϕ}.

Following Farah [25] we say that an analytic P-ideal I on ω is non-pathological ifthere is a lscsm ϕ such that I = Exh(ϕ) = Exh(ϕ).

We define the degree of pathology of a submeasure ϕ on X such that ϕ(X) < ∞by

P (ϕ) =ϕ(X)

sup{μ(X) : μ is a measure dominated by ϕ} .

Kelley’s covering number [61] of a family of sets is defined as follows: Let Fbe a set and B ⊆ P(F ). For any finite sequence S = 〈S0, . . . Sn〉 of (not necessarilydistinct) elements of B let

m(S) = min {|{i ≤ n : x ∈ Si}| : x ∈ F} .

The covering number C(B) is defined as

C(B) = sup

{m(S)

|S| : S ∈ B<ω

}

.

The fundamental theorem of Kelley which links the covering number with sub-measures is the following. Recall, that a (sub)measure ϕ on a set X is normalizedif ϕ(X) = 1.

Theorem 5.24 (Kelley [61]). For each non-empty family B of P(F ) the cov-ering number C(B) is the minimum of the numbers sup{μ(A) : A ∈ B}, where theminimum is taken over all normalized measures μ on P(F ). �

Using Kelley’s theorem, one can deduce the following lemma, crucial in ourproof of the Measure dichotomy. It can be seen as a finite (atomic) version ofa theorem of Christensen [21] who showed that a submeasure ϕ on an atomlessBoolean algebra is pathological if and only if the Fubini theorem for ϕ fails.

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Lemma 5.25 (Quantitative version of Christensen’s lemma). Let F be a finiteset, ε > 0, ϕ a normalized submeasure on P(F ) and Aε = {A ⊆ F : ϕ(A) < ε}.Then

C(Aε) ≥ 1− 1

εP (ϕ).

One should note that, in this context, the Kelley’s covering number “measures”the failure of the Fubini theorem: C(Aε) > δ if and only if there is an N < ω andthere is a set A ⊆ F ×N such that all horizontal sections of A have submeasure < εwhile all vertical sections have normalized counting measure > δ. It is easily seen,that the finite N can be replaced by the Cantor set and the counting measure byLebesgue (Haar) measure. Interpreted in this way, the lemma says that “the morepathological is the submeasure, the worse the Fubini theorem for ϕ fails”.

Corollary 5.26. If I is an analytic P-ideal then the following conditions areequivalent:

(a) I � X ≤K Z for every I-positive set X,(b) S �≤K I � X, for every I-positive set X,(c) I has the Fubini property and(d) I is non-pathological.

Recall that cov∗(S) = non(N ) and cov∗(Z) is a close relative of cov(N ). Notealso that the Measure dichotomy does not hold for all Borel ideals (for instancefin× fin is a counterexample). We do not know whether it holds for Fσ ideals. It iseven conceivable, though unlikely, that the measure dichotomy could be extendedto a trichotomy for all Borel ideals as follows: Let I be a Borel ideal. Then, eitherI ≤K Z or there is X ∈ I+ such that S ≤K I � X or there is X ∈ I+ such thatfin× fin ≤K I � X.

6. Tukey order

The Tukey order on directed partial orders (i.e. partially ordered sets whereany two elements have a common upper bound) was introduced by J. W. Tukey[103] in order to study the Moore-Smith convergence in topology.

Given two directed partial orders P and Q a function f : P −→ Q is a Tukeymap (or Tukey reduction) if f maps unbounded sets to unbounded sets or, equiva-lently, if pre-images of bounded sets are bounded. We say that P is Tukey reducibleto Q (P ≤T Q) if there is a Tukey map f : P −→ Q. The existence of a Tukey mapfrom P to Q is equivalent to the existence of a convergent map from Q to P , i.e. amap sending cofinal subsets to cofinal subsets.

Two partially ordered sets P,Q are cofinally similar or of the same cofinaltype if there is another partially ordered set such that both P and Q are cofinalsubsets of it. Tukey noticed that directed partial orders are of the same cofinaltype if and only if they are Tukey-equivalent (Tukey-bi-reducible). The Tukeyorder was further studied by J. Isbell in a series of papers [38, 53, 54, 55], wherebasic concepts, such as the notion of a weakly bounded set, were introduced andfundamental open problems were raised. Answers to many of these were providedby Todorcevic in [98, 101]. In particular he showed that:

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26 MICHAEL HRUSAK

Theorem 6.1 (Todorcevic [98]). Assuming the Proper forcing axiom PFA, anydirected set of size ℵ1 is cofinally similar to one of the following: 1, ω, ω1, ω × ω1

and [ω1]<ω.

On the other hand, he has also shown that there are 2ℵ1 many distinct cofinaltypes of directed orders of size c. Therefore, an attempt to classify cofinal typesof directed orders of size c is bound to fail. There could be a better chance for aclassification for definable structures.

The interest in Tukey order restricted to definable partial orders was at leastpartially motivated by the Bartoszynski-Raisonnier-Stern [4, 5, 83] results aboutcardinal invariants of measure and category, which can be concisely expressed inthe language of Tukey order:

Theorem 6.2 ([5]). M ≤T N .

Another reason for such study was the fact that many ideals and directed setsnaturally arising in analysis are simply definable. The study of the Tukey order ondefinable directed sets was initiated by D. Fremlin [36, 37], S. Todorcevic [101]and P. Vojtas [107].

6.1. Tukey order on analytic ideals. Cofinal types of Borel (or analytic)ideals were mentioned already by Isbell in [55], where he considered the densityzero ideal Z. Also, Fremlin [36, 37] and Todorcevic [101] dealt with some idealson a countable set, though their focus was on σ-ideals of Borel subsets of Polishspaces and structures of size ℵ1, respectively.

The first papers dedicated to the study of analytic ideals on ω ([99, 102] and[70]) provided the fundamental structural theorems for the Tukey order on definableideals.

Let I be an ideal. A subset X ⊆ I is weakly bounded8 if every infinite sequenceof elements of X has a bounded subsequence (i.e. a subsequence whose union is inI). Dually, a subset X ⊆ I is strongly unbounded if no union of infinitely manymembers of X is in I. Note that if X ⊆ I is weakly bounded then its closure X iscontained in I.

Note that any two ideals having a strongly unbounded set of size c are Tukey-equivalent and are Tukey-above any ideal on ω. The strongest property in this senseis for an ideal to have a perfect strongly unbounded subset. It was conjectured byLouveau and Velickovic in [70] that any analytic ideal which has an uncountablestrongly unbounded set should have a perfect one. This conjecture has recentlybeen refuted by T. Matrai [74], who showed that there is (in ZFC) an analytic idealwhich has a strongly unbounded subset of size ℵ1 but not a strongly unboundedsubset of size ℵ2 and, in particular, does not have a perfect strongly unboundedsubset.

The following fundamental theorem shows that the ideal ∅ × fin is the leastanalytic ideal which is not Fσ.

Theorem 6.3 (Louveau-Velickovic [70]). Let I be an analytic ideal. Theneither ∅ × fin ≤T I or I is Fσ.

8Pseudobounded according to Isbell [55].

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Proof. Consider the following two player game: In the n-th inning of the gameplayer I plays a weakly bounded set Xn ⊆ I and player II responds by playing ana finite subset of ω not in Xn. Player II wins if

⋃n∈ω an ∈ I.

The existence of a winning strategy for player I implies that I can be coveredby the closures of countably many weakly bounded sets, hence is Fσ.

If, on the other hand, player II has a winning strategy, then there is a treeT ⊆ ([ω]<ω)<ω such that

(1) for every t ∈ T the set of a ∈ [ω]<ω t�a ∈ T is strongly unbounded and(2)

⋃rng(f) ∈ I for every branch f ∈ [T ].

This readily implies that ωω �T ∅ × fin ≤T I.The game as stated is easily seen to be determined for Borel ideals and a simple

modification turns it into a closed game for any analytic ideal. �

Theorem 6.4 (Louveau-Velickovic [70]). Let I be an analytic ideal such thatI ≤T ∅ × fin. Then either ∅ × fin �T I or I is countably generated.

The focus of both the Louveau-Velickovic and Todorcevic papers was on ana-lytic P-ideals. We state the results and sketch alternative proofs based on Solecki’stheorem 1.6. Let us first make the following simple observation:

Claim 6.5. Let I = Exh(ϕ) be an uncountably generated analytic P-ideal.Then there is a pairwise disjoint family {am,n : m,n ∈ ω} of finite subsets of ωsuch that 2−n−1 ≤ ϕ(am,n) ≤ 2−n.

Theorem 6.6 (Todorcevic [99]). Let I be an analytic P-ideal. Then either Iis countably generated or ∅ × fin ≤T I.

Proof. Let I = Exh(ϕ) and {am,n : m,n ∈ ω} be as in the claim. The ideal∅ × fin is naturally Tukey equivalent with ωω ordered pointwise. For every f ∈ ωω

let ψ(f) =⋃

n∈ω af(n),n. Then ψ is the required Tukey map. This follows directlyfrom the fact that for a fixed n the set {am,n : m ∈ ω} is strongly unbounded.

�

Theorem 6.7 (Todorcevic [102]). Let I be an analytic P-ideal. Then I ≤T I 1n.

Proof. Let I = Exh(ϕ). Fix {ab,n : b ∈ [ω]<ω, n ∈ ω} a pairwise disjointfamily of finite subsets of ω such that 2−n−1 ≤

∑j∈ab,n

1j ≤ 2−n. For every I ∈ I

let

fI(n) = min{k : ϕ(I \ k) ≤ 1

2n}.

By exhaustivity, fI ∈ ωω is well defined. Let ψ(I) =⋃

n∈ω aI∩fI(n),n. Then ψ isthe required Tukey map. �

This result could also be attributed to Louveau and Velickovic, as they werethe first to notice that there is a top ideal among analytic P-ideals in the Tukeyorder. Louveau and Velickovic in their article also showed that the Tukey order onanalytic P-ideals is complex.

Theorem 6.8 (Louveau-Velickovic [70]). There is an embedding of P(ω)/fininto analytic P-ideals ordered by the Tukey order.

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28 MICHAEL HRUSAK

They also pointed out that there are, indeed, uncountably generated Fσ idealswhich are not above ∅ × fin in the Tukey order. They introduced the ideal ofpolynomial growth

P = {A ⊆ ω : (∃k ∈ ω)(∀n ∈ ω) |A ∩ 2n| ≤ nk}and showed that it is σ-weakly bounded, hence not Tukey-above ∅ × fin.

Inspired by this example, with Zapletal and Rojas [51] we have introduced andstudied a class of Fσ ideals disjoint from the class of analytic P-ideals.

If I is an Fσ ideal and ϕ is a lscsm such that I = Fin(ϕ), denote by Iϕk (or

simply Ik) the set {A ∈ I : ϕ(A) ≤ k}. Clearly, for all k ∈ ω, Iϕk is closed, and

I =⋃

k Iϕk .

Definition 6.9 ([51]). Let I be an Fσ ideal on ω. The ideal I is said to befragmented if there are a lscsm ϕ and a partition {ai : i ∈ ω} of ω into finite sets,such that for every k ∈ ω,

Iϕk = {A ∈ I : (∀i ∈ ω)(ϕ(A ∩ ai) ≤ k)}.

The ideal P of polynomial growth is an example of a fragmented σ-weaklybounded ideal (under ϕ(A) = supn∈ω{min{k : |A ∩ [2n, 2n+1)| ≤ nk}]). Otherexamples of fragmented ideals are: EDfin and the ideal L = {A ⊆ ω : (∃k ∈ω)(∀n ∈ ω) |A ∩ 2n| ≤ n · k} of linear growth. The last two ideals both have aperfect strongly unbounded subset as opposed to the first example, which is σ-weakly bounded. The essential difference between the first and the last two idealsis that the growth of the submeasure in the fragments of the first can be controlledin the following sense:

Definition 6.10 ([51]). An ideal I is gradually fragmented if it is fragmented(via ϕ) and, moreover,

∀k∃m∀l∀∞j(∀B ∈[P(aj)∩ Iϕ

k

]l)(∪B ∈ Iϕ

m)

There is a dichotomy for fragmented Fσ ideals.

Theorem 6.11 ([51]). Let I = Fin(ϕ) be a fragmented ideal. Then:

• Either I is gradually fragmented, or• I contains a perfect strongly unbounded subset.

K. Mazur in [77] has (essentially) shown that also the Tukey order on graduallyfragmented Fσ-ideals is complex:

Theorem 6.12 (Mazur [77]). There is an order embedding of P(ω)/fin intogradually fragmented Fσ-ideals ordered by the Tukey order.

Solecki and Todorcevic in [91] showed that every analytic ideal has a cofinalGδ set, improving on a theorem of Zafrany [108]. They also showed that amonganalytic basic orders (a class of orders that includes all analytic P-ideals and relativeσ-ideals of compact sets) the Tukey order reduces to the existence of a continuouscofinal map, hence is absolute. There is a large body of work on (relative) σ-idealsof compact sets and Tukey order which we do not include, and we refer the reader to[36, 60, 71, 91]. We only mention recent results of T. Matrai [74, 75] who showedthat the Tukey order on relative σ-ideals of compact sets is also complex (embedsP(ω)/fin) and that the ideal nwd is not an upper bound for relative σ-ideals ofcompact sets.

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6.2. Cofinalities of Borel ideals. There are very few “standard” cofinalitiesof Borel ideals:

• ω . . . the cofinality of fin,• d . . . the cofinality of ∅ × fin and fin× fin,• cof(M) . . . the cofinality of nwd,• cof(N ) . . . the cofinality of Z and I 1

n.

• c . . . the cofinality of any ideal having a perfect strongly unbounded set.

Proposition 6.13. Let I be an uncountably generated analytic ideal. Thencof(I) ≥ cov(M).

Given that there does not seem to be a definable ideal of cof(I) = cov(M)and in light of the Louveau-Velickovic theorem 6.3, one has to wonder: Is d alower bound for cofinalities of all uncountably generated Borel ideals? Are thereonly finitely many distinct cofinalities of Borel ideals? It turns out that the answerto both questions is in the negative.

A quick glance at the definition of gradually fragmented ideals reveals thattheir cofinalities are preserved by any proper forcing having the Laver property (see[5]) and consequently:

Theorem 6.14 ([51]). It is consistent that b = ω2 and cof(I) = ω1 for allgradually fragmented ideals I.

There is a natural forcing associated to every Borel ideal I, which adds a newelement of I not contained in any ground model set in I defined as follows: Let Ibe a Borel ideal. Let J be the σ-ideal on I generated by the family {P(I) : I ∈ I}.Denote by PI the forcing Borel(I)/J .

We say that a forcing notion P adds an unbounded element of a Borel ideal Iif there is a P-name τ such that �P “τ ∈ I and τ �⊆ I for any ground model I ∈ I”.

General theorems of Zapletal [109] and simple genericity arguments give:

Proposition 6.15. Let I be a Borel ideal and let PI be the correspondingforcing. Then:

• PI is proper.• PI preserves non(M).• PI preserves cof(M), provided that I is Fσ.• PI adds an unbounded element of I.

A simple consequence is:

Theorem 6.16 ([51]). It is consistent that cof(M) = ω1 and cof(I) = ω2 forall uncountably generated Fσ ideals I.

Using a countable support product of the forcings of type PI for a carefullychosen family of gradually fragmented Fσ ideals we were able to prove:

Theorem 6.17 ([51]). It is consistent with ZFC that there are uncountablymany pairwise distinct cofinalities of gradually fragmented Fσ ideals.

It would be interesting to know how combinatorial properties of an ideal Iimpact preservation properties of the corresponding forcing PI . When does PIpreserve cof(M)? cof(N )? outer measure?

There is a close relationship between Borel ideals I and the correspondingforcing notions PI and a variant of the Tukey order, considered e.g. by Fremlin

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30 MICHAEL HRUSAK

in [37]: Given two ideals I and I ′ we say that I is ω-Tukey reducible to I ′

(I ≤ωT I ′) if there is a function f : I → I ′ such that the pre-images of boundedsets are σ-bounded. It can be easily seen that the forcing PI adds an unboundedelement of an ideal I if and only if I ≤ωT I ′, where the witnessing function f issufficiently definable (piece-wise Borel in the following sense: Let J be the σ-idealcorresponding to I. For every J -positive set B there is an J -positive set B ⊆ Csuch that f � C is Borel.)

There is a natural (trivial) characterization of the situation when a properforcing of the type PJ (= Borel(X)/J) adds an unbounded element of a Borel idealI. It is if and only if there is a Borel function f : X → I such that f−1[P(I)] ∈ Jfor any I ∈ I. This seems to be particularly interesting for the Sacks and Millerforcing, i.e. for J being the σ-ideal of countable, or σ-compact sets.

Proposition 6.18. Let I be a Borel ideal. Then:

• The Sacks forcing adds an unbounded element of I if and only if there isa perfect set P ⊆ I such that any element of I contains only countablymany elements of P (i.e. I �ωT [2ω]<ω).

• The Miller forcing adds an unbounded element of I if and only if ∅ ×fin ≤ωT I with Borel witnessing map.

This simple observation raises the following natural questions, the second ofwhich was asked in a stronger form (for Tukey order) in [70]:

Question 6.19. Let I be a Borel ideal. Is it true that either I ≤ωT I 1n

or

I �ωT [2ω]<ω?

Question 6.20. Let I be a Borel ideal. Is it true that either ∅ × fin ≤ωT I orI is σ-weakly bounded?

It is also not clear whether every σ-weakly bounded ideal has cofinality consis-tently strictly below d. In fact, there does not seem to be any known example of aσ-weakly bounded ideal which is not gradually fragmented.

Another natural question concerns additivities. While we have shown thatthere are consistently many distinct cofinalities of Borel ideals, there are still onlythree (ω, add(N ) and b) known distinct additivities of Borel (analytic) ideals.

Question 6.21. Is the additivity of every analytic P-ideal equal to either add(N )or b?

Dual question is also open for cofinalities of analytic P-ideals. In particular:

Question 6.22 (Solecki-Todorcevic [91]). Are there two Tukey non-equivalentFσ P-ideals?

There are several non-reducibility results. Some of them can be deduced fromthe proofs of the consistency results about distinct cofinalities, e.g. there is nouncountably generated Fσ ideal Tukey reducible to nwd and Z �≤T nwd [37]. Otherresults are more involved, e.g. I 1

n�≤T Z [70] and some of them still open:

Question 6.23 (Fremlin [37]). Is nwd �≤T Z?9

9This has been recently solved by T. Matrai [73], and S. Solecki and S. Todorcevic [90],independently.

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6.3. Tukey order on ultrafilters. The Tukey order is also interesting whenrestricted to maximal ideals (or ultrafilters ordered by reverse inclusion). Isbell in[53] proved that there is an ultrafilter U of the maximal cofinal type, i.e. U �T

[2ω]<ω and asked:

Question 6.24 (Isbell [53]). How many cofinal types of ultrafilters are there?

In fact, it is still an open question, whether in ZFC there is a free ultrafilter ofa cofinal type different from [2ω]<ω. There are, of course many consistency results.Any ultrafilter of character less than c is an example, as is any P-point. D. Milovichin [79] has shown that there is consistently a non-P-point U such that U <T [2ω]<ω.Further research in the area is being done by Dobrinen and Todorcevic [23].

The study of cofinal types of ultrafilters is clearly related to the following clas-sical problem known as the Katowice problem [81]:

Question 6.25. Can the Cech-Stone remainders ω∗ and ω∗1 be consistently

homeomorphic?

The question is, via Stone duality, equivalent to the question of whether theBoolean algebras P(ω)/fin and P(ω1)/fin can be consistently isomorphic. It is,moreover, equivalent to the question whether a free (or, equivalently, any free)ultrafilter on ω can be (as partially ordered set) isomorphic to a free ultrafilter ofω1. In particular, should the answer be positive, the cofinal types of ultrafilters onω and the cofinal types of ultrafilters on ω1 would have to coincide.

Question 6.26. What are the cofinal types of ultrafilters on ω1?

7. Comparison game

In this section we propose a “rough” classification of Borel ideals based on asimple two player game. The game induces an order which is coarser than theRudin-Keisler order, in fact coarser than the “monotone Tukey order”. We hopethat the order could provide insight into the structure of Borel ideals of low com-plexity, in particular, into the internal structure of Fσδ ideals.

Definition 7.1 ([47]). Let I and J be ideals on ω. The Comparison Gamefor I and J denoted by G(I,J ) is defined as follows: In step n, Player I choosesan element In of I and Player II chooses an element Jn of J . Player II wins if⋃

n In ∈ I if and only if⋃

n Jn ∈ J ; otherwise, Player I wins.

The comparison game defines an order on ideals on ω.

Definition 7.2. Let I and J be ideals on ω. We say I � J if Player II hasa winning strategy in the comparison game G(I,J ). We say that I � J if I � Jand J � I.

Note that the relation � is reflexive and transitive, but not antisymmetric; andthe relation � is an equivalence relation.

It is easy to see that the comparison game on Borel ideals is determined, infact, it naturally reduces to Wadge degrees. Putting

X = {x ∈ ωω : rng(x) ∈ X}

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32 MICHAEL HRUSAK

for a subset X of P(ω), one readily sees that

I � J if and only if I ≤Wadge J .

Just like Wadge degrees, the comparison game order on Borel ideals is well-founded and “almost” linear.

Proposition 7.3. (1) If I, J and K are Borel ideals, I �� J and J �� Kthen K � I.

(2) Let I and J be two �-incomparable ideals. Then, for any ideal K on ω,K � I iff K � J or I � K iff J � K. �

We do not know whether � is linear. We also do not know whether it respectsBorel complexity. What we do know is that it “almost” respects it.

Proposition 7.4. If I is a Σ0α or Π0

α ideal then I is a Σ0α+1 or Π0

α+1 set,respectively.

Note also that if I ≤RK J (in fact, the existence of a monotone Tukey-mapf : I −→ J suffices) then I � J .

Question 7.5. Is the order � linear (a well order)?

Next we state some basic facts about the low levels of the order. Given f ∈ 2ω,denote by Af = {f � n : n < ω} the branch of the tree 2<ω corresponding to f .The ideal I0 is the ideal on 2<ω generated by the family of sets Af where f ∈ 2ω

is not eventually zero.

Theorem 7.6 ([47]). Let I be an ideal on ω. Then:

• fin � I,• I � fin if and only if I is Fσ,• I is not an Fσ ideal if and only if I0 � I,• ∅ × fin �� I0.

Both the ideal I0 and ∅ × fin are Fσδ, so unlike in the case of Fσ ideals, thereare at least two classes of Fσδ ideals.

Farah in [25] asked whether every Fσδ ideal I is of the following canonical form:There is a family of compact hereditary sets {Cn : n < ω} such that

I = {A ⊆ ω : (∀n < ω)(∃m < ω)(A \ [0,m) ∈ Cn)}.

We will say that I is a Farah ideal if it is of this form. Obviously, every Farah idealI is an Fσδ ideal. One can easily see that an ideal I is Farah if and only if thereis a sequence {Fn : n < ω} of hereditary Fσ sets closed under finite changes suchthat I =

⋂n Fn. With some extra work, one can show that:

Theorem 7.7 ([47]). Let I be an ideal on ω. Then, I is Farah if and only ifthere is a sequence {Fn : n < ω} of Fσ sets closed under finite changes such thatI =

⋂n Fn.

We call an Fσδ ideal I weakly Farah if there is a sequence 〈Fn : n < ω〉 ofhereditary Fσ sets such that I =

⋂n Fn.

Proposition 7.8 ([47]). If I is a weakly Farah ideal then I � ∅ × fin.

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Every analytic P-ideal is either equivalent with fin (if and only if it is Fσ) orequivalent with ∅× fin, so the class of analytic P-ideals skips the intermediate classof I0. Most known Fσδ ideals are equivalent with ∅×fin, e.g. Z, nwd. On the otherhand, the Fσδσ ideal fin× fin is strictly above ∅× fin. We conclude with three moreopen problems:

Question 7.9. (1) Is every Fσδ ideal (weakly) Farah?(2) Are there exactly two classes of Fσδ ideals?(3) How many classes of Fσδσ ideals are there?

8. Quotient algebras P(ω)/IWe now turn our attention to the study of the quotient Boolean algebras of the

form P(ω)/I for definable ideals I.

8.1. Rigidity phenomena. The starting point of any considerations in thisarea has to be the celebrated result of S. Shelah:

Theorem 8.1 (Shelah [84]). It is consistent that all automorphisms of P(ω)/finare trivial.

An automorphism is trivial if it is induced by an almost permutation of ω, i.e.a bijection between two co-finite subsets of ω. Shelah’s original argument used theoracle c.c. method. Later it was shown by Shelah and Steprans [85] that the resultis true assuming PFA. A careful analysis by Velickovic [106, 105] revealed thatthe proof can be naturally split into two somewhat independent parts concerningliftings of homomorphisms between quotients.

Definition 8.2. Let Φ : P(ω)/I → P(ω)/J be a homomorphism. A functionϕ : P(ω) → P(ω) is a lifting of Φ if [ϕ(X)]J = Φ([X]I) for every X ⊆ ω.

Note that the function ϕ is not required to be a homomorphism.

The two parts of the proof are:

(1) using forcing or some strong axiom (PFA, OCA, . . . ) show that every au-tomorphism has a nicely definable (continuous, Baire-measurable) lifting,and

(2) an automorphism which has a definable lifting is trivial (i.e. has a com-pletely additive lifting, see the definition below).

The rigidity conjectures of Farah and Todorcevic, roughly speaking, assert thatthe same phenomenon occurs for any homomorphism (isomorphism) between quo-tients by definable ideals. It was quickly noticed that any homomorphism thathas a Baire-measurable or a Lebesgue-measurable lifting has, in fact, a continuouslifting [105, 100, 58].

Definition 8.3 (Farah [25]). An ideal I has the Radon-Nikodym (RN) prop-erty if every homomorphism Φ : P(ω)/fin → P(ω)/I with a continuous lifting hasa completely additive lifting.

By a completely additive lifting we mean a lifting of the form ϕ(A) = h−1[A] forsome function h from ω to ω. Todorcevic in [100] conjectured that every analyticP-ideal has the RN property. This has been partially confirmed by Farah

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34 MICHAEL HRUSAK

Theorem 8.4 (Farah [25]). Every non-pathological analytic P-ideal has theRadon-Nikodym property.

and extended by Kanovei and Reeken

Theorem 8.5 (Kanovei-Reeken [57, 58]). Every analytic ideal having the Fu-bini property has the Radon-Nikodym property.

However, it turned out that not all analytic P-ideals are RN [25], so someconstraint is necessary for the positive answer to Todorcevic’s conjecture. Also notall RN ideals are Fubini (nwd is a counterexample [28]). It is still an open problemto find a combinatorial characterization of RN ideals. This is open even for analyticP-ideals:

Question 8.6 (Farah [25]). Is there a pathological analytic P-ideal with theRN property?

Not surprisingly, Farah’s Rigidity conjecture has two components:

(1) asks whether two quotients over Borel ideals are Baire-isomorphic (i.e.there is an isomorphism with a Baire-measurable lifting) if and only if theideals are Rudin-Keisler equivalent,

(2) asks whether assuming PFA (or Martin’s Maximum,. . . ) every isomor-phism between quotients over Borel ideals has a Baire-measurable lifting.

This has also been partially confirmed. Recall the definition of a Farah ideal fromsection 7.

Theorem 8.7 (Farah [28]). Assume PFA. If I and J are analytic ideals andat least one of them is Farah then every isomorphism between their quotients has acontinuous lifting.

We refer the interested reader to [25, 28, 29, 57, 58, 100, 102] for moreinformation on this deep subject.

8.2. Gap spectra of analytic quotients. A development parallel to thestudy of the rigidity phenomena was the study of gap spectra of analytic quotients.Given an ideal I, we call two families A,B of subsets of ω I-orthogonal if A∩B ∈ Ifor every A ∈ A and B ∈ B. Two I-orthogonal families A,B form a gap if there isno C ⊆ ω such that A \ C ∈ I for all A ∈ A and C ∩ B ∈ I for all B ∈ B. A gap(A,B) is Hausdorff if both A and B are σ-directed under inclusion mod I.

Todorcevic in [100] showed that any Baire-embedding of P(ω)/fin into an ana-lytic quotient preserves Hausdorff gaps and that any Baire-embedding of P(ω)/fininto a quotient by an analytic P-ideal preserves all gaps. In particular, he showedthat

Theorem 8.8 (Todorcevic [100]). Let I be an analytic ideal. Then:

(1) P(ω)/I contains an (ω1, ω1)-gap.(2) If, moreover, I is a P-ideal then P(ω)/I contains both an (ω1, ω1)-gap

and an (ω, b)-gap.

and asked to: Determine the gap spectrum of P(ω)/I for every analytic ideal I onω.

No Hausdorff gap in P(ω)/fin is analytic [99]. Therefore, it came as a surprisethat

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Theorem 8.9 (Farah [26]). There is an analytic Hausdorff gap in any quotientover an uncountably generated Fσ P-ideal.

Another surprising fact was proved recently by J. Brendle [13]:

Theorem 8.10 (Brendle [13]). There is an (ω1, ω)-gap in P(ω)/EDfin.

8.3. How many quotients are there? The title of this subsection is bor-rowed from [27]. Let us start with a problem from [25].

Question 8.11 (Farah [25]). Are there infinitely (even uncountably) manyanalytic P-ideals whose quotients are provably in ZFC pairwise non-isomorphic?

Farah also asked the same question for arbitrary analytic and even definableideals. He has also shown that assuming CH many classes of distinct ideals haveisomorphic quotients [27]. The first general theorem of this kind was proved (usingParovicenko’s theorem) by Just and Krawczyk:

Theorem 8.12 (Just and Krawczyk [56]). Assuming CH all quotients over Fσ

ideals are pairwise isomorphic.

On the other hand, Steprans in [93] produced uncountably many pairwise forc-ing non-equivalent (not just non-isomorphic) quotients over co-analytic ideals. Inretrospect, his ideals are trace ideals, so his construction could be seen as a partic-ular case of our theorem 2.5. The question was later answered completely by M.Oliver [82], who showed that:

Theorem 8.13 (Oliver [82]). There are c-many pairwise non-isomorphic quo-tients over analytic P-ideals.

However, his method does not seem to produce quotients which are distinct asforcing notions. So we propose to reformulate the original question:

Question 8.14. Are there infinitely (uncountably) many analytic (P-)idealswhose quotients are not forcing equivalent?

As of now, very few algebras of the form P(ω)/I for analytic ideal I are suffi-ciently well understood as forcing notions:

• (Farah [30]) P(ω)/Z is equivalent to the iteration P(ω)/fin∗B(2ω), whereB(2ω) denotes the measure algebra for adding c- many random reals.

• ([45]) P(ω)/tr(N ) is proper and equivalent to the iteration B(ω) ∗ Q,where Q does not add reals.

• ([45]) There is an analytic P-ideal whose quotient is not proper.

So there are at least four forcing non-equivalent quotients over analytic P-ideals,the fourth being any quotient over an Fσ P-ideal. There is also P(Q)/nwd, which isan iteration of Cohen forcing and a forcing not adding reals. The analysis of traceideals gives one more candidate (or class of candidates). Let us consider again theMathias-Prikry forcing MJ associated to an ideal J .

Proposition 8.15 ([45]). Let J be an ideal on ω. The forcing MJ has thecontinuous reading of names if and only if J is a P-ideal.

Proposition 8.16 ([45]). Let J be an analytic P-ideal, and let I be the σ-idealassociated with the Mathias-Prikry forcing MJ . Then the following are equivalent:

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36 MICHAEL HRUSAK

(1) J is Fσ,(2) the ideal tr(I) is Borel.

So, by theorem 2.5 the quotient P(ω<ω)/tr(I) is a proper forcing adding a realfor every Fσ P-ideal J , where I is the natural σ-ideal associated with the forcingMJ

Question 8.17. Are there uncountably many pairwise forcing non-equivalentquotients over the trace ideals corresponding to the Mathias-Prikry forcing with Fσ

P-ideals?

Various of our theorems about the Katetov order on Borel ideals required thatthe corresponding quotient be proper. We still do not know whether this assumptionis essential in any of those results. However, it would be useful to have a betterunderstanding of properness in this context. We also have only one example of anon-proper quotient and that example is over an analytic P-ideal. We need morenon-proper Borel quotients against which we could test our conjectures.

8.4. Cardinal invariants of P(ω)/I. Another useful way of studying thequotient algebra P(ω)/I is via its cardinal characteristics.

Recall some of the standard cardinal invariants corresponding to the algebraP(ω)/fin which form the so called van Douwen’s diagram (see [104, 11] for defini-tions and more information).

We denote by pI , tI , hI , rI , sI their direct analogues for the case of the algebraP(ω)/I.

These cardinal invariants have so far been calculated only for a very short listof ideals. Typically, for Borel quotients that add a new real, the name for the realessentially witnesses sI = ℵ0. This is the case for the ideals nwd, Z and tr(N ), soin these cases, the only one of the cardinal invariants introduced of interest is thereaping number.

r d

b

��

��s

��

h

��

��

t

��

p

��

ℵ1

��

van Douwen’s Diagram

6464


Proposition 8.18. (1) (Steprans [94]) rZ = c.(2) ([1]) max{r, cof(M)} ≤ rnwd ≤ i.

Recall that i is the minimal cardinality of a maximal independent family.

The situation is more interesting for ideals such that their quotients are σ-closed, in particular for Fσ ideals. The quotient over the ideal fin×fin was consideredby Brendle, Szymanski and Zhou, and Hernandez [13, 41, 97]:

Proposition 8.19. (1) (Szymanski and Zhou [97]) tfin×fin = ω1.(2) (Brendle [13]) sfin×fin = s.(3) (Hernandez [41]) It is consistent with ZFC that hfin×fin < h.

There are also some results concerning aI , the minimal size of an uncountablemaximal antichain in P(ω)/I. Note that uncountable is important; for many quo-tients there are countably infinite maximal antichains. It is well known that b ≤ a.This has been extended to the quotients over Fσ P-ideals by Farkas and Soukup[31]. However, it is not true in general, not even for Fσ ideals:

Proposition 8.20. (1) (Farkas and Soukup [31]) b ≤ aI for all Fσ P-ideals I.

(2) (Steprans [96]) It is consistent with ZFC that anwd < b.(3) (Brendle [13]) It is consistent with ZFC that aEDfin

< b.

Of course, any two Boolean algebras with distinct cardinal characteristics arenon-isomorphic. Since we are interested mostly in forcing properties of the quo-tients P(ω)/I, the most interesting of the cardinal invariants introduced is thedistributivity number hI . It is particularly interesting for quotients over Fσ ideals.While all quotients over Fσ ideals are isomorphic under CH there seems to be astrong rigidity phenomenon of consistency results:

Question 8.21. Let I and J be Fσ ideals and suppose that there is no regularembedding of P(ω)/I into P(ω)/J with a completely additive lifting. Is it thenconsistent that hI < hJ ?

If not, is the following true?

Question 8.22. Are there infinitely (uncountably many) Fσ ideals which haveconsistently pairwise different distributivity numbers?

It is easy to see that P(ω)/fin regularly embeds into P(ω)/I and, hence, hI ≤ h

for any fragmented Fσ ideal. On the other hand Brendle [13] announced

Theorem 8.23 (Brendle [13]). hEDfin< h is consistent.

and asked

Question 8.24 (Brendle [13]). Are hI < h and h < hI consistent for anysummable ideal I?

It is also not clear whether all summable ideals have the same distributivitynumber.

A particular instance of a theorem of Balcar, Simon and Pelant [2, 3] showsthat any quotient over an Fσ ideal has a base tree of height hI . It follows that ifthe forcing P(ω)/I is homogeneous in density then it collapses c to hI . Where doesthe forcing P(ω)/I collapse c for any Borel ideal I? In particular:

6565

38 MICHAEL HRUSAK

Question 8.25. (1) Where does P(Q)/nwd collapse c?(2) Where does P(ω)/tr(N ) collapse c?(3) Does every non-proper Borel quotient collapse c to ω? More precisely, is

there an I-positive set X such that P(X)/I � X collapses c to ω?

There is very little known about the pI and tI .

Question 8.26. Is there a Borel ideal I such that pI < tI is consistent? IspI = p (tI = t) for every Fσ ideal I?

One last question is rather ad hoc. The cardinal characteristic h is equal tothe minimal size of a family of tall ideals whose intersection is not tall. One cananalogously define hanalytic, hBorel,. . . , hFσ

as the minimal size of a family of tallanalytic (Borel, . . . , Fσ) ideals whose intersection is not tall. Obviously,

h ≤ hanalytic ≤ hBorel ≤ · · · ≤ hFσ≤ min{b, s}.

Question 8.27. Which of the above inequalities are consistently strict?

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Michoacan, Mexico.


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More structural consequences of AD

Richard Ketchersid

Abstract. Woodin and Steel showed that under AD + DCR the Suslin car-dinals are closed below their supremum; Woodin devised an argument basedon the notion of strong ∞-Borel code which is presented here. A consequenceof the closure of the Suslin cardinals below their supremum is that the Suslincardinals and the reliable cardinals coincide, the proof of this fact is also in-cluded.

Woodin’s argument yields that AD+ implies that the Suslin cardinals areclosed below Θ. It turns out that this characterizes AD+. We include a sketchof this argument as well.

Contents

1. Introduction2. Preliminaries3. Cone ultrafilters and ultraproducts.4. Compatibility of constructibility degrees. 805. Constructibility degrees and the structure of AD models 846. Strong ∞-Borel codes 957. Equivalence of AD+ with the closure of the Suslin cardinals below Θ8. Appendix 99References

1. Introduction

This is an expository paper based on notes from the set theory seminar at UCBerkeley in Fall 1994 and Spring 1995, personal communications with Woodin, andhandwritten notes from a seminar on AD+ at UCLA given by John Steel. Theresults here are due to many people including, but not limited to Howard Becker,Steve Jackson, Alexander Kechris, Tony Martin, Yiannis Moschovakis, John Steel,Robert Solovay, and Hugh Woodin. I will make attributions when known, however,

2010 Mathematics Subject Classification. 03E60, 03E25, 03C20.Key words and phrases. Determinacy, AD+, ADR, ∞-Borel sets, ordinal determinacy, Suslin

cardinals.


1



71

98

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104



71

2 RICHARD KETCHERSID

I do not have a full account of the history and hence cannot attribute every fact, non-attributed facts are, as far as I know, folklore, come from private communication,or are unattributed in various hand written notes. The goal is to present some ofthe theory of AD+ which, heretofore, has not been published, although see [Ike10].References given are typically the ones most readily available to the author, andmost likely the reader as well, and almost certainly are not the original sources.

I have placed some of the more technical descriptive set theoretic material to anappendix so as not to deter the reader less familiar with descriptive set theory. Theprimary tools used are forcing, ultraproducts, and absoluteness arguments. Thispaper, with the exception of §7, is intended to be self contained for those familiarwith set theory in general and the axiom of determinacy in particular. I will beginwith a few preliminaries in §2.

§3 concerns degree measures and associated ultrapowers and ultraproducts.Here we will also discuss generic ultrapowers formed by forcing with the positivesets associated to a filter and see that this forcing is essentially a version of Sacksforcing.

§4 introduces the hierarchy of degree notions.§5 discusses many structural consequences of the degree structure developed in

§4. In particular, the cases of having or not having a maximal degree are considered.Various implications concerning AD, AD+, and ADR are considered. This sectionalso contains a proof due to Woodin that AD+ uniformization implies that all setsare ∞-Borel.

§6 develops the critical notion, due to Woodin, of strong infinity Borel code.The main results concerning the closure of the Suslin cardinals below their supre-mum, under AD + DCR, and closure below Θ, under AD+, are discussed in thissection. Subsection §6.1 contains the short proof that the reliable cardinals and theSuslin cardinals are the same. The key to this result is the closure from §6.

§7 contains a sketch of the equivalence between AD+ and the closure of theSuslin cardinals below Θ leaving the existence of a maximal model of AD+ whichcontains all the Suslin sets, part of Woodin’s derived model theorem, as a blackbox.

1.1. Acknowledgments. I want to thank Hugh Woodin for his permissionto publish these results here. I also want to thank the referee for being both quickand thorough, and for making many good suggestions and corrections.

2. Preliminaries

Throughout we will be assuming ZF+AD+DCR. As is typical, in this area, theterm real is used to mean either an element of Baire space, ωω, or of Cantor space,2ω; in general the spaces Xω admit simple pairing functions that the Euclideanreals do not posses. I will use x to indicate a ≤ω-sequence from a set X, i.e.,x = 〈xi : i < |x|〉. For a sequence, 〈xi : i < n〉 ∈ (Xω)n, I will often identify thesequence with its corresponding element in Xω via the canonical isomorphism ofXω with (Xω)n and I will write (x)ni for the element in Xω so that 〈(x)ni :i < n〉 = xvia this identification. Finally, when n is understood from context, I will write (x)irather than (x)ni .

Given a set A ⊆ Xω, the game GX(A) consists of ω-rounds, where in the ith

round player I plays x2i ∈ X and then player II plays x2i+1 ∈ X. When the game

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MORE STRUCTURAL CONSEQUENCES OF AD 3

is over, a play x ∈ Xω has been constructed. I wins if x ∈ A, otherwise II wins. Iwill refer to (x)0 ∈ Xω as I’s play and (x)1 ∈ Xω as II’s play.

A strategy σ for player I (II) is a function σ : X<ω → X telling player I (II)how to move. For f ∈ Xω and σ a strategy for I (II) write σ(f) for the playresulting from II (I) playing f and I (II) using σ. The strategy σ is winning forI iff σ(f) ∈ A for every valid play f by II; similarly define σ is winning for II. IfX ∈ {ω, 2}, then a strategy is easily coded by a real.

The game GX(A) is determined provided one of the players has a winningstrategy. AD is the assertion that for all A ⊆ R, G(A) is determined.

Following on the heals of joint work with Martin, Moschovakis, and Kechris[KKMW81, §2], Woodin showed that if M ⊇ R is a transitive model of ZF andevery set of reals in M is Suslin in some larger transitive model of ZF + AD, thenthe following hold in M :

• (InfBorel) All sets are ∞-Borel.• (OrdDet) <Θ-ordinal determinacy.• DCR.

Woodin eventually isolated the theory AD+, which is now taken to be

ZF + OrdDet + InfBorel + DCR.

AD+ is intended to axiomatize those sentences ϕ that hold in M where M is atransitive model of ZF containing R and such that every set of reals of M is Suslinin some possibly larger transitive model of ZF + AD with the same reals. Thisdownward absoluteness is discussed below (see page 33 and Lemma 2.5). One im-mediate consequence is that AD+ all sets are Suslin =⇒ AD+. It is open whetherAD =⇒ AD+. The arguments contained here should illustrate how DCR, ∞-Borelrepresentations for sets of reals, and ordinal determinacy are used to investigatethe structure of models of AD+.

For facts about Descriptive Set Theory and models of AD refer to [Mos09] andto [Jac10]. For facts and references concerning AD+ see [Woo99, CK09, Ste94].I will define and discuss the Suslin sets and cardinals and ∞-Borel sets and codesin the following two subsections. I have placed many of the more technical factsinvolving descriptive set theory in models of AD in an appendix to be referred towhen needed. The key results in this paper use forcing, ultrapowers/products, andabsoluteness, the reader should not be deterred by the descriptive set theory thatenters in now and again.

2.1. Suslin sets and cardinals. A tree on X is a subset of X<ω closed underrestriction. For T a tree on X, [T ] is the set of all infinite branches through T .Topologically, if X is a discrete space and Xω is given the usual product topology,then the closed subsets of Xω are of the form [T ] for some tree T on X. Inparticular the closed subsets of ωω or 2ω are of this form. For s ∈ X<ω, Ts ={t ∈ T : s ⊆ t ∨ t ⊆ s}.

For T a tree on X × Y , and s ∈ X<ω, Ts = {(t, u) ∈ T : t ⊆ s ∨ s ⊆ t} and forf ∈ Xω set Tf = {u : ∃n (f |n, u) ∈ T }. The set

p[T ] = {f ∈ Xω : ∃g ∈ Y ω (f, g) ∈ [T ]} = {f ∈ Xω : Tf is illfounded}is the projection of T . Here I am identifying sequences 〈(x0, y0), . . . , (xn, yn)〉 ∈(X × Y )≤ω with sequences s = (〈x0, . . . , xn〉, 〈y0, . . . , yn〉) ∈ X≤ω × Y ≤ω such thatlh(s0) = lh(s1).

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A set A ⊆ Xω is Y -Suslin iff A = p[T ] for some tree on X × Y .We will only be interested in the case where Y is wellorderable. If <Y is a

wellordering of Y and T is a tree on Y , with [Y ] �= ∅, then bT is the leftmost branchof T (w.r.t. <Y ) and is defined by

bT |i =<lexY -least s such that Ts is illfounded

From now on just take Y to be an ordinal.If T is a tree on X × κ, then define ϕT

i (x) = bTx(i) for x ∈ p[T ]. So ϕTi (x) :

p[T ]→ κ. The sequence {ϕTi }i is the semiscale associated to T . In general we have

Definition 2.1. A sequence of functions, {θi : A→ OR}i, is a semiscale on Aiff whenever

(1) xi ∈ A for all i ∈ ω,(2) limi xi = x, i.e., 〈xi(j) : i ∈ ω〉 is eventually constant, and(3) ∀j∃λj∀∞i θj(xi) = λj , i.e., 〈θj(xi) : i ∈ ω〉 is eventually constant,

then x ∈ A. �

In other-words, {θi}i a semiscale on A iff A = p[T {θi}i ], where

T {θi}idf= {(x|n, 〈θ0(x), . . . , θn−1(x)〉) : n ∈ ω ∧ x ∈ A}.

A semiscale is regular if for each i, θi : Aonto−−→ κi. Any semiscale {θi}i generates

a regular semiscale {θi}i by collapsing the range. If {θi}i with θi : Aonto−−→ κi is a

regular semiscale and A ⊆ R, then κi < Θ, being the rank of a prewellordering onA. Similarly, supi κi ≤

∑i κi < Θ, since defining (i, x) ≤ (j, y)

df⇐⇒ i < j ∨ [i =j ∧ θi(x) ≤ θi(y)] is a prewellordering of ω × A. As a consequence, we see that ifA = p[T ] for T a tree on ω× κ, then A = p[T ′] for T ′ a tree on ω× κ′ with κ′ < Θ.In the future κ-Suslin will entail κ < Θ.

Define Sκ to be the collection of all κ-Suslin sets. A set is co-κ-Suslin if itscomplement is κ-Suslin. Sκ is closed under continuous preimages, real existentialquantification (projection), in fact closed under ∃f ∈ κω. Under AD + DCR, Sκ isalso closed under countable union and intersection, where the coding lemma 8.4 isused in conjunction with DCR to pick a countable sequence of Suslin representationsgiven a countable sequence of Suslin sets.

It might seem as though Sκ should be closed under κ-length wellordered unions;however this would require the ability to pick a κ-sequence of κ-Suslin representa-tions and as we shall see, this amount of choice fails under AD.

A cardinal κ is a Suslin cardinal if Sκ \S<κ �= ∅, where S<κ=

⋃λ<κ Sλ. Notice

κ a Suslin cardinal implies κ < Θ. DC implies that the Suslin cardinals forman omega-club, since one needs only to pick a suitable sequence of trees. Defineκ∞ = sup{κ < Θ : κ is a Suslin cardinal} ≤ Θ and S∞ to be the set of all Suslinsets. Thus κ∞ is a Suslin cardinal iff S∞ \ S<κ∞ �= ∅. If κ∞ < Θ, then DCR

suffices to show that the Suslin cardinals form an ω-club. Again, the coding lemmais used to reduce DC to DCR. Under DCR it is possible for cf(κ∞) = ω, but onlyif κ∞ = Θ.

Suslin subsets of R2 are uniformizable as follows. Suppose A ⊆ R2 with A =p[T ] for T a tree on ω2 × κ, then define A∗(x, y) ⇐⇒ (bTx)0 = y, that is, the lefthand side of the leftmost branch through Tx is y. A∗ is a uniformization of A.This shows that being Suslin entails a bit of choice. Since choice conflicts with

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determinacy, AD together with “there are a lot of Suslin sets”, should be strongerthan just AD and this is indeed the case.

Borel sets are precisely the ω-Suslin/co-ω-Suslin sets. Consider WOα = {x :x codes a wellorder of rank α}. Each WOα is Borel; hence has a ω-Suslin represen-tation. However, assuming AD, there is no sequence of trees Tα (on any ordinal)witnessing WOα is Suslin. If such a sequence existed, then setting xα = bTα

0 wouldgive an uncountable sequence of distinct reals. This example shows that the pas-sage from Borel representation to Suslin representations is non-trivial. One of themain goals of this paper is to understand the passage from ∞-Borel code of aset to a Suslin representation, if such exists. Conversely, passing from Suslin to∞-Borel representations is relatively straightforward; an old result of Sierpiński isthat any co-κ-Suslin set is the κ+-union of <κ+-Borel sets and the passage fromSuslin representation to Borel representation is simply definable.

We will need the following fact regarding the rank of λ-Suslin wellfoundedrelations.

Theorem 2.2 (Kunen-Martin [Mos09, Jac10]). Let≺ be a λ-Suslin wellfoundedrelation, then || ≺ || < λ+. �

2.2. ∞-Borel sets. In section 2.2.3 of [CK09], ∞-Borel codes are defined aswell as several equivalent notions. We need two of these here. The official definitiontakes an ∞-Borel code to be a wellfounded tree, T , which describes how to build aset of reals, AT , via well ordered unions, complements, etc., beginning with basicopen sets. Let BCκ be the collection of ∞-Borel codes where the tree is on κ. Itshould be fairly clear that for α ≥ ω, BCα is closed under continuous substitutionsince given π : ωω → ωω continuous we just need for each i, j ∈ ω a code forπ−1[{x : x(i) = j}].

Our official definition of ∞-Borel code is equivalent to considering the infini-tary propositional calculus, L∞(x), with basic propositions xi,j with intended in-terpretation being {x ∈ ωω : x(i) = j}, so instead of writing xi,j I write x(i) = j.Negation and wellordered conjunction/disjunction is allowed. The standard defi-nition of z |= S(x) is used and clearly z |= S(x) ⇐⇒ z ∈ AS . I will utilize thisnotation and write “S(z)” in place of “z |= S(x)” or “z ∈ AS” when useful. SimilarlyL∞(x0, . . . , xn−1) is used to describe subsets of Rn with L<ω

∞ ({xi : i ∈ ω}) being theunion of the Ln

∞. This allows easy manipulation of variables to derive new codesfrom old and I will utilize this when useful.

There is a fixed Σ1-formula, Φ, so that for T ∈ BCκ,

x ∈ AT ⇐⇒ LαT,x[T, x] |= Φ(T, x)

where αT,x is the least α so that Lα(T, x) is a model of Kripke-Platek (KP) settheory [Bar75].

Consequently, one variant of the coding is to take a code to be a triple (α, ϕ, S),for S ⊆ OR and ϕ a formula of set theory, and sets

x ∈ A(α,S,ϕ) ⇐⇒ Lα[S, x] |= ϕ(S, x).

Call (α, ϕ, S) an ∞-Borel∗ code and let BC∗κ be those codes (α, ϕ, S) with S ⊆ α ⊆

κ. The relative sizes of these two notions of code and in which inner models theyexists will be used.

Lemma 2.3. For any κ, BCκ ⊆ BC∗κ+ while for any κ such that ωκ = κ, BC∗

κ ⊆BCκ.

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Proof. For S ∈ BCκ let αS = supx∈RαS,x, then

x ∈ AS ⇐⇒ LαS[S, x] |= Φ(S, x).

Here we use the fact that Φ is Σ1. Letting ακ = supS∈P(S) αS and αS,x < κ+ forS ⊆ κ so αS ≤ κ+ and we have

BCκ ⊆ BC∗ακ⊆ BC∗

κ+ .

The other direction requires quite a significant detour and will be omitted here.The point is essentially that Jα[S] can “see” Jα[S, x] (as a collection of “uninterpretednames/descriptions for sets”), where the J-hierarchy is Jensen’s version of L (see[Jen72].) ❑

The following two lemmas say something about how far one must look to in-terpret a given ∞-Borel code and conversely how far one must look to find a codefor a given ∞-Borel set. It would be good to consult Lemma 8.3 for the definitionsand facts used concerning δ(A), Π(A), and Δ(A).

Lemma 2.4. For any set A, B<δ(A) ⊆ Δ(A).

Proof. If S ∈ BCκ and ≤ is are prewellorder of rank κ, then δ(A) is measurable,hence regular, and Δ(A) is closed under <δ(A)-wellordered unions, so B<δ(A) ⊆Δ(A). ❑

Lemma 2.5. Any ∞-Borel set, A, has a code in Δ(A), with respect to any Π(A)-norm on a complete Π(A) set.

Proof. Let S be a code for A and consider the relation ∼ on BC∞ given byT ∼ T ′ ⇐⇒ AT = AT ′ . Let M = L[∼, S] and look at BCM

∞/ ∼. If BCM<δ(A) = BCM

∞ ,then there is a code S′ of size < δ(A) with S′ ∼ S and such a code is in Δ(A) .

If BCM<δ(A) �= BCM

∞ , then there is a δ(A)-antichain, 〈Sα :α < δ(A)〉. This gives aprewellorder ≤∗ of length δ(A) with a code S∗ =

∨α≤β Sα×Sβ in BCδ(A) and thus

has a Δ(A) code. Clearly, ≤∗�w A, since otherwise || ≤∗ || < δ(A). Since ≤∗� A,either A ≤w≤∗ or A ≤w�∗. Since BCα is closed under continuous substitution forα ≥ ω. ❑

This gives the downward absoluteness of being ∞-Borel.It is shown in [CK09], under AD + DCR, or more precisely, assuming there

is a countably complete fine ultrafilter on Pω1(R) and DCR, that A is ∞-Borel iff

A appears in a model of the form L(S,R) for some S ⊆ OR. In fact Woodin hasshown more:

Theorem 2.6 (Woodin). Work in ZF. Suppose μ is a fine measure on Pω1(P(γ))

and∏

OR/μ is wellfounded. Let S ⊆ OR and A ⊆ R with A ∈ ODL(S,P(γ))S,T for

T ⊆ γ, then A is ∞-Borel with code in ODVS,T,μ. �

Assuming AD for each γ < Θ there is an OD, fine, σ-complete measure, μγ , onPω1

(P(γ)). So μ can be dropped in the definability estimate for the code in thiscase.

This theorem shows that under AD+DCR+V = L(P(R)), L(B∞,R)∩P(R) =B∞ so there is a largest model of AD + DCR + InfBorel. The point is that ifB∞ �= P(R), then B∞ = Bλ for a λ < Θ and thus B∞ ⊆ L(P(γ)).

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3. Cone ultrafilters and ultraproducts.

Letx ≤S y ⇐⇒ L[S, x] ⊆ L[S, y]

be the partial order of S-constructibility degrees. Let

x ≡S y ⇐⇒ x ≤S y & y ≤S x

be the corresponding equivalence relation with classes [x]S = {y : x ≡S y} and letDS be the set of S-degrees. We will use x ≤ y to mean x is Turing reducible to y.Notice, in particular, that x ≤∅ y is different from x ≤ y.

A set of reals, A, is S-invariant iff

x ∈ A & x ≡S y =⇒ y ∈ A

An S-invariant set can be viewed as a subset of DS . A function f : R → V isS-invariant if

x ≡S y =⇒ f(x) = f(y)

so f can be viewed as f : DS → V . More generally, for a formula ϕ, define ϕ isS-invariant iff

x ≡S y =⇒(ϕ(x)⇐⇒ ϕ(y)

).

A set, A, of reals contains an S-cone iff ∃x0 ∀x ≥S x0 (x ∈ A). I will write thisas ∀∗Sx (x ∈ A). More generally, write ∀∗Sxϕ(x) for a formula ϕ and say ϕ holds onan S-cone, if ∃x0∀x ≥S x0 ϕ(x).

The collection of sets of reals containing S-cones forms a σ-complete filter underDCR, denoted μS and called the S-cone filter or Martin measure on S-degrees. μwill denote the Martin measure on Turing degrees, so μ and μ∅ are distinct.

Theorem 3.1 (Martin). μS when restricted to S-invariant sets is an ultrafilter.

Proof. Let A be S-invariant. Play the game where I plays x and II plays y (bitby bit). Player II wins if y ≥S x and y ∈ A. If I wins with strategy σ, then whenII plays y ≥S σ, x = σ(y) /∈ A since x ≤S σ ⊕ y ≡S y and A is S-invariant. Thisshows that the S-cone above σ is contained in ¬A.

If σ is a II winning strategy, then for x ≥S σ, σ(x) ∈ A, but σ(x) ≡S x sinceσ(x) ≥S x, so x ∈ A by the S-invariance of A. ❑

Clearly for S-invariant ϕ

∀∗xϕ(x)⇐⇒ ∀∗Sxϕ(x)and we will use this without mention throughout. In particular we do not haveto be careful about using the expression “S-cone” and can just use “cone” in mostcases.

We give a partial order to P(OR) as follows:

S � T ⇐⇒ ∀∗x (L[S, x] ∩ R ⊆ L[T, x])

S ≈ T ⇐⇒ ∀∗x (L[S, x] ∩ R = L[T, x] ∩ R)

I will refer to S/ ≈ as the degree notion corresponding to S.When looking at reduced products of S-invariant functions, whether μ or μS is

used is irrelevant; what does matter is the class of functions used. For S-invariantf : R→ OR define [f ]S recursively by

[f ]S =∏

S

f/μ = {[g]S : g is S-invariant & ∀∗x g(x) < f(x)}

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One gets the same notion if μ is replaced by μS and ∀∗ is replaced by ∀∗S .If S � T , then πS,T (μS) = μT , where πS,T : DS → DT is given by πS,T ([x]S) =

[x]T , this is extended to the measures by defining X ∈ πS,T (μS)df⇐⇒ π−1

S,T [X] ∈ μS .If f is S-invariant and S ≺ T , then [f ]T ≤ [f ]S , but equality will not hold in general,for example, Lemma 3.3 shows

∏T ω

L[T,x]1 = ωV

1 while∏

S ωL[T,x]1 >

∏S ω

L(S,x)1 ,

since ωL(S,x)1 < ω

L(T,x)1 on a S-cone of x by Theorem 4.4.

I will assume DC−, the statement that∏

μ OR/μ is wellfounded, for the re-mainder of the paper. It is a non-trivial fact due to Woodin that DC− follows fromAD+. It is clear that DC− sits somewhere between DCR and DC. Since all of theother measures we use reduce to μ, DC− gives

∏S OR/μS is wellfounded. Assum-

ing DC− could be avoided throughout a large portion of the paper. Assuming DCR

we have that L(A,R) is a model of DC and where we can get by working locally ina model of the form L(A,R), we could get by also with just DCR.

The notation [f ]S can be extended to S-invariant functions f : R→ P(OR) andmore generally to the situation where we have S-invariant map x �→ (Mx,�x) whereMx is a transitive (set- or proper class-sized) structure which carries a natural wellordering �Mx

. Form∏

S Mx/μ using S-invariant functions f such that f(x) ∈Mx

on a cone of x. This will yield

M∞S =

∏

S

Mx/μ

a transitive structure with well ordering �M∞and Łos’s Lemma will hold:

M∞S |= ϕ([f ]S)⇐⇒ ∀∗xMx |= ϕ(f(x)),

for all formulas ϕ. This uses DC or DCR in some appropriate model as describedabove.

Let [T ]S be the object corresponding to the constant function with value T ,and let jS : P(OR) → P(OR) be the function T �→ [T ]S . The critical point of thisembedding is ωV

1 .Extending slightly the notation from [CK09], for S � T , set

Hx(T ) = HODL[T,x]T

and setH∞

S (T ) =∏

S

Hx(T )/μ.

More generally, I will use S∞ for jS(S) and T∞S ambiguously for jS(T ) or [x �→ T x]S

where x �→ T x is S-invariant.We will see below that H∞

S (T ) can be viewed as Hx∞S (T∞

S ) = HODL[T∞S ,x∞

S ]T∞S

for a generic real x∞S which can be viewed as a kind of Sacks generic over V .

Set δxS = ωL[S,x]2 and δ∞S =

∏S δxS/μ. Recall from [CK09] that on a cone of x,

GCH holds below ωV1 in L[S, x], ωV

1 is inaccessible in L[S, x], and δxS is inaccessible(in fact Woodin [KW10]) in Hx(S). Let GCH∗ denote “GCH holds below theleast inaccessible” (ω2, “least measurable”, etc. would work just as well as “leastinaccessible” here.) We will primarily use

GCH∗ =⇒ 2ω = ℵ1 and 2ω1 = ℵ2.

We will show below that δ∞S depends on S, see Theorem 5.16 and Corol-lary 6.4. In contrast, we have

∏S ω

L[S,x]1 = ωV

1 . First we need the following:

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Lemma 3.2. Suppose f : R → OR is S-invariant, then either f is monotonicallyincreasing on a cone, i.e., for a cone of x, y > x → f(y) > f(x), or else, f isconstant on a cone.

Proof. If monotonically increasing fails on a cone, then on a cone of x there isy ≥ x, with f(y) ≤ f(x). Since there is not an infinite descending sequence ofordinals we get a cone on which f is constant. ❑

Lemma 3.3. For all S,∏

S ωL[S,x]1 /μ = ωV

1 .

Proof. Let f ∈∏

S ωL[S,x]1 . If f is constantly α on a cone, then [f ]S = α, so

suppose f is monotonic increasing (see Lemma 3.2). Consider the game where I

plays x and II plays y, z. II wins iff y ≥S x and z ∈ WOL[S,y] codes f(y), whereWO is just the collection of reals that code ordinals below ω1.

Suppose I follows a strategy σ and II plays y, z with y above σ and such thaty is in the cone on which f is monotonic, and z ∈ WOL[S,y] coding f(y). This isclearly a win for II, so no strategy is winning for I.

Let σ be a II winning strategy and set Y = {(σ(x))1 : x ∈ R}. This is a Σ˜11

subset of WO and hence Y ⊆WOα for some α < ω1. Let x ≥S σ, then (σ(x))0 ≡S xand f(x) = f((σ(x))0) = ||(σ(x))1||. This shows that on a cone of x, f(x) < α, butthen f must be constant on a cone. ❑

Viewing μS as a filter on P(R) rather than an ultrafilter on S-invariant sets wehave A ⊆ R is S-positive provided ∀∗x [x]S ∩ A �= ∅. Equivalently, A is S-positiveif [A]S contains a cone. Let PS be the notion of forcing with conditions being S-positive sets and with A ≤PS

B iff A ⊆ B. PS-generics are V -ultrafilters on P(R)V .The next lemma shows that the map jG : V → ult(V,G) agrees with jS on P(OR).Recall ult(V,G) is formed in V [G] using functions f : R→ V in V . Without choicein V , jG need not be elementary and we will be more interested in ultraproductsof canonically well-ordered structures like L[T, x] where T ⊆ OR.

Lemma 3.4. For G ⊆ PS generic, the map k([f ]S) = [f ]G is an isomorphism of∏S OR/μ with

∏R

OR/G.

Proof. It is clear that k([f ]S) = [f ]G is an embedding. We want to see that k isonto. Let f : A → OR for A S-positive. Let B ≤PS

A and define f : [B]S → ORby f([x]S) = inf f [[x]S ∩B].

Define C = {x ∈ B : f(x) = inf f [[x]S ∩B]}, then C is an S-positive subset ofB and C �PS

k([f ]S) = [f ]G. Since B ≤S A is arbitrary, A � [f ]G ∈ rng(k). ❑

If G is PS-generic, then define

x∞S (G) =

⋃{s : [s] ∈ G}

where [s] = {x ∈ R : x ⊃ s}. When G is understood, I will simply write x∞S . We

have, by Łos’s Lemma, that∏

R

〈L[T, x], Hx(T ), T, x〉/G = 〈L[T∞S , x∞

S ], H∞S (T ), T∞

S , x∞S 〉

and

L[T∞S , x∞

S ] |= H∞S (T ) = HODL[T∞

S ,x∞S ]

T∞S

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In particular, H∞S (T ) = Hx∞

S (T∞S ).

We will use the fact that PS can be recast as a version of Sacks forcing. Calla tree a on ω S-pointed perfect iff a is perfect and, identifying a with a real in anatural way, a ≤S x for every branch x ∈ [a]. A proof of the following appears in[Kec88].

Theorem 3.5 (Martin). For A ⊆ R, A is S-positive iff there is an S-pointedperfect tree a such that [a] ⊆ A.

Proof. Players I and II play x and y respectively. Player II wins if x ≤S y andy ∈ A. If σ is a winning strategy for II, then σ[R] contains a perfect subset and atree, a, witnessing this can be found in L[S, σ]. (In fact a is very simply definablefrom σ, but one still needs to go to L[S, y] to compute a from y ∈ [a].) One way tosee this as follows. In L[S, σ] define s �→ ts, ns where ts =

(xs ⊕ σ

)|ns for some xs

subject to the constraints that(1) lh(s) = lh(s′)→ ns = ns′

(2) s ⊥ s′ → σ(ts) ⊥ σ(t′s)

Let a be the tree {σ(ts) : s ∈ 2<ω}. For y ∈ [a] there is b ∈ 2ω and xb|i so thatlimi xb|i ⊕ σ = xb ⊕ σ and y = σ(xb⊕ σ). Since σ is winning for II, xb ⊕ σ ≤S y soin particular σ ≤S y for y ∈ [a] and so a ≤S y since a ≤S σ.

If player I wins with σ, then if y ≥S σ we have y ≥S σ(y) so y /∈ A. This shows¬A contains a cone. ❑

So forcing with PS is equivalent to the version of Sacks forcing where S-pointedperfect trees are used, call this SS . If G is a generic ultrafilter of S-positive sets,then x∞

S (G) as defined previously is the corresponding Sacks generic.The next theorem will be used later and is our main application of the fact that

PS is essentially S-pointed perfect Sacks forcing, the point being that literally S isnot necessarily in L(S∞,R). By passing to the S-pointed Sacks forcing, however,we can see that if x∞

S is SS generic, then x∞S is SS∞ generic over L(S∞,R).

Theorem 3.6. Let x∞S be PS-generic, then δ∞S ≤ Θ and x∞

S is SS∞-generic overL(S∞,R).

Proof. First notice that S∞ ≈ S since for any y, x ∈ R:

y ∈ L[S∞, x]⇐⇒ ∀∗z(y ∈ L[S, x]

)↔ y ∈ L[S, x]

So, while L(S∞,R) might not see S, it does see the corresponding S∞-pointedperfect forcing and this is equivalent to S-pointed perfect forcing. So x∞

S is SS∞-generic over L(S∞,R).

In L(S∞,R)[x∞S ] there is no map from RV onto ΘV since if τ were a name for

such a map, look at Bα = {(a, x) : a � τ (x) = α}. The sequence Bα determines aprewellordering, ≤df

=⋃

α≤β<ΘV Bα ×Bβ, in V , of length ΘV , which cannot exist.

If Θ < δ∞S = ωL[S∞,x∞

S ]2 , then |Θ|L[S∞,x∞

S ] = ωL[S∞,x∞

S ]1 = ωV

1 by Lemma 3.3.But, then L(S∞,R)[x∞

S ] |= |Θ| = ωV1 contradicting the previous paragraph, since

L(S∞,R) has a map from RV onto ωV1 . ❑

4. Compatibility of constructibility degrees.

In this section we reproduce Woodin’s proof that for any two sets of ordinals Sand T either L[S, x] sees a large initial segment of L[T, x] or else L[T, x] sees a large

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initial segment of L[S, x] in H(ω1)L[T,x]. For this Woodin utilized a generalized

notion of Prikry forcing.Work in ZF and let U be an ultrafilter on a set X. A condition in PU is a pair

p = (p0, p1) where p0 ∈ X<ω and p1 : X<ω → U with order defined by p ≤PUq iff

(1) p0 ⊇ q0,(2) for all i ∈ dom(p0) \ dom(q0), p0(i) ∈ q1(p0 � i), and(3) p1(t) ⊆ q1(t) for all t ∈ X<ω.

We may, and do, work with conditions with the property that for all subsequencess of t, p1(t) ⊆ p1(s). (Here I mean subsequences, not initial segments.) Only finiteadditivity is required. This is almost a tree forcing with p1 determining U -largesplitting above a trunk p0, however, for the Mathias Condition below we require p1to be defined on X<ω and not just a subtree. Using finite additivity of the measureit is clear that any two conditions p, p′ with p0 = p′0 are compatible.

A generic, G, determines an element, g, of Xω with g =⋃{p0 : p ∈ G}. Con-

versely, G = {p : p0 ⊆ g and ∀i > |p0| g(i) ∈ p1(g � i)}. We shall refer to g as thegeneric sequence from X.

Two key properties of Prikry forcing extend to this setting, these are the PrikryProperty and the Mathias Property. We follow Woodin and use a rank function inthe proofs. In the case of the Prikry Property, the rank function turns out to onlytake the value 0 or ∞.

Theorem 4.1 (Prikry Property). Given any condition p and sentence ϕ of theforcing language, there is a condition p′ ≤PU

p such that p0 = p′0 and p′ decides ϕ.

Proof. It suffices to show that there is some f so that (∅, f) decides ϕ, for thenwe can take p′1(t) = f(t) ∩ p1(t).

Define a rank function ρϕ : X<ω → OR ∪ {∞}ρϕ(t) = 0⇐⇒ there is p such that p0 = t and p � ϕ

ρϕ(t) = α⇐⇒ {x : ρϕ(tx) < α} ∈ U and ρϕ(t) ≮ α

ρϕ(t) =∞⇐⇒ ρϕ(t) �= α for any α

Define

fϕ(s) =

⎧⎪⎨

⎪⎩

{x : ρϕ(sx) = 0} if ρϕ(s) = 0

{x : ρϕ(sx) < ρϕ(s)} if 0 < ρϕ(s) <∞{x : ρϕ(sx) =∞} if ρϕ(s) =∞

So (∅, fϕ) ∈ PU and we will see (∅, fϕ) decides ϕ.If ρϕ(∅) < ∞, then take q ≤PU

(∅, fϕ) with q deciding ϕ. For i ∈ dom(q0),q0(i) ∈ fϕ(q0 � i), so ρϕ(q0) �=∞, so ρϕ(q0|i) ≥ ρϕ(q0|i+1) and equality only occursif both values are 0. We can extend q0 to q′0 so that ρϕ(q′0) = 0 and set q′ = (q′0, q1)so q′ ≤PU

q and q′ � ϕ. This means that q � ϕ.If ρϕ(∅) =∞, then take q ≤PU

(∅, fϕ) which decides ϕ. In this case it must bethat q � ¬ϕ since ρϕ(q0) =∞. In this case (∅, fϕ) � ¬ϕ. ❑

Clearly the proof shows that for all t

ρϕ(t) = 0⇐⇒ {x : ρϕ(tx) = 0} ∈ U

ρϕ(t) =∞⇐⇒ {x : ρϕ(tx) =∞} ∈ U

So for all t, either ρϕ(t) = 0 or ρϕ(t) =∞.

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Notice we have shown that to each ϕ there is a canonical fϕ so that lettingpϕ,t = (t, fϕ), ρϕ(t) < ∞ → pϕ,t � ϕ and ρϕ(t) = ∞ → pϕ,t � ¬ϕ. The mapt �→ fϕ requires no choice.

Say that g ∈ Xω has the Mathias Condition if for any (∅, f) ∈ PU , there is isuch that ∀j ≥ i, g(j) ∈ f(g � i).

Theorem 4.2 (The Mathias Property). Assuming some choice, any g ∈ Xω withthe Mathias condition is generic.

Proof. The proof is as in the Prikry property. Define a rank function ρD(t) forD an open dense set. The base case is

ρD(t) = 0⇐⇒ ∃p ∈ D(p0 = t

)

ρD(t) = α⇐⇒ {x : ρD(tx) < ρD(t)} ∈ U and ρD(t) ≮ α

ρD(t) =∞⇐⇒ ρD(t) is undefined

Define fD analogous with fϕ above, except when s ∈ D∗, where

s ∈ D∗ df⇐⇒ ρD(s) = 0 ∧ ∀i ∈ dom(s) ρD(s|i) > 0

Here we need enough choice to choose one member of Pu for each member of D∗.

fD(s) =

⎧⎪⎨

⎪⎩

ft(s) if t ⊆ s ∧ t ∈ D∗

{x : ρD(sx) < ρϕ(s)} if 0 < ρD(s) <∞{x : ρD(sx) =∞} if ρD(s) =∞

First we show that ρD(t) �= ∞ for all t. If ρD(t) = ∞ let pD,t = (t, fD). Letq ≤PU

pD,t with q ∈ D. Then ρD(q0) = 0; however, the fact that q ≤PUpD,t implies

ρD(q0) =∞. This contradiction shows ρD(t) �=∞ to begin with.The amount of choice here is ACD∗

PUand |PU | =

∣∣∣(2X

)X<ω ∣∣∣ = 2X

<ω

so ACX<ω

2X<ω

suffices. For X = ω, which is the case we use, ACωR

is what is requires. This isweaker than DCR.

Fix i so that for all k ≥ i, g(k) ∈ pD,1(g � k). By definition of fD we have thatρD(g � k + 1) < ρD(g � k) if ρD(g � k) �= 0. Thus ρD(g � k) = 0 for some k ≥ i. Wehave for all l ≥ k, g(l) ∈ fg�k(g � l) and thus (g � k, fg�k) ∈ D so Gg ∩D �= ∅. ❑

From the Mathias property it follows that if g is PU -generic and g′ is anyinfinite subsequence of g, then g′ is also PU -generic. This is why we restricted PU

to conditions satisfying p1(s) ⊆ p1(t) for t a subsequence of s.These facts are all that we shall require concerning generalized Prikry forcing.For the next theorem we need the following lemma due to Hausdorff.

Lemma 4.3. There is a recursive Lipschitz continuous π : 2ω → [ω]ω so that ifx1, . . . , xn are distinct elements of 2ω, then π(x1), . . . , π(xn) are mutually indepen-dent.

Proof. a0, . . . , an−1 ∈ [ω]ω are mutually independent if⋂

i<n as(i)i is infinite for

all s ∈ 2n where a1i = ai and a0i = ω \ ai. One way to accomplish this is to defineπn : 2n → [mn]

<ω so that these functions cohere reasonably, i.e., πn+1(t) ∩mn =πn(t|n) and ∣

∣∣∣∣

⋂

s∈2n

πn(s)t(s)

∣∣∣∣∣≥ n

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where t ∈ 22n

, πn(s)1 = πn(s), and πn(s)

0 = mn \ πn(s). Suppose πn is defined,then we take mn+1 = mn+22

n+2

and enumerate all σ : 2n+1 → 2 by i ∈ [mn,mn+1).For σi we put i ∈ πn+1(t) iff σi(t) = 1. ❑

Define the filter Fπ as follows:

a ∈ Fπ ⇐⇒ {y : a ⊇∗ π(y) ∨ a ⊇∗ ω \ π(y)} is infinite

where a ⊇∗ b iff |b \ a| < ω. Let a1, . . . , an ∈ Fπ and let y1, . . . , yn be so thatai ⊃∗ πni(yi) where ni ∈ 2 and π1(yi) = π(yi) while π0(yi) = ω \ π(yi). Now⋂ai ⊇∗ ⋂

πni(yi) and⋂πni(yi) is infinite. So Fπ generates a filter on P(ω) all of

whose elements are infinite. The following characterizes being Fπ-positive:

a is Fπ-positive ⇐⇒ {y : a ⊆∗ π(y) ∨ a ⊆∗ ω \ π(y)} is finite.

So being Fπ-positive is Π11 in a fixed parameter y1, . . . , yn and thus is absolute.

The following theorem is the main theorem of this section. The theorem statesthat either (on a cone) L[S, x] and L[T, x] agree for quite a while, or else, oneof the models is “much” larger than the other (on a cone). Let κS

x be the leastinaccessible of L[S, x] (or ωL[S,x]

2 , or least Mahlo of L[S, x], or any other uniformly,in x, definable “large” cardinal.)

Theorem 4.4. Let S and T be sets of ordinals. Then one of the following holdon a Turing cone:

(1) κ = κSx = κT

x and H(κ+)L[S,x] = H(κ+)L[T,x] or(2) L[S, x] ∩ P(κS

x ) ∈ H(ω1)L[T,x], or else,

(3) L[T, x] ∩ P(κTx ) ∈ H(ω1)

L[S,x], or

Proof. First suppose {x : L[S, x] ∩ R ⊆ L[T, x]} contains a cone. Let x0 be thebase of such a cone and x ≥ x0. Let g be L[T, x] generic for collapsing κS

x toω. Since every a ⊆ κS

x can be coded by a real, ag, using g, we essentially have,L[S, x] ∩ P(κS

x ) ⊆ L[S, x, g] ∩ R ⊆ L[T, x, g]. Since g is any L[T, x]-generic (in V )we have L[S, x] ∩ P(κS

x ) ⊆ L[T, x]. Notice that this also shows κSx ≤ κT

x .If it is the case that {x : L[S, x] ∩ R = L[T, x] ∩ R} contains a cone, then the

same argument shows that on a cone, κ = κSx = κT

x and H(κ+)L[T,x] = H(κ+)L[S,x].If it is not the case that {x : L[S, x] ∩ R = L[T, x] ∩ R} contains a cone, then

either on a cone {x :L[S, x] ∩ R � L[T, x]} or on a cone {x :L[T, x] ∩ R � L[S, x]}.Suppose the former.

Fix x0 so that for x ≥ x0, L[S, x]∩R � L[T, x]. Fix π as in the previous lemma.Let U be an ultrafilter in L[T, x] extending the filter FL[T,x]

π . For a ∈ U are positiveand since this is absolute a is positive in V . So if z /∈ L[S, x], π(z)∩ a and a∩ π(z)are both infinite for all a ∈ U .

Build g, PL[T,x]U generic over L[T, x] such that g ∩ π(z) and g ∩ ω \ π(z) are

both infinite. For any b ∈ 2ω we can shrink g to gb so that b(i) = 1 ⇐⇒ gb(i) ∈π(z). In this way we get b ∈ L[S, x, gb]. Of course b can be chosen so as to codeL[T, x0] ∩ P(κT

x ) and thus L[T, x] ∩ P(κTx ) ∈ L[S, x, gb] and countable there.

Consider L[T, x, gb]. In this model gb is PU -generic so κ = κTx = κT

x,gband

H(κ+)L[T,x][gb] = H(κ+)L[T,x,gb] so H(κ+)L[T,x,gb] ∈ H(ω1)L[S,x,gb]. We have

shown that either (2) or (3) hold for some y ≥ x, for all x ≥ x0 (y = x ⊕ gbabove). This implies that either (2) or (3) holds on a cone. ❑

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Corollary 4.5. (ZF + AD) For all S, T ⊆ OR

S � T → on a cone of x, L[S, x] ∩ P(κSx ) ⊆ L[T, x]

S ≺ T → on a cone of x, L[S, x] ∩ P(κSx ) ∈ H(ω1)

L[T,x]

From here on κxS = δxS = ω

L[S,x]2 , so I will dispense with κx

S in favor of δxS .

5. Constructibility degrees and the structure of AD models

5.1. Uniformization and the non-existence of a maximal degree no-

tion. Uniformization (UNIF) is the assertion that for all R ⊆ R × R, there is R∗

so that∃y R(x, y)⇐⇒ ∃!y R∗(x, y).

UNIF(Γ,Γ′) is the same assertion, except R is taken from the pointclass Γ and R∗

can be found in the pointclass Γ′.The relation R is uniformizable on a cone means that there is R∗ so that for a

cone of x,∃y R(x, y)⇐⇒ ∃!y R∗(x, y).

Write UNIF∗(Γ,Γ′) if for all relations R in Γ, there is a relation in Γ′ that uni-formizes R on a cone. The following shows that there is essentially no differencebetween uniformization and uniformization on a cone.

Lemma 5.1. For Γ and Γ′ pointclasses

UNIF(Γ,Γ′)⇐⇒ UNIF∗(Γ,Γ′)

Proof. Let R ∈ Γ and let R∗ ∈ Γ′ uniformize R on a cone. Set

S(u⊕ x, y)⇐⇒ R(x, y)

So S ∈ Γ. Let S∗ ∈ Γ′ uniformize S on the cone above a. Then R∗(x, y) ⇐⇒S∗(a⊕ x, y) uniformizes R. ❑

Because of this fact I will not distinguish between uniformization and uni-formization on a cone.

The method of proof in Theorem 4.4 can be extended to yield uniformizationon ∞-Borel relations whose codes are not maximal. The general idea is as follows,suppose S ≺ T and S is an ∞-Borel code of a relation. We will show that on acone of x

∃y AS(x, y)⇐⇒ ∃y ∈ L[T, x](AS(x, y))

This gives a canonical uniformization of AS using the natural well ordering ofL[T, x]. For this we need to review the construction of the ∞-Borel code ∃RS for∃RAS described in [CK09] and reviewed below. It turns out that for the argumentS ≺ T does not suffice, we must replace S with S∞

∗ described below.For each z consider the “Vopenka-like” algebra

Qz(S)df= BCHz(S)

∞ / ∼zS

where for T, T ′ ∈ BCHz(S)∞ ,

T ∼zS T ′ ⇐⇒ (AT )

L[S,z] = (AT ′)L[S,z].

It is shown in [CK09] that, for a cone of z,

Qz(S) = BCHz(S)δzS

/ ∼zS .

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Because of this last fact, on a cone of z, Qz(S) can be viewed as a complete Booleanalgebra on δzS in Hz(S).

Every Qz(S)-generic, G, generates a real xG with xG(i) = j ⇐⇒ ci,j/ ∼zS∈ G,

where ci,j is the code for the basic open set Aci,j = {x : x(i) = j}. It can beshown that G = {T/ ∼z

S : xG ∈ AT }, so Hz(S)[xG] = Hz(S)[G]. Moreover aswith the usual Vopenka algebra, every real, x, in L[S, z], generates a Qz(S)-genericGx = {T/ ∼z

S∈ Qz(S) : x ∈ AL[S,z]T }. That Gx is generic follows from the fact that

A ⊆ Qz(S) is predense ⇐⇒⋃

p∈A

AL[S,z]p = RL[S,z].

To put this in context, recall Vopenka’s algebra (see, e.g., [Jec03]) Qz(S)∗

takes ODL[S,z] subsets of R as conditions (actually a Hz(S) copy is used.) EveryQz(S)∗-generic G∗ gives rise to a real xG∗ just as above and every real, x, in L[S, z]

gives rise to a generic G∗x and Hz(S)[G∗

x] = Hz(S, x) = HODL[S,z]S,x .

The two algebras are related as follows: Qz(S) is the complete subalgebra ofQz(S)∗ generated by the basic open sets [s] for s ∈ ω<ω. For x ∈ L[S, x],

Hz(S)[Gx] = Hz(S)[x] = HODL[S,z]S [x] ⊆ HODL[S,z]

S,x = Hz(S, x) = Hz(S)[G∗x]

Below I use M |κ to mean V Mκ , so L[S, x]|κ does not mean Lκ[S, x]. The fol-

lowing facts are discussed thoroughly in [CK09]:• For all x, x is Qz(S)-generic over Hz(S) for all z ≥S x.• On a cone of z:

– δzS = ωL[S,z]2 is inaccessible in Hz(S).

– Qz(S) ⊆ Hz(S)|δzS and hence, essentially, Qz(S) ⊆ δzS .– Qz(S) is δzS-cc in Hz(S).

So if Dz(S) is the collection of maximal antichains of Qz(S) in Hz(S), then

Dz(S) ⊆ Hz(S)|δzSand so the structure, 〈Qz(S),Dz(S)〉, is definable in 〈Hz(S)|δzS,∼z

S 〉. Let Sz bethe least ∞-Borel code in Hz(S) so that S ∼z

S Sz. Set

Nz(S) = 〈Hz(S)|δzS,∼zS , Sz 〉

andSz∗ =

∧

A∈Dz(S)

∧

T,T ′∈A

¬(T ∧ T ′) ∧ Sz ∧∧

A∈Dz(S)

∨A

The code Sz∗ is a member of BCHz(S)

δzS

, here we are using the fact that Hz(S) has

a canonical function e : δzSonto−−→ Dz(S)∪BCHz(S)

δzS

which depends on L[S, z] and S,but not on z. So z �→ Sz

∗ is S-invariant. Take S∞∗ to be the corresponding object

in the ultra power. A similar comment applies to Nz(S) and we define N∞(S).Notice that for x (in any wellfounded model) there is a Σ1-formula, Φ, so that

Lα(Nz(S),x)[Nz(S), x] |= Φ(x,Nz(S))⇐⇒ x ∈ ASz

∗

and thus for any x (in V ), x is generic over L[N∞(S)] and

AS(x)⇐⇒ Lα(N∞(S),x) |= Φ(x,N∞(S))

⇐⇒ Lδ∞S

+ |= Φ(x,N∞(S)).

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So (δ∞S+, N∞(S),Φ) is an ∞-Borel∗ code which is essentially the same as S∞

∗ ; asis made explicit in the next paragraph.

Clearly, Sz∗ ∈ Hz(Nz(S)) and so S∞

∗ ∈ H∞(N∞(S)). The converse is alsotrue, but this requires a bit of work. For now it suffices to notice that for any x,on a cone of z, x ∈ L[Sz

∗ , z]⇐⇒ x ∈ L[Nz(S), z]. This amounts to showing

x ∈ L[Sz∗ , z]⇐⇒ x ∈ HODL[S,z]

S

The left-to-right direction is clear. Conversely, if x ∈ HODL[S,z]S , then x “is” a code

for itself in Qz(S) and thus all we need is that Qz(S) ⊆ L[Sz∗ , z] on a cone. I leave

this as an exercise. So we have• N∞(S) ≈ S∞

∗ , in fact, L[N∞(S)] = L[S∞∗ ].

• In V , AS = AS∞∗ = A(δ∞

S+,N∞(S),Φ).

It would be nice to see that T ≈ S =⇒ T∞∗ ≈ S∞

∗ , however, I do not see howto show this. It is possible to prove something slightly weaker.

Lemma 5.2. If S ∈ Hz(T ) and T ∈ Hz(S) on a cone of z, so that Hz(S) = Hz(T )on a cone of z, then S∞

∗ ≈ T∞∗ .

Proof. Notice that the hypothesis readily implies L[T, x] = L[S, x] on a coneso this is a strengthening of S ≈ T . To prove S∞

∗ ≈ T∞∗ it suffices to show that for

a cone of zx ∈ L[Sz

∗ , z]⇐⇒ x ∈ L[T z∗ , z]

We will actually see that for a cone of z, L[Sz∗ , z] = L[T z

∗ , z]. As explained above wemay show that on a cone of z, L[Nz(S), z] = L[Nz(T ), z]. Since Hz(S) = Hz(T ),this is trivial. ❑

The following lemma appears in [CK09]

Lemma 5.3. The ∞-Borel code Sz∗ satisfies

x ∈ ASz∗ ⇐⇒ x is Qz(S)-generic over Hz(S) and Hz(S)[x] |= x ∈ AS .

Correspondingly,

x ∈ AS∞∗ ⇐⇒ x is Q∞(S)-generic over H∞(S) and H∞(S)[x] |= x ∈ AS∞ .

In both x can come from any transitive model containing S∞∗ and H∞(S) or Sz

∗and Hz(S) respectively. �

Lemma 5.4. For all reals, x, in V :• x is Q∞(S)-generic over L[S∞

∗ ]. This by Łos’ Lemma together with thefact that Nz(S) is a rank initial segment of L[Nz(S)].

• AS = AS∞∗

• S∞∗ ⊆ δ∞S

• S � S∞∗ .

Proof. The first three facts are immediate. For the last fact show S∞ � S∞∗ and

use S ≈ S∞ (see the first line in the proof of Theorem 3.6.) We want to see thaton a cone of z, L[S∞, z] ∩ R ⊆ L[S∞

∗ , z]. In fact, for all z, for a cone of x ≥ z,L[S, z] ∩ R ⊆ L[Sx

∗ , z], since L[Sx∗ , z] = L[Hx(S)|δxS][z] and L[Hx(S)|δxS][z]|δxS =

Hx(S)[z]|δxS . Of course Hx(S)[z] ⊇ L[S, z] and Hx(S)[z]|δxS ⊇ R ∩Hx(S)[z]. ❑

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Arguments reminiscent of those here appear in [KMS83, §13]. In particular,it is shown that if T is a tree on a complete Π1

2n+1 set, so that T as a degree notionis essentially C2n+2, then T∞

∗ is essentially Q2n+3 as a degree notion, so in thesecases T ≺ T∞

∗ .By collapsing δzS to be countable we can produce Hz(S)-generic subsets of

Qz(S). It is shown in [CK09, Lemma 4.6] that:

∃y AS(t, y)⇐⇒ ∀∗z L[Sz∗ ][t]

Col(ω,δzS) |= ∃y ASz∗ (t, y)

⇐⇒ L[S∞∗ ][t]Col(ω,δ∞S ) |= ∃y AS∞

∗ (x, y)

If g is Col(ω, δ∞S )-generic over L[S∞∗ , t], then “∃y AS∞

∗ (x, y)” is Σ11(x, code for S∞

∗ )

and so absolute to L[S∞, t][g]|ωL[S∞,t][g]1 . Since

ωL[S∞,t][g]1 =

(δ∞S

+)L[S∞,t]<

(δ∞S

+)V ,

the supremum of these ordinals is ≤ δ∞S+ and so

∃y AS(t, y)⇐⇒ Lδ∞S

+ [S∞∗ , t]Col(ω,δ∞S ) |= ∃y AS∞

∗ (x, y).

From Lemma 2.3 this gives an ∞-Borel code ∃RS ∈ BCδ∞S

+ ∈ L[S∞∗ ] such that

∃y AS(x, y)⇐⇒ A∃RS(x).

This shows that ∃RS � S∞∗ and that if κ is a limit cardinal such that for all

S ⊆ λ < κ, δ∞S < κ, then B<κ is closed under real quantification.Finally we can prove the promised uniformization result which is in essence just

a variant of the construction of the code ∃RS.

Theorem 5.5. Assume S ≺ T , then

AS(x, y)⇐⇒ ∃y ∈ H∞(S, T )[x]AS(x, y)

⇐⇒ ∃y ∈ L[N∞(S, T )][x]AS(x, y)

This produces an ∞-Borel uniformization of AS, with code in BCδ∞S,T

+ .

Proof. Clearly∀∗z ∃y ∈ L[S, z]AS(x, y)

Recall every real in L[S, z] is Qz(S)-generic over Hz(S). Let g ⊆ Col(ω, δzS) beHz(S)[x]-generic. I claim that in Hz(S)[x][g] there is y so that AS(x, y). If not,then this is forced by some p ∈ Col(ω, δzS). We can in V build a Hz(S)[x] genericthrough p which would allow us to build a Hz(S)-generic, y, for Qz(S) such thatAS(x, y). So we have

∀∗zHz(S)[x]Col(ω,δzS) |= ∃y AS(x, y).

Since Hz(T, S)[x] can find Hz(S)[x] generics for Col(ω, δzS) on a cone, we have thatif ∃y AS(x, y), then

∀∗zHz(S, T )[x] |= ∃y AS(x, y),

hence,H∞(S, T )[x] |= ∃y AS∞(x, y).

This produces an S-invariant uniformization of AS using the canonical wellorderof H∞(S, T ). The argument where Hz(S, T )[x] is replaced by L[Nz(S, T ), x] is thesame. ❑

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If the relation AS starts out with countable slices, then we get a slightly betteruniformization result.

Lemma 5.6. Suppose S is an∞-Borel code for a many-countable relation, then forall x, for a cone of z, (AS)x ⊆ Hz(S)[x]. Consequently, for all x, (AS)x ⊆ H∞(S)[x].

Proof. For any x, fix z0 so that for any z ≥ z0, (AS)x ∈ H(ω1)L[S,z]. We can find a

g ⊆ Col(ω, δzS) generic g over Hz(S)[x] and name u so that (AS)x = {(u[g])i :i ∈ ω}.Assume there is i so that �Hz(S)[x]

Col(ω,δzS) AS(x, (u)i)∧ (u)i /∈ V . Then we could use (u)i

to build a perfect set of y so that AS(x, y). So for all z ≥ z0, (AS)x ⊆ Hz(S)[x].We can replace Hz(S) and H∞(S) by Nz(S) and N∞(S) respectively and thus

get (AS)x ⊆ L[N∞(S), x] = L[S∞∗ , x]. ❑

It essentially follows from this that S ≺ T iff H∞(S, T ) (or L[N∞(S, T )]) can“uniformize” y /∈ L[S∞

∗ , x] in a sense made precise below. Set

DS(x, y)df⇐⇒ y /∈ L[S, x].

Theorem 5.7. If T is an ∞-Borel code for a uniformization of DS , then S ≺ T∞∗ .

If S ≺ T , then, (S, T )∞∗ gives a code for a uniformization of DS .

Proof. For the first claim, just apply the preceding lemma to get that on a coneof z

(DS)x ⊆ L[T∞∗ , x]

So L[T∞∗ , x] ∩ R �⊂ L[S∞

∗ , x] on a cone of x, i.e., T∞∗ � S∞

∗ , and hence S∞∗ ≺ T∞

∗ .For the converse, note that there is code S′ for DS so that S′ ∈ L[S] so

S′ � S ≺ T and Theorem 5.5 gives N∞(S′, T ) ≈ (S′, T )∞∗ yields a code for auniformization of S′. ❑

While it need not be the case that S ≈ S∞∗ (recall S ≈ S∞), it is true that if

S is a non-maximal degree, then S∞∗ is also non-maximal. This gives the following

corollary:

Corollary 5.8. The following are equivalent(1) S is a non-maximal degree.(2) DS is uniformized by an ∞-Borel set.(3) δ∞S < ΘL(B∞,R). (See Theorem 5.13 and Theorem 5.16) �

5.2. The extent of Suslin sets under the non-existence of a maximal

degree. Following Becker [Bec85], a strongly closed pointclass, Λ (recall δΛ =wΛ), of countable Wadge cofinality has the Kunen-Martin property iff

(δΛ)+ = δ11(Λ)

or equivalently ifw(A)+ = δ11(A)

for A =⊕

i∈ω Ai where 〈Ai : i ∈ ω〉 is Wadge cofinal in Λ.If δΛ is a Suslin cardinal, then Π1

1(Λ) is scaled and SδΛ = Σ11(Λ). The Kunen-

Martin theorem gives that σ11(Λ) = (δΛ)

+, recall δ11(Λ) = σ11(Λ) in this case, so

the Kunen-Martin property holds. Becker showed in [Bec85], assuming some ad-ditional closure for Λ, that

Λ-Uniformization + the Kunen-Martin property holds ⇐⇒ δΛ is Suslin

The following question is apparently still open and relevant for part of our analysis.

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Question. Does the Kunen-Martin property hold for all type I hierarchies?

Woodin answered this question assuming Λ ⊆ B∞.

Theorem 5.9 (Woodin). For Λ strongly closed with countable cofinality, if Λ ⊆B∞, then the Kunen-Martin property holds, i.e., δ11(Λ) = (δΛ)

+. �Woodin also showed that AD+uniformization =⇒ all sets are ∞-Borel, so that

the Kunen-Martin property holds for all strongly closed pointclasses of countablecofinality. The proof, as far as I know, does not appear in print and its techniquesare similar to those being discussed here, so I am including it. His argument worksin a more general setting: Rather than AD + DC we shall assume the existence ofa fine σ-complete measure, μ, on Pω1

(R), uniformization, and that∏

Pω1(R) ω1/μ is

wellfounded.The existence of a fine measure on Pω1

(R) is guaranteed by Turing-determinacy,that is, determinacy for all Turing invariant sets. Turing determinacy is equivalentto the cone filter on Turing degrees being an ultrafilter and this ultrafilter readilyinduces a fine σ-complete measure on Pω1

(R), under DCR. The assumption that“∏

Pω1(R) ω1/μ is wellfounded” is implied by DC and implies DCR, but not DC.

This is discussed in [CK09]. The existence of a fine measure on Pω1(R) implies

that ω1 is measurable, hence choice fails, and that ωV1 is Mahlo in any inner model

of choice. This is also discussed in [CK09], where these same hypotheses are usedto derive several results.

For P a poset let the collection of canonical names for reals, RP, be the collectionof P-names satisfying:

(1) For all n, for densely many p there is m ∈ ω with (p, (n,m)) ∈ τ .(2) For (p, (n,m)), (q, (n,m′)) ∈ τ with p ‖

Pq =⇒ m = m′.

If τ is any P-name for a real, then τ∗ = {(p, (n,m)):p � τ (n) = m} ∈ RP, conversely,whenever g ⊆ P is a “sufficiently generic” filter and τ ∈ RP, then τg ∈ R, here gneed only meet each Dτ

n = {p : ∃m (p, (n,m)) ∈ τ}.For A ⊆ R, the term relation for A is defined as

A(P, p, τ )df⇐⇒(1)P a poset

(2) τ ∈ RP

(3) ∀∗g ⊆ P(p ∈ g → τg ∈ A

),

where ∀∗g ⊆ Pϕ(g) means that there is a countable collection of P-dense sets, D,in V , such that if g is a D-generic filter, then ϕ(g) holds, in V . For M a transitiveinner model, the M -term relation for A is defined by A

M df= A ∩M . In general,

there is no reason for AM

to be in M (or amenable to M) and when this occurs Mis called weakly A-closed. Similarly, M is weakly A,P-closed iff AP ∩M ∈M .

If M is weakly A,P-closed and weakly ¬A,P-closed, and

(†) for all τ ∈ RP ∩M , DAP,τ = {p : p �∗

Pτ ∈ A ∨ p �∗

Pτ /∈ A} is dense in P,

then∀∗g ⊆ PAM

P[g] = A ∩M [g].

Above “p �∗Pτ ∈ A” means there is a countable collection, D, of P-dense sets so

that for any D-generic filter, τg ∈ A, similarly for “ /∈”.

Claim. The condition (†) is guaranteed by all sets having the Baire property.

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Proof. This is essentially a standard forcing fact. There is a dense embeddingπ : ω<ω onto−−→ P0 ⊆ P with P0 a dense sub-poset of P, and a dense Gδ set E ⊆ ωω

so that π : Eonto−−→ XP where XP is the set of P-filters. Now define fτ : E → R by

x �→ τπ(x). For p ∈ P0, look at Eτ,p = {x ∈ E : p ∈ π(x) ∧ fτ (x) ∈ A}. If this setis comeager in [π−1(p)], then let E be a countable collection of open dense sets inω<ω so that if x is E-generic, then x ∈ E and fτ (x) ∈ A. Let D = π[E], then if g isD-generic, then τg ∈ A, and hence D witnesses p �∗

Pτ ∈ A. If Eτ,p is not comeager,

then there is q ≤P p so that ¬Eτ,p is comeager in [π−1[q]] and one argues as abovethat there is D so that if g is D-generic, then q ∈ g → τg /∈ A. ❑

If we strengthen weak A,P-closure to

AM

P∈M and for all M -generic g ⊆ P (in V ), AM

P[g] = A ∩M [g],

then call M strongly A,P-closed. M is said to be weakly (strongly) A-closed iff Mis weakly (strongly) A,P-closed for all P ∈M .

It is a relatively simple matter to produce weakly A-closed structures, namely,

NAx = Lω1

[A, x] and MAx = HODNA

x

Aare such, since A

NAx = A ∩NA

x .To produce strongly A-closed structures is trickier since we must ensure that

the model has “enough” dense sets, this is where uniformization is used. Notice, inthe argument that follows, essentially it is proved that assuming

(1) ZF,(2) all sets have the Baire property,(3) the existence of a fine measure, μ, on Pω1

(R), and(4)

∏Pω1

(R) ω1/μ is wellfounded,

then the following are equivalent:(1) A is ∞-Borel.(2) There is a uniform sequence 〈Mσ : σ ∈ Pω1

(R)〉 of strongly A,¬A-closedtransitive substructures of H(ω1), such that each y ∈ σ is generic over Mσ

for a poset Pσ ∈Mσ.

Theorem 5.10 (Woodin). Work in ZF. Suppose all subsets of R have the prop-erty of Baire, there is a fine measure , μ, on Pω1

(R), and that∏

Pω1(R) ω1/μ is

wellfounded. Then UNIF implies that all sets are ∞-Borel.

Proof. Fix a set A ⊆ R with the aim being to show that A is ∞-Borel. Definethe relation BA on reals by BA(x, y) iff x codes (P, τ ), where P is a countable posetand τ is a canonical P-name for a real, and y codes D with DA

P,τ ∈ D such that forg that is D-generic and p ∈ g ∩DA

P,τ

τg ∈ A⇐⇒ p �∗Pτ ∈ A (and so also τg /∈ A⇐⇒ p �∗

Pτ /∈ A),

or, equivalently, for g ⊆ P that is D-generic,

(‡τ ) τg ∈ A⇐⇒ ∃p ∈ g AP(p, τ ) and τg /∈ A⇐⇒ ∃p ∈ g ¬AP(p, τ ).

Uniformization of BA can be used to select witnesses to weak closure. Let B∗A

uniformize BA and set A∗(x, i, j)df⇐⇒ B∗

A(x)(i) = j.Set

Nσ = Lω1[A,¬A, A∗

, σ] and Mσ = HODNσ

A,¬A,A∗ .

These are model of ZFC since ω1 is measurable in V .

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Claim. Mσ is strongly A-closed.

Proof. Let P ∈ Mσ and let Q = Col(ω,P) and fix τ ∈ RP ∩Mσ. Let z be theinduced generic coding of (P, τ ). Pick Eτ a countable collection of dense subsets ofQ so that for any Eτ -generic, G, for all i ∈ ω, there is j ∈ ω such that

(A∗)Mσ

P[G] ⊆ A∗ ∩Mσ[G] and (A

∗)Mσ [G](zG, i, j)

This is essentially just the weak A∗,Q-closure of Mσ. For G that is both Mσ-genericand Eτ -generic, B∗

A(zG) ∈Mσ[G].Let DG be the countable collection of P-dense sets coded by B∗

A(zG), thenDG ∈Mσ[G]. If g ⊆ P is Mσ[G]-generic, then as DG ⊆Mσ[G], g is DG-generic andthus (‡τ ) holds for every canonical P-name, τ , in Mσ.

The order of the forcings can be inverted so that, in Mσ[g][G], (‡τ ) holds forall τ ∈ RP ∩Mσ. This is independent of G and thus (‡τ ) holds in Mσ[g] for allτ ∈ RP ∩Mσ. Thus for g ⊆ P which are Mσ-generic, for all τ ∈ RP ∩Mσ,

τg ∈ A⇐⇒ ∃p ∈ g A(P, p, τ ) and τg /∈ A⇐⇒ ∃p ∈ g ¬A(P, p, τ )so

AMσ

P[g] = A ∩Mσ[g]

and hence Mσ is strongly A,P-closed. ❑

Now let Pσ = BMσ∞ / ∼σ where for T, T ′ ∈ BMσ

∞ , T ∼σ T ′ df⇐⇒ ANσ

T = ANσ

T ′ .This is just the ∞-Borel version of the Vopenka algebra, and as with the Vopenkaalgebra, for any y ∈ Nσ, Gy

σ = {T/ ∼σ : y ∈ AMσ

T } is Mσ-generic. For any y

with y ∈ Nσ, we have from the claim that, y ∈ A ⇐⇒ y ∈ AMσ

Pσ[Gy]. Letting

TσA =

∨{T ∈ Pσ : A

Mσ

Pσ(T, τ )}, it follows that

y ∈ AMσ

Pσ[Gy]⇐⇒ y ∈ ATσ

A

and hence A ∩Nσ = ATσA∩Nσ so that , locally anyway, A is ∞-Borel.

Let (M∞,P∞, T∞A ) =

∏σ(Mσ,Pσ, T

σA), then for y ∈ V , y is M∞-generic for P∞

andy ∈ A⇐⇒M∞[y] |= y ∈ AT∞

A

Thus T∞A is an ∞-Borel code for A. ❑

Working in ZF + AD + uniformization, let μ be the measure induced from thecone measure on Turing degrees, then the ultrapower taken in the proof could betaken in L(R, A,A∗) and since uniformization implies DCR, and since DCR in Vgives DC in L(R, A,A∗), it follows that

ZF + AD + uniformization =⇒ all sets are ∞-Borel

Corollary 5.11. In the case that there is no maximal degree notion, L(B∞,R)is the maximal model of AD + uniformization. �

Corollary 5.12. The following are equivalent under ZF + DC:(1) AD + uniformization.(2) AD + all sets are Suslin.(3) AD+ + all sets are Suslin.(4) ADR.

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Proof. (2) =⇒ (3) follows from the discussion of AD+ in the preliminary section,namely the discussion of downward absoluteness (see page 33 and Lemma 2.5).That (2) =⇒ (4) (and hence (3) =⇒ (4)) is due independently to Martin andWoodin (unpublished). (4) =⇒ (1) is a simple exercise, in fact each player needonly make a single real move.

Only (1) =⇒ (2) needs discussion here. Given any set A, uniformization givesthat we can find a minimal strongly closed pointclass Λ containing A such that Λ-uniformization holds. DC implies w(Λ) < Θ and by the results of Woodin, Λ ⊆ B∞and Λ has the Kunen-Martin property. So Becker’s result gives that w(Λ) is a Suslincardinal. ❑

5.3. Bounds on δ∞S . Let f be an S-invariant function with f(x) = Mx ∈H(ω1)

L[T,x] a transitive structure (on a cone). Say that y codes Mx iff (ω,∈y) #(Mx,∈) and set

Rf (x, y)df⇐⇒ y ∈ R ∩ L[T, x] codes f(x).

Rf can be uniformized as follows:

R∗f (x, y)

df⇐⇒ y is the L[T, x]-least y, such that Rf (x, y).

Let f∗(x) = y ⇐⇒ R∗f (x, y), f∗ need not be S-invariant. If R∗

f (x, y) let πx :

(ω,∈f∗(x)) # Mx be the induced isomorphism and set nx = πx(n) ∈ Mx. For hany S-invariant function such that on a cone, h(x) ∈ Mx, there n ∈ ω, such thatA = {x : nf(x) = h(x)} is S-positive. This n need not be unique. Together with f ,the pair (n,A) “codes” h since for a cone of x,

h(x) = nf(z) for any z ∈ [x]S ∩A.

A can be replaced by an S-pointed perfect tree, a, such that [a] ⊆ A, so that thecodes are reals. Let Cf be the set of all such codes, that is,

(n, a) ∈ Cf ⇐⇒ a is an S-pointed perfect tree andfor x, y ∈ [a], if x ≡S y, then nx = ny

For (n, a) ∈ Cf set h(n,a)([x]S) = nz, where z ∈ [x]S ∩ [a]. Set (n, a) ∼ (n′, a′) iffh(n,a) = h(n′,a′) on a cone. Let [n, a]f be the equivalence class of (n, a) and set

[n, a]f ∈f [n′, a′]fdf⇐⇒ ∀∗x (h(n,a)(x) ∈ h(n′,a)′(x))

Then (Cf/ ∼,∈f ) is isomorphic to∏

S Mx/μ = M∞S . By looking at the complexity

of this coding we get bounds on δ∞S .

Theorem 5.13. If S ≺ T with S, T ⊆ γ < λ with λ a measurable cardinal and≤ is a prewellordering of length λ, then δ∞S ≤ δ(≤).

Proof. Let f(x) = HxS , then f(x) ∈ H(ω1)

L[T,x] and R∗f is ∞-Borel since

R∗f (x, y)⇐⇒ L[S, T, x] |= “ y =<L -least z such that (ω,∈z) # (Hx

S ,∈) ”

Let (ϕ, S, T ) be the corresponding code in BC∗γ ⊆ BCκ. Since Bκ ⊆ B<δ(≤) ⊆ Δ(≤),

the relation R∗f (x, y) is Δ(≤). Similarly, x ≤S y is Δ(≤), since x ≤S y ⇐⇒ x ∈

L[S, y]⇐⇒ L[S, x, y] |= x ∈ L[S, y].First compute the complexity of (n, a) ∈ Cf :(1) a is S-pointed perfect.(2) x, y ∈ [a] and x ≡S y → nx = ny. (S-invariance.)

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For (1): a is S-pointed perfect iff

∀s ∈ [a]∃s′, s′′ ∈ [a] (s′ ⊥ s) & ∀x (x ∈ [a]→ a ≤S x)

For (2):

∀x, y[(x, y ∈ [a] and x ≡S y

)→

∃u, v (R∗f (x, u) & R∗

f (y, v) and (ω,∈x) # (ω,∈y)))]

So both (1) and (2) are Π(≤) and thus Cf ∈ Π(≤).Similarly, ∈f is Π(≤) in the codes. Restricting to codes for ordinals, i.e.,

h(x) ∈ δxS , produces a Π(≤) norm on Cf and thus δ∞S ≤ δ(≤). ❑

All we actually use below is the following:

Corollary 5.14. If Λ is strongly closed and there is no maximal degree in Λ,then for all S ∈ Λ, δ∞S < δΛ. If S is not a maximal degree, then δ∞S < Θ. �

Corollary 5.15. If κ is a limit of Suslin cardinals, then for all S ⊆ λ < κ,δ∞S < κ.

Proof. The point is that S<κ = Λ is strongly closed and each A ∈ Λ has a Suslinrepresentation in Λ. If S ∈ Λ, then DS(x, y)

df⇐⇒ y /∈ L[S, x] is in Λ. So DS isSuslin by a tree in Λ and hence uniformized in Λ. Let D∗

S be the uniformizationand let S′ be a tree in Λ projecting to D∗

S . We see S ≺ S′ since for any x,∃y (x, y) ∈ [S′]⇐⇒ ∃y ∈ L[S′, x] (x, y) ∈ [S′]. (This is similar to 5.7.) ❑

Compare these corollaries to Theorem 5.16 which shows that if S is maximal,then δ∞S = Θ and Corollary 6.4 which shows that if κ < κ∞ is Suslin, thenδ∞S ≥ λ where λ is the next Suslin past κ.

5.4. Maximal degree. Call a degree notion, S, strongly maximal iff

For all A ⊆ R, on a cone of xA ∩ L[S, x] ∈ L[S, x]

If S is maximal and every set of reals is ∞-Borel, then S is strongly maximal.Conversely, if S is strongly maximal, then L(P(R)) = L(S∞,R) and so all sets are∞-Borel.

Theorem 5.16. If S is strongly maximal, then L(P(R)) = L(S∞,R) and δ∞S =Θ.

Proof. We have that ∀∗x (A ∩ L[S, x] ∈ L[S, x]). Set A(x) = L[S, x] ∩ A, this isan S-invariant function. Unfortunately the well ordering of A(x) is not S-invariant.Let hL[S,x]|δxS : δxS

onto−−→ L[S, x]|δxS be the canonical Σ1-Skolem function. GCH∗

holds on a cone so A(x) = h(α(x)) for some minimal α(x) < δxS . The function α(x)is not S-invariant, since the well ordering of L[S, x] depends on x. So let G be PS

generic and let α∞S = [x �→ α(x)]G. We want to see that

hL[S∞,x∞S ]|δ∞S (α∞

S ) ∩ RV = A

For y ∈ RV , we have

y ∈ hL[S∞,x∞S ]|δIS (α∞

S )⇐⇒ {x : y ∈ hL[S,x]|δ∞S (α(x))} ∈ G

⇐⇒ {x : y ∈ A ∩ L[S, x]} ∈ G

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Since {x : y ∈ A ∩ L[S, x]} is S-invariant we have

{x : y ∈ A ∩ L[S, x]} ∈ G⇐⇒ y ∈ A

This is what we wanted to see.We have shown here that for all sets of reals A, for any S∞-pointed perfect

Sacks real x∞S , A ∈ L(S∞,R)[x∞

S ] and thus A ∈ L(S∞,R) to begin with. SoL(P(R)) = L(S∞,R).

From Theorem 3.6, δ∞S = ωL[S∞,x∞

S ]2 ≤ Θ and GCH∗ holds in L[S∞, x∞

S ].For each A ⊆ R (in V ) we have A = A∞ ∩ RV for some A∞ ∈ L[S∞, x∞

S ] and so|P(R)V | ≤ |P(R)L[S∞,x∞

S ]| = ωL[S∞,x∞

S ]2 = δ∞S and thus Θ ≤ δ∞S . ❑

The following theorem is due to Woodin and appears in [Ste94].

Theorem 5.17 (Woodin). If V = L(S,R) and ZF+AD+DCR holds, then thereis T ⊆ Θ such that HODS,x = L[T, S, x] for all x. �

As a consequence of this theorem T, S is a largest degree notion in L(S,R). ThisDS,T (x, y) ⇐⇒ y /∈ HODL(S,R)

S,x can not be uniformized, since any uniformization,

F , must be ODL(S,R)S,x for some x and hence F (x) ∈ HODL(S,R)

S,x . This yields

Corollary 5.18. (ZF+DCR +AD) If there is no maximal degree notion, thenV �= L(S,R) for any S ⊆ OR. �

For S a maximal degree, Theorem 5.16 and variants give that δ∞S is large:(1) Assuming S is a “strongly maximal degree notion” in the sense that A ∩

L[S, x] ∈ L[S, x] for all A ⊆ R we have δ∞S = Θ.(2) If V = L(T,R), then every set is ∞-Borel so V = L(S∞,R) and δ∞S = Θ

by (1).(3) In general, if there is a maximal degree notion S, then B∞ ⊆ L(S∞,R)

and, conversely, L(S∞,R) is a model of all sets are ∞-Borel. HenceL(B∞,R) = L(S∞,R) and letting ΘB∞ = ΘL(B∞,R), we have δ∞S = ΘB∞ .

In general, if there is a largest degree notion S, then it need not be the case thatS has size κ∞, however, if κ∞ is a Suslin cardinal, then any tree, S, on ω × κ∞witnessing this is strongly maximal. This follows from the following theorem.

Theorem 5.19 (Woodin). Suppose κ∞ is Suslin and S is a tree on ω × κ∞witnessing this, then S is strongly maximal.

This will require some results of Martin which appear in [Jac10]. For a non-selfdual pointclass Γ and A ∈ Γκ set

N(A) = {A : ∀σ ∈ Pω1(R)∃α < κ (A ∩ σ = Aα ∩ σ)}

andEnv(Γ, κ) = {B :B ≤w A such that ∃A ∈ Γκ A ∈ N(A)}.

Call Γ nice if Γ has the prewellordering property and is closed under ∀R and ∨. Fornice Γ set Env(Γ) = Env(Γ, δΓ). It is shown in [Jac10] that for nice Γ

Env(Δ) = Env(Γ) = Env(∃RΓ).

Let ϕ : Conto−−→ δΓ be a Γ-norm where C ∈ Γ \ ¬Γ. Let U be universal ∃RΓ and D

be the set of codes for subsets of δΓ; so for t ∈ D

(1) U(t, x)→ x ∈ C and

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(2) U(t, x) & ϕ(x) = ϕ(x′)→ U(t, x′).For t ∈ D set π(t) = {ϕ(x) : U(t, x)}. For U a measure on δΓ set

U∗ = {t : π(t) ∈ U}It is shown in [Jac10] that U∗ ∈ Env(Γ) for nice Γ.

For S a set of ordinals define

Env(S) = {A ⊆ R : ∀∗xA ∩ L[S, x] ∈ L[S, x]}.Since GCH∗ holds on a cone, we have

Env(S) = {A ⊆ R : ∀∗xA ∩ L[S, x]|δxS ∈ L[S, x]|δxS}.For nice Γ such that ∃RΓ is scaled, ∃RΓ = Sκ for κ = δΓ and

Env(S) = Env(Γ)

where S is the tree of a scale on a complete Γ-set.Letting meas(κ) be the set of measures on κ and let meas∗(κ) be the set of

codes, we have:

Lemma 5.20. Suppose κ is a Suslin cardinal, Sκ = ∃RΓ where Γ is nice. Thenmeas∗(κ) ⊆ L(S∞,R), for S the tree of a scale on a complete Γ set. �

Martin and Woodin [MW08] have shown that if meas∗(κ) is bounded in theWadge degrees, then any tree on κ is weakly homogeneous and thus κ is not thelargest Suslin cardinal.

Proof of Theorem 5.19. Assume κ∞ is Suslin and let Γ = Sκ∞ , then Γ is niceand ∃RΓ = Γ. If S is a tree on ω × κ∞ witnessing κ∞ is Suslin, then δ∞S = Θ.This means that ΘL(S∞,R) = Θ and thus L(S∞,R) = L(P(R)). Thus S is stronglymaximal. ❑

6. Strong ∞-Borel codes

Definition 6.1. An ∞-Borel code S ⊆ κ is strong if player II wins the followinggame Gstrong(S). Player I and II take turns playing ordinals below κ. In the endf ∈ κω is played and we let Sf be the collapse of (f [ω], S ∩ f [ω]). II wins if Sf isa Borel code, i.e., in BC<ω1

and ASf⊆ AS as computed in V .

In a world with the axiom of choice this amounts to saying that a club ofσ ∈ [κ]ω satisfies ASσ

⊆ A. Without choice, the club must be witnessed by astrategy.

The game described can be cast appropriately so that ordinal determinacy willyield the determinacy of the game either assuming AD+ or that κ is below thesupremum of the Suslin cardinals. See [CK09, §2.2.4] for more on AD+, ordinaldeterminacy and references. The point is that the map f �→ Sf is continuous whereSf is the canonical coding of Sf by a real given f as input, i.e., an enumeration off [ω]. The winning condition is that Sf be a Borel code and ASf

⊆ AS .Clearly if T is a tree on ω × κ, then T is a strong ∞-Borel code for p[T ] since

II need only ensure that T ∩ σ be sufficiently elementary in S, in this case, T ∩ σmust be a tree on ω × σ. Then p[Tσ] = p[T ∩ σ] ⊆ p[T ] just by absoluteness. Thatthe converse holds is the content of the next theorem.

Theorem 6.2. If S ⊆ κ is a strong ∞-Borel code, then there is a tree T on κwith p[T ] = AS, in particular, AS is κ-Suslin.

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Remark. The map S �→ T depends on a winning strategy in the game describedabove and there is no uniform way to produce those winning strategies.

Proof. Let σ : κ<ω → κ be II’s winning strategy in Gstrong(S). Then let (x, f) ∈[T ] iff S∩f0[ω] is σ-closed and f1 witnesses x ∈ ASf0

. We need to show x ∈ AS ⇐⇒x ∈ p[T ].

If x ∈ p[T ], then let (x, f) ∈ [T ] and let g be a play of the game with IIusing σ and g[ω] = f0[ω]. Then x ∈ ASf0

⊆ AS . If x ∈ AS , then take α > κ sothat Lα[S, x, σ] is a model of some reasonable fragment of ZFC and Lα[S, x, σ] |=“ S is an ∞-Borel code & x ∈ AS ”. Let {S, x, σ} ⊆ N ≺ Lα[S, x, σ]. Choose f0 sothat f0[ω] = N ∩ κ. Since Lα[S, x, σ] |= “ S is an ∞-Borel code & x ∈ AS ”, wehave S = Sf0[ω], Sf0[ω] is an ∞-Borel code, and x ∈ Sf0[ω], so we can choose f1witnessing this. This f satisfies (x, f) ∈ [T ], as desired. ❑

While the passage from a strong ∞-Borel code to the corresponding Suslinrepresentation is not uniform, it turns out that the passage from ∞-Borel code tocorresponding strong code is uniform and this will yield the closure of the Suslincardinals. Recall the codes Sx

∗ and S∗ from Lemma 5.3.

Theorem 6.3. If δ∞S -determinacy holds, then S∗ is strong.

Proof. By assumption Gstrong(S∗) is determined so we need only show that I doesnot win. Suppose σ is a I winning strategy. We aim to produce a play f consistentwith σ so that (S∗)f = Sx

∗ for some x. Since ASx∗ ⊆ AS = AS∗ this will yield a

contradiction.On a cone of x, consider the closed game Gx where in round i player I plays

α2i < δxS and II plays α2i+1 < δxS , βi < δ∞S , and ki ∈ 2. In the end f ∈ (δxS)ω,

g ∈ (δ∞S )ω, and x ∈ 2ω are played. For II to win, g[ω] must be σ-closed, and xmust determine a map πx : f [ω]→ g[ω] which must be an embedding of Sx

∗ ∩ f [ω]into S∗. This game is closed for II.

Let G∞ be the corresponding game played in H∞S (σ∞

S , S∗, S∞∗ ). Let G be

generic over V for collapsing δ∞S to ω and have II play so that f [ω] = δ∞S , g[ω] =jS [δ

∞S ], and x codes jS : δ∞S

onto−−→1-1

jS [δ∞S ]. This play is winning against any play by

I. By absoluteness for winning a closed game, II wins G∞ in H∞S (σ∞

S , S∗, S∞∗ ). So

by Łos’ lemma, II wins Gx on a cone of x with canonical strategy τx.Now fix x in the cone where II wins and have II play τx against an enumeration

of δxS and let π : δxS → δ∞S be II’s isomorphism and g ∈ (δ∞S )ω be II’s enumerationof π[δxS ]. Since g[ω] is σ-closed, σ(g)[ω] = g[ω] and so Sx

∗ #π S∗. So g is a play byII that defeats σ. ❑

Corollary 6.4. If κ < κ∞ is Suslin as witnessed by a tree S on ω × κ, thenλ ≤ δ∞S where λ is the next Suslin cardinal after κ.

Proof. Since κ < κ∞, δ∞S < κ∞ by Theorem 5.13 and the corollaries followingit. Set x ∈ A ⇐⇒ x /∈ p[S], and note that A is not κ-Suslin. Fix ϕ so thatA = A(ϕ,S) and let S be the corresponding ∞-Borel code. Then S∗ is a strong∞-Borel code of size δ∞S so A is δ∞S -Suslin and so δ∞S ≥ λ. ❑

Theorem 6.5. Suppose that κ is a limit of Suslin cardinals and that κ-ordinaldeterminacy holds, then κ is Suslin.

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The hypotheses are satisfied if either κ < κ∞ or κ < Θ and AD+ holds. TheSuslin cardinals do form an ω-club, so if cf(κ) = ω, there is nothing to do, henceassume cf(κ) > ω. The following lemma is reminiscent of 2.5.

Lemma 6.6. Let λ be a cardinal and suppose B<λ � B∞. Then B<λ has aλ-length antichain.

Proof. First suppose λ is regular. Let ∼ be the relation on BC∞ given by T ∼T ′ df⇐⇒ AT = AT ′ . Let γ be least such that Bγ \B<λ �= ∅ and let S ∈ BCγ \BC<λ

code such a set. Work now in L[S,∼]. For readability let Q<λ = BCL[S,∼]<λ / ∼

and let Q = BCL[S,∼]λ / ∼. The point is that Q is a countably generated complete

Boolean algebra in L[S,∼] and hence for any cardinal λ of L[S,∼], the “λth level”of Q, essentially Q<λ, is either already complete or has a cfL[S,∼](λ)-sized antichainunbounded in Q<λ.

In L[S,∼], λ is regular and there is a least γ′ ≤ γ so that L[S,∼] |= Qγ′ \Q<λ �= ∅. Choose S′ ∈ BCL[S,∼]

γ′ witnessing this. Then S′/ ∼=∨

α<γ′ Sα/ ∼ with

Sα ∈ BCL[S,∼]<λ . Now inside L[S,∼] take S′

α ∈ BCL[S,∼]<λ so that S′

α/ ∼=∨

α′<α Sα.We can thin out this sequence to a strictly increasing sequence in Q<λ, 〈S′

αξ:ξ < ρ〉,

where ρ ≥ cf(λ). This gives an antichain of length cfL[S,∼](λ) of codes in BCL[S,∼]<λ .

Since λ is regular in L[S,∼] this does it.The preceding paragraph actually showed that for any λ, if Q<λ �= Q, then a

sequence 〈Sα ∈ BCλα: α < ρ〉, where ρ = cfL[S,∼](λ), can be found in L[S,∼] such

that Sα/ ∼ and Sα′/ ∼ are incompatible for α < α′ < ρ.Suppose now that λ is singular in L[S,∼] with supα<ρ λα = λ λα < λα′ for

α < α′ < ρ where ρ = cfL[S,∼](λ). We may assume the Sα from the precedingparagraph are of the form Sα =

∨γ<λα

Sα,γ with 〈Sα,γ : γ < λα〉 an antichain inQλα

. This gives us an antichain in Q<λ of length λ. ❑

Corollary 6.7. If κ < κ∞ is a limit of Suslin cardinals, then there is a κ-lengthantichain in B<κ.

Proof. The only point is that B<κ �= B∞. ❑

Lemma 6.8. If Sα ∈ BC<κ is a strong code for all α < κ, then S =∨Sα is strong.

Proof. All we need to do is see that player I cannot win Gstrong(S). Suppose Idid win with σ. Since cf(κ) > ω take α closed under σ so that Sα ∈ BCα. Nowhave II play a winning strategy σα in Gstrong(Sα) against σ. Let f ∈ αω be theresulting play. A(Sα)f ⊆ ASα

⊆ AS . So this is a win for II. This contradictionshows that II must win Gstrong(S). ❑

Now we can easily prove the theorem:

Proof of Theorem 6.5. Let κ be a limit of Suslin cardinals and assume κ-ordinal determinacy holds and cf(κ) > ω. Let 〈Sα : α < κ〉 be an antichain byLemma 6.6. For each α let Sα,∗ be the associated strong code. Sα,∗ ∈ BC<κ byCorollary 5.15. By Lemma 6.8, T =

∨β<κ

(∨α<β(Sα,∗ × Sβ,∗)

)is strong and

hence AT is κ-Suslin. AT is a prewellordering of length κ and hence is not λ-Suslinfor any λ < κ, by the Kunen-Martin theorem. So AT witnesses that κ is a Suslincardinal. ❑

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6.1. Equivalence of Suslin cardinals and reliable cardinals. Recall Def-inition 2.1 and the discussion around it. For a tree T on ω × κ set T ′ df

= T {ϕTi }i ,

that is, T ′ ⊆ T is the tree induced by the semiscale associated to T , i.e.,

T ′ = {(x|i, bTx |i) : x ∈ p[T ] and i ∈ ω}A tree T , on ω × κ, is a tree of a scale if T = T ′. A cardinal κ is called reliable ifthere is a tree on ω×κ such that T is the tree of a semiscale and |T | = κ. Call a treeT , as in the definition of reliability, a witness to the reliability of κ. If κ is reliable,then it is possible to find a reliability witness such that ∀α < κ∃x ∈ p[T ] bTx (0) = α,i.e., ϕT

0 : p[T ]onto−−→ κ, call such a reliability witness a good witness to the reliability

of κ.A Suslin cardinal is reliable; just take T on ω×κ witnessing that κ is Suslin, then

T ′ is a reliability witness. The closure of the Suslin cardinals below their supremum,Theorem 6.5 yields that every reliable cardinal is Suslin, thus providing a direct“structural” way of recognizing the Suslin cardinals.

Theorem 6.9. Every reliable cardinal is Suslin.

Proof. Suppose there is a reliable cardinal, κ < κ∞, that is not a Suslin cardinal.By the closure of the Suslin cardinals below κ∞, κ is not a limit of Suslin/reliablecardinals and hence there is a largest Suslin cardinal λ < κ and Sλ = Sκ.

If γ is the next Suslin cardinal after λ, then λ < κ < γ, so γ �= λ+ andthus [Jac10, Lemma 3.7] gives that cf(γ) = ω and [Jac10, Theorem 3.28] yieldsthat λ is regular and Scale(Sλ). In particular Sλ has the prewellordering propertyand hence is closed under arbitrary wellordered unions. This in turn means thatthere can be no λ+-sequence of mutually disjoint sequence of sets in Sλ, since if〈Aα :α < λ+〉 were such a sequence, x ≺ y

df⇐⇒ (x, y) ∈⋃

β<ξ<λ+ Aβ×Aξ would bea Sλ wellfounded relation of rank λ+ and this violates the Kunen-Martin theorem2.2.

Let T be a good reliability witness for κ. Then Aα = {x ∈ p[T ] : ϕT0 (x) = α}

is a sequence of disjoint non-empty sets of length κ in Sκ = Sλ. This contradictsthe preceding paragraph. ❑

7. Equivalence of AD+ with the closure of the Suslin cardinals below Θ

Recall, working in ZFC for the moment, that for δ a strong limit cardinal andG ⊆ Col(ω,<δ) generic, the set R∗

G =⋃

α<δ RV [G|α] is called the set of symmetric

reals for Col(ω,<δ). That a certain set of reals R∗ is the symmetric reals for somegeneric G ⊆ Col(ω,<δ) can be axiomatized as follows:

(1) Every real x ∈ R∗ is in V [g] for generic g ⊆ P for some P ∈ Vδ.(2) sup{||x|| : x ∈ WO ∩ R∗} = δ.(3) For x, y ∈ R∗, L[x, y] ∩ R ⊆ R∗.

If R∗ is the set of symmetric reals for Col(ω,<δ), then V (R∗) ∩ R = R∗ is a modelof ZF. The following is known as the Derived Model Theorem.

Theorem 7.1 (Woodin). Assume ZFC and that δ is a limit of Woodin cardinalsand R∗ is the set of symmetric reals for Col(ω,<δ). Define

Γ∗AD+ = {A ⊆ R∗ :A ∈ V (R∗) and L(A,R∗) |= AD+}

Then L(Γ∗AD+ ,R∗) |= AD+. �

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The model L(Γ∗AD+ ,R∗) is called the derived model. For R∗ the symmetric

reals for Col(ω,<δ), in V (R∗) define

A ∈ Hom∗ df⇐⇒ A = p[T ] = R \ p[S] for T, S trees on ω × δ

Then Hom∗ = SL(Γ∗

AD+ ,R∗)<∞ .

The derived model theorem gives a way of producing models of AD+ from largecardinals. Starting with a model of ZF + AD + DCR, define

ΓAD+ = {A : L(A,R) |= AD+}.Woodin produced an inner model N in a generic extension of V , so that

(1) δ = ωV1 is a limit of Woodin cardinals in N ,

(2) RV is a set of symmetric reals over N for Col(ω,<δ),(3) (Γ∗

AD+)N(R) = (ΓAD+)V , and(4) (S∞)V = (Hom∗)N(R)

(5) For S ∈ (Γ∗AD+)N(R),

∏S ω

L(S,x)2 /μ is the same computed in V or in

L(Γ∗AD+ ,R)N(R).

The item (5) is a little technical, but we need it below. In particular, it impliesthat if S is a tree in L(Γ∗

AD+ ,R)N(R) witnessing that κ∞ is Suslin, then L(S∞,R)

is the same computed in V or in L(Γ∗AD+ ,R)N(R).

These results imply that every AD+ model is a derived model and more gen-erally, and more importantly for us here, every model of AD contains a maximalclass inner model of AD+ containing the reals, and this maximal model of AD+

contains all of the Suslin sets. So we have the following:

L(S∞,R) ⊆ L(ΓAD+ ,R) ⊆ L(B∞,R) ⊆ L(P(R)),

where in the case that there is no largest degree notion, L(B∞,R) is also themaximal model of AD+uniformization. Moreover, if κ∞ is Suslin, then L(S∞,R) =L(ΓAD+ ,R).

The desired characterization of AD+ models follows almost immediately:

Theorem 7.2 (Woodin). The following are equivalent under ZF + DCR

(1) AD+

(2) AD + The Suslin cardinals are closed below Θ.

Proof. (1) =⇒ (2) has already been discussed. So assume (2) holds. If κ∞ = Θ,then we have AD+ all sets are Suslin and this easily gives AD+. So assume κ∞ <Θ. Fix a tree, S, witnessing κ∞ is Suslin. Theorem 5.19 shows that S is a stronglymaximal degree and hence L(S∞,R) = L(P(R)) for S a tree on ω×κ∞ witnessingκ∞ is Suslin. From the derived model theorem, L(S∞,R) is the maximal model ofAD+. ❑

8. Appendix

Under AD, P(R) has a fair amount of structure. One facet of this is the Wadgehierarchy. For A and B sets of reals, A is Wadge reducible to B, denoted A ≤w B,if there is a continuous reduction of A to B, that is, there is continuous f : R→ Rso that A = f−1[B]. Wadge showed that, assuming AD,

either A ≤w B or ¬B ≤w A

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where ¬A = R \A. This gives a quasi-linear-order to P(R) with classes

[A]w = {B :B =w A or B =w ¬A}A is selfdual if A =w ¬A, otherwise A is non-selfdual.

Martin showed that <w is wellfounded. Set w(A) to be the rank of A in <w.Notice w(A) < w(B) ⇐⇒ A <w B, but w(A) ≤ w(B) ⇐⇒ A ≤w B or A ≤w ¬B.Van Wesep showed that for A, cf(w(A)) = ω iff A is selfdual [VW78].

The height of the Wadge hierarchy is denoted Θ; this ordinal can be alterna-tively defined by

Θ = sup{α : ∃f : Ronto−−→ α}.

A pointclass (Wadge class), is any collection of sets of reals closed under continuousreduction, in other words, a Wadge initial segment of P(R). To any pointclass Γ let¬Γ = {¬A :A ∈ Γ} and ΔΓ = Γ∩¬Γ. A pointclass, Γ, is selfdual if Γ = ¬Γ(= ΔΓ).

A set A ∈ Γ is called Γ-complete iff Γ = {B : B ≤w A}. In the case that Γis non-selfdual, then Γ not only has a complete set, but even has a universal set,that is, a set U ⊆ R × R such that A ∈ Γ ⇐⇒ A = Ux where Ux = {y : U(x, y)}.There are even nice collections of universal sets that have the “s-n-m” property (see[Jac10]) and thus “light face”, or “effective”, arguments from descriptive set theorylift to Γ, once a collection of nice universal sets is fixed. Selfdual classes can havecomplete sets, but a diagonal argument shows that they can never have a universalset.

For any pointclass Γ, let wΓ = sup{w(A) + 1 : A ∈ Γ}. If Λ is selfdual andcf(wΛ) = ω, then for any sequence Ai Wadge cofinal in Λ, letting A =

⊕i Ai

we have w(A) = wΛ and Λ′ = {B : B ≤w A} is the first pointclass past Λ and isselfdual. If cf(wΛ) > ω, then there is a non-selfdual Γ so that ΔΓ = Λ. This followsfrom the result of Van Wesep mentioned above.

I will use ∨, ∧, etc., to operate on pointclasses. So

Γ ∧ Γ′ = {A ∩A′ :A ∈ Γ & B ∈ Γ′},∧

κ Γ ={⋂

α<κ Aα : 〈Aα : α < κ〉 ∈ Γκ},

etc. For example, Γ is closed under finite unions iff Γ∨Γ ⊆ Γ, and Γ is closed undercountable unions iff

∨ω Γ ⊆ Γ. Notice that as long as Γ has a complete set, then

closure under∨

ω is equivalent to closure under ∃ω and I will use these two notionsof closure interchangeably.

There are several ordinals other than wΓ associated to pointclasses, two impor-tant ones are:

δΓ = sup{|| ≤ || :≤∈ ΔΓ is a prewellordering}σΓ = sup{|| ≺ || :≺∈ Γ is a wellfounded relation}

Here I use || ≺ || to mean the ordinal rank of ≺. I will write ||x||≺ for the rank of xin ≺. For a pointclass Γ, if Δ∧Δ ⊆ Δ, then δΓ ≤ σΓ, since for any prewellordering≤∈ Δ we can define x ≺ y

df⇐⇒ x ≤ y ∧ y � x.

8.1. The generalized projective hierarchy. A pointclass Λ is strongly clo-sed if it is closed under real quantification and finite Boolean operations. Notice thatif Λ is strongly closed, then Λ is selfdual. The smallest strongly closed pointclassis the pointclass of projective sets,

⋃i Σ

1i . If Λ is strongly closed, then all three

ordinals wΛ, δΛ, and σΛ are the same [KSS81].We will use the hierarchy of Levy classes more than the Wadge hierarchy. A

pointclass is a Levy class if it is non-selfdual and closed under one or both of the

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real quantifiers. If Γ is a Levy class closed under ∃R and not ∀R, then Γ is a Σ-Levy class and its complement is the corresponding Π-Levy class. In [Ste81b],Steel shows that for any non-selfdual pointclass Γ exactly one of Γ or ¬Γ satisfiesseparation, we will use this to distinguish Σ-Levy classes and Π-Levy classes in caseΓ is closed under both real quantifiers. If a Levy class, Γ, is closed under both realquantifiers, then take the Σ-Levy class to be the side on which separation fails; itturns out that the Σ-class actually has the prewellordering property. Facts aboutLevy classes and the associated ordinals are taken from [KSS81, Ste81a, Bec85];a good source is [Jac10, “Wadge degrees and abstract pointclasses”].

By Wadge comparability we get a generalized projective hierarchy (Σ1α,Π

1α),

for α < Θ, where Σ1α is the αth Σ-Levy class and Π1

α is the corresponding αth

Π-Levy class. When discussing the general theory of Levy classes we will assumethat if Λ is strongly closed, then there is κ so that

∨κ Λ � Λ. Steel showed that

the least such κ is cf(wΛ) [Ste81b]. If cf(wΛ) = ω, then∨

ω Λ � Λ. Under theassumption, if cf(wΛ) > ω, then Λ = ΔΓ for some pointclass Γ by the result of VanWesep mentioned above; Steel showed that Γ is a Levy class closed under ∀R withthe prewellordering property [Ste81b, Ste81a, Jac10]. Conversely, if Γ is non-selfdual and has the prewellordering property, then

∨κ ΔΓ � ΔΓ for κ least such

that there is a Γ-norm of length κ on a set in Γ \ΔΓ. If Λ is selfdual and properlycontained in the ∞-Borel sets, then, Λ is not closed under arbitrary wellorderedunions. From results below, if Λ is strongly closed and properly contained in the∞-Borel sets, then a fair amount of the Levy hierarchy above Λ is also containedin the ∞-Borel sets.

We will quickly review a few of the relevant facts concerning the generalizedprojective hierarchy. Set

δ1α = δΣ1α

σ1α = σΣ1

αw1

α = wΔ1α

For λ a limit I will also use δ1<λ, σ1<λ, and w1

<λ in the obvious way corresponding tothe class Δ1

<λ =⋃

α<λ Δ1α. Recall δ1<λ = σ1

<λ = w1<λ since Δ1

<λ is strongly closed; Iwill use δ1<λ when there is no particular reason to use one of the other two. Noticeδ1α = δΠ1

α= δΔ1

α, but, in general, σ1

α �= σΠ1α. Clearly, δ1α ≤ σ1

α and w1α ≤ δ1α+1,

since for A ∈ Δ1α, the relation x ≤ y iff f−1

x [A] ≤w f−1y [A] can be seen to be Δ1

α+1

where fx : R → R is a (Lipschitz) continuous function coded by x. The followinglemma summarizes several properties of the generalized projective sets.

Lemma 8.1. For α < Θ the following hold:(1) If Σ1

α has the prewellordering property, then it is closed under arbitrarywellordered unions.

(2) If Σ1α is closed under finite intersections, then there is a δ1α complete

measure on σ1α. (The argument for this essentially appears in [Kec78,

§5].)(3) If Π1

α is closed under finite unions and has the prewellordering property,then(a) δ1α = || ≤ ||, where ≤ is any Π1

α norm on a complete Π1α set.

(b) σ1α = δ1α and so δ1α is measurable.

(c) Δ1α is closed under <δ1α-wellordered unions. �

To determine whether or not, for example, Π1α is closed under finite unions, has

the prewellordering property, etc., depends on the nature of the projective hierarchyto which Π1

α belongs.

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For λ a limit, Δ1<λ is the largest strongly closed pointclass contained in Δ1

λ;Δ1

<λ is called the base of the projective hierarchy Projλ = 〈(Σ1λ+n,Π

1λ+n) : n ∈ ω〉.

By First Periodicity, the behavior of the prewellordering property in the λth pro-jective hierarchy, Projλ, is determined by what happens on (Σ1

λ,Π1λ) and alternates

between the Σ-side and the Π-side.The projective hierarchy Projλ is type I if cf(λ) = cf(w1

<λ) = ω. In this case,Δ1

<λ �⊕

ω Δ1<λ � Δ1

λ. In type I hierarchies:

Σ1λ =

∨ωΔ

1<λ = Σ0

1(Δ1<λ) = Σ0

1(A) and Σ1λ+1 = Σ1

1(Δ1<λ) = Σ1

1(A)

for any A =⊕

i Ai with 〈Ai : i ∈ ω〉 Wadge cofinal in Δ1<λ, where Σ0

1(Λ) is thesmallest non-selfdual pointclass containing Λ and closed under ∃ω, Σ1

1(Λ) is thesmallest pointclass containing Λ and closed under ∃R and ∀ω, and where Σi

j(A) =

Σij({A}). Σ1

λ has the prewellordering property and so it is closed under arbitrarywellordered unions and, moreover, it is closed under finite intersections, but not∀ω.

In [Bec85], Becker proved the following:

Lemma 8.2. For λ a limit of countable cofinality

δ1λ+1 = σ1λ = w1

λ.

equivalentlyδ11(A) = w(Δ0

1(A)) = σΣ01(A)

where A =⊕

i∈ω Ai for some 〈Ai : i ∈ ω〉 Wadge cofinal in Δ1<λ. �

Of course we already knew that w(Σ01(A)) ≤ δ11(A) and that σΣ0

1(A) ≤ σ11(A) =

δ11(A). Becker arguesσ11(A) ≤ σΣ0

1(A) ≤ w(Σ01(A)).

This shows that Δ1λ is quite a bit larger than Δ1

<λ at least as far as the Wadgehierarchy is concerned (recall δ1λ+1 is measurable).

In the case that cf(λ) > ω, then, as mentioned above, there is a Levy class Γ sothat ΔΓ = Δ1

<λ; if chosen to be on the side that separation fails, Γ is closed under∀R and satisfies the prewellordering property. It follows that Γ = Π1

λ, if Γ is notclosed under ∃R, and Γ = Σ1

λ, otherwise. So for λ a limit of uncountable cofinality,Δ1

<λ = Δ1λ and hence δ1λ = w1

λ = w1<λ = δ1<λ and cf(λ) = cf(w1

<λ) = cf(w1λ). There

are three subcases for the hierarchy when cf(λ) > ω.If Π1

λ is not closed under ∃ω, and hence not under ∃R, then Projλ is called typeII. In [Ste81b], Steel showed that Π1

λ is closed under ∃ω iff Π1λ is closed under finite

unions. In fact, Steel showed that is separation fails for Γ, then Γ ∨ Γ ⊆ Γ ⇐⇒∨ω Γ ⊆ Γ.

If w1λ is singular, then Π1

λ ∨ Δ1λ � Π1

λ and thus Projλ is type II [Ste81a, pg150]. Thus in the non-type II case w1

λ is regular and λ ≥ cf(λ) = cf(w1λ) = w1

λ ≥ λ,so λ = w1

λ. Also w1λ = δ1λ, the last equality having already been discussed.

The projective hierarchy, Projλ is called type III if Π1λ is not closed under ∃R,

but is closed under finite unions, equivalently, countable unions. In the type IIIcase, λ = δ1λ = w1

λ is regular, even measurable.It is open whether or not w1

λ regular implies that Π1λ ∨ Π1

λ ⊆ Π1λ. If so, then

whether or not w1λ is regular or not would determine the type of Projλ.

Finally, if cf(λ) > ω and Π1λ is closed under both real quantifiers, then prewell-

ordering holds on the Σ-side. In this case, Projλ is a type IV hierarchy. As in the

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type III case, δ1λ is measurable. In a type IV hierarchy, it is the pair (Σ1λ+1,Π

1λ+1)

that has weak closure properties. Π1λ+1 = Σ1

λ ∧Π1λ. This pointclass is closed under

∀R, but not finite unions, it has the prewellordering property.For A a set of reals let Π(A) be whichever of Π1

1(A) or Π12(A) has the prewell-

ordering property. Let Σ(A), Δ(A), σ(A), and δ(A) have the obvious definitions.The following lemma, a corollary of Lemma 8.1, summarizes what we will need.

Lemma 8.3. Let A be a set of reals,

(1) δ(A) is the rank of any Π(A)-norm on a complete Π(A) set.(2) σ(A) = δ(A) is measurable.(3) Δ(A) is closed under <δ(A)-wellordered unions/intersections. �

8.2. Coding Lemma. One version of the Moschovakis Coding Lemma, see[Mos09, Jac10], is as follows:

Theorem 8.4 (Moschovakis). Suppose Γ is a Σ-Levy class closed under finiteintersection and let ≺ be a wellfounded relation in Γ of rank γ. Let U ⊆ R× R beuniversal for Γ. For any f : γ → P(R), there is ε so that for all β < γ:

(1) If f(β) �= ∅, then ∃x, y ||x||≺ = β & Uε(x, y).(2) For all x, y, Uε(x, y) implies x ∈ field(≺) and y ∈ f(||x||≺).

ε is called a code of f . �

If Γ is a Σ-Levy class closed under finite intersections and ≺ is the strict partof a prewellordering � of length γ with both ≺ and � in Γ, then for any S ⊆ γthere is a � invariant code for C ∈ Γ for S, i.e., for x ∈ field(�):

C(x)⇐⇒ S(||x||≺)

Since the same is true for γ \ S we have S is Δ in �. The following variant of thecoding lemma is discussed in [Jac10, §2.2].

Theorem 8.5. If Γ is a Π-Levy class closed under ∨ with the prewellorderingproperty, e.g., Π(A) for any set A, then any subset of δΓ has a ΔΓ-code (ratherthan a Δ∃RΓ-code) with respect to a fixed Γ-norm on a complete Γ-set. �

There are several other variants of the coding lemma. I will use the terminologyS ∈ Γ to mean that there is a prewellordering ≤ in Γ so that {x : ||x||≤ ∈ S} ∈ Γ.So for example, the preceding theorem can be stated as P(δΓ) ⊆ ΔΓ.

One way the coding lemma will be used is given by the following corollary.

Corollary 8.6. If M and N are models of ZF+AD, then for all γ < ΘN ∩ΘM ,P(γ) ∩M = P(γ) ∩N . �

In [KKMW81] it is shown that AD implies its own strengthening to <κ∞-ordinal determinacy. What is actually shown is that if γ < Θ and A ⊆ γω isκ-Suslin/κ-co-Suslin for some κ < Θ, then the game on γ, Gγ(A) is determined.The determinacy of of Gγ(A) is absolute between any M and N modeling AD andhaving the same reals provided γ < ΘM ∩ ΘN and the set A ∈ M ∩ N , since theCoding Lemma will guarantee that these two models will have the same strategies,since a strategy “is” a subset of γ. This gives the downward absoluteness of ordinaldeterminacy mentioned in the introduction.

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References

[Bar75] Jon Barwise, Admissible sets and structures, Springer-Verlag, Berlin, 1975, An ap-proach to definability theory, Perspectives in Mathematical Logic. MR MR0424560(54 #12519)

[Bec85] Howard Becker, A property equivalent to the existence of scales, Trans. Amer. Math.Soc. 287 (1985), no. 2, 591–612. MR MR768727 (86g:03085)

[CK09] Andrés Caicedo and Richard Ketchersid, A trichotomy theorem in natural models ofAD+, http://unixgen.muohio.edu/~ketchero/preprints, 2009.

[Ike10] Daisuke Ikegami, Games in Set Theory and Logic, Ph.D. thesis, Institiute for Logic,Language, and Computation; universiteit nan Amsterdam, 2010.

[Jac10] Stephen Jackson, Structural consequences of AD, Handbook of Set Theory, Springer,Berlin, 2010, pp. 1753–1876.

[Jec03] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag,Berlin, 2003, The third millennium edition, revised and expanded. MR MR1940513(2004g:03071)

[Jen72] R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic4 (1972), 229–308; erratum, ibid. 4 (1972), 443, With a section by Jack Silver.MR MR0309729 (46 #8834)

[Kec78] Alexander S. Kechris, AD and projective ordinals, Cabal Seminar 76–77 (Proc.Caltech-UCLA Logic Sem., 1976–77), Lecture Notes in Math., vol. 689, Springer,Berlin, 1978, pp. 91–132. MR MR526915 (80j:03069)

[Kec88] , “AD + UNIFORMIZATION” is equivalent to “half ADR”, Cabal Semi-nar 81–85, Lecture Notes in Math., vol. 1333, Springer, Berlin, 1988, pp. 98–102.MR MR960897 (89i:03093)

[KKMW81] Alexander S. Kechris, Eugene M. Kleinberg, Yiannis N. Moschovakis, and W. HughWoodin, The axiom of determinacy, strong partition properties and nonsingularmeasures, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79), Lec-ture Notes in Math., vol. 839, Springer, Berlin, 1981, pp. 75–99. MR MR611168(83f:03047)

[KMS83] Alexander S. Kechris, Donald A. Martin, and Robert M. Solovay, Introduction toQ-theory, Cabal seminar 79–81, Lecture Notes in Math., vol. 1019, Springer, Berlin,1983, pp. 199–282. MR MR730595

[KSS81] Alexander S. Kechris, Robert M. Solovay, and John R. Steel, The axiom of de-terminacy and the prewellordering property, Cabal Seminar 77–79 (Proc. Caltech-UCLA Logic Sem., 1977–79), Lecture Notes in Math., vol. 839, Springer, Berlin,1981, pp. 101–125. MR MR611169 (83f:03042)

[KW10] Peter Koellner and Hugh Woodin, Large cardinals from determinacy, Handbook ofSet Theory, Springer, Berlin, 2010, pp. 1951–2120.

[Mos09] Yiannis N. Moschovakis, Descriptive set theory, second ed., Mathematical Surveysand Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009.MR MR2526093

[MW08] Donald A. Martin and W. Hugh Woodin, Weakly homogeneous trees, Games, scales,and Suslin cardinals. The Cabal Seminar. Vol. I, Lect. Notes Log., vol. 31, Assoc.Symbol. Logic, Chicago, IL, 2008, pp. 421–438. MR MR2463621

[Ste81a] John R. Steel, Closure properties of pointclasses, Cabal Seminar 77–79 (Proc.Caltech-UCLA Logic Sem., 1977–79), Lecture Notes in Math., vol. 839, Springer,Berlin, 1981, pp. 147–163. MR MR611171 (84b:03066)

[Ste81b] , Determinateness and the separation property, J. Symbolic Logic 46 (1981),no. 1, 41–44. MR MR604876 (83d:03058)

[Ste94] , Notes on AD+, 1994.[VW78] Robert Van Wesep, Wadge degrees and descriptive set theory, Cabal Seminar 76–

77 (Proc. Caltech-UCLA Logic Sem., 1976–77), Lecture Notes in Math., vol. 689,Springer, Berlin, 1978, pp. 151–170. MR MR526917 (80i:03058)

[Woo99] W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationaryideal, de Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter &Co., Berlin, 1999. MR MR1713438 (2001e:03001)

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Department of Mathematics, Boise State University, 1910 University Drive,

Boise, ID 83725-1555

URL: http://sites.google.com/site/richardketchersid/E-mail address: [email protected]

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αi-selection principles and games

Ljubisa D.R. Kocinac

Abstract. In this survey we review selected results in the theory of selectionprinciples restricting ourselves to αi-selection principles and related games andtheir applications to various areas of topology and real analysis. Several newresults are proved, and several open problems are formulated.

1. Introduction

In a paper published in 1874, Georg (Ferdinand Ludwig Philipp) Cantor (1845–1918) proved that a set of real numbers is not countable. His celebrated methodof diagonalization used in that proof generated investigation in mathematics of thefollowing sort: given a sequence (An : n ∈ N) of mathematical objects of one kind,choose some elements from each An so that the chosen elements form an object ofsome desired (same or different) kind. In [55] (see also [36]) Scheepers systemized,unified and initiated further investigation regarding this matter, and after that thefield of mathematics that uses this method is usually called Selection PrinciplesTheory.

This sort of investigation has a long history going back to 1920’s and 1930’sto works by (Felix Edouard Justin) Emile Borel (1871–1956) [7], Fritz Rothberger(1902–2000) [52], Karl Menger (1902–1985) [45], Witold Hurewicz (1904–1956)[33], [34], Wac�law Franciszek Sierpinski (1882–1969) [61], [62]. Nowadays, anincreasing number of mathematicians work in this beautiful area of mathematics,and in recent years many nice, deep results were obtained and many innovationswere done.

In this paper we review some results in this mathematical discipline restrictingourselves mostly to αi-selection principles and related games and their applicationsto some areas of topology and real analysis. Several new results will be proved anda number of open problems will be formulated.

Three classical selection principles are defined as follows [55].Let A and B be families of subsets of an infinite set X. Then the following

selection principles are defined:

1991 Mathematics Subject Classification. Primary: 54-02, 54D20; Secondary: 26A12, 54A20,54B20, 54C35, 91A44.

Partially supported by MN RS..


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2 LJ.D.R. KOCINAC

Sfin(A,B): For each sequence (An : n ∈ N) of elements of A there is a sequence(Bn : n ∈ N) of finite sets such that for each n, Bn ⊂ An, and

⋃n∈N

Bn ∈ B.Ufin(A,B): For each sequence (An : n ∈ N) of elements of A there is a sequence

(Bn : n ∈ N) of finite sets such that for each n, Bn ⊂ An, and {⋃Bn : n ∈ N} ∈ B.

S1(A,B): For each sequence (An : n ∈ N) of elements of A there is a sequence(bn : n ∈ N) such that for each n, bn ∈ An, and {bn : n ∈ N} is an element of B.

There are games which are naturally associated to the above selection princi-ples. We define only the game associated to the principle S1(A,B), because thegames associated to other principles are defined similarly.

The symbol G1(A,B) denotes the following game: Two players, ONE andTWO, play a round for each positive integer. In the n-th round ONE choosesa set An ∈ A, and TWO responds by choosing an element bn ∈ An. TWO wins aplay (A1, b1; · · · ;An, bn; · · · ) if {bn : n ∈ N} ∈ B; otherwise, ONE wins.

A strategy of a player is a function σ from the set of all finite sequences ofmoves of the opponent into the set of (legal) moves of the strategy owner.

A strategy σ for the player TWO is a coding strategy if TWO remembers onlythe most recent move by ONE and by TWO before his next move i.e. the movesof TWO are: b1 = σ(A1, ∅); bn = σ(An, bn−1), n ≥ 2.

In this article we deal mainly with S1-type selection principles and correspond-ing games, because they are connected with our main topic, αi-selection principles.

Our topological terminology and notation are standard as in Engelking’s book[26]; at the beginning of each section, if it becomes necessary, we give additionalinformation about notions we are dealing with in that section. We suppose thatconsidered spaces are Hausdorff and non-compact (see Section 2). For survey papersconcerning selection principles and their relationships with other fields of mathe-matics we refer the reader to [38], [40], [59], [64].

Several spaces with nice combinatorial properties play an important role in thestudy of selection principles. We need here only one such space. The Baire spaceNN is the Tychonoff product of countably many copies of the discrete space N.There is a natural pre-order ≤∗ defined on NN: f ≤∗ g means that f(n) ≤ g(n) forall but finitely many n. A subset B of NN is called bounded if there is a g ∈ NN

such that f ≤∗ g for each f ∈ B. The symbol b denotes the minimal cardinality ofan unbounded set in (NN,≤∗).

The paper is organized in the following way. We end this introductory materialby definitions of αi-selection principles. In Section 2 we discuss in detail theseprinciples in general topological spaces. Sections 3, 4 and 5 deal with applicationsof αi-selection properties to topological groups, function spaces and hyperspaces,respectively. Applications of selection principles to asymptotic analysis of divergentprocesses will be discussed in Section 6. Section 7 gives some indications how theidea of statistical convergence can be used in investigation of αi-selection principles.

1.1. αi-selection principles. In 2005 (at 2005 International General Topol-ogy Symposium in Zhangzhou, May 25-28, 2005, Zhangzhou, China; published in[42]), the author introduced and began a systematic investigation of the followingselection principles; A and B are as above.

A spaceX has property αi(A,B), i = 1, 2, 3, 4, if for every sequence (An : n ∈ N)in A there exists B ∈ B with:

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αi-SELECTION PRINCIPLES 3

α1(A,B): for each n ∈ N, |An \B| < ω;α2(A,B): for each n ∈ N, An ∩B is infinite;α3(A,B): for infinitely many n ∈ N, An ∩B is infinite;α4(A,B): for infinitely many n ∈ N, An ∩B �= ∅.

For a topological space X and a point x ∈ X, let Σx denote the family ofsequences in X converging to x. Then the properties αi(Σx,Σx), i = 1, 2, 3, 4, areexactly the Arhangel’skii local αi-properties introduced and first studied in [1] (seealso [2]). A space X is said to be an αi-space, i = 1, 2, 3, 4, if it satisfies αi(Σx,Σx)for each x ∈ X.

It is known that the four properties αi(Σx,Σx) are different from each other [1],[49] and that the same holds in topological groups [60], [50]. However, in functionspaces Cp(X) (see Section 4) and some hyperspaces (see Section 5) the propertiesα2, α3 and α4 are equivalent to each other and to the corresponding S1 property.

EvidentlyS1(A,B) ⇒ α4(A,B).

If all members of A are infinite, then

α1(A,B) ⇒ α2(A,B) ⇒ α3(A,B) ⇒ α4(A,B).If A contains finite members, then α3(A,B) fails.

2. General topological spaces

This section contains some results about αi-selection principles in general topo-logical spaces. We also prove some new results concerning classes of topologicalspaces defined in terms of α1-selection properties and their relatives.

In [42] we were interested in αi(A,B) properties when A and B are collectionsof certain open covers of a topological space, preferably in coincideness of suchproperties with the classical properties S1(A,B) (see some of the results below).

We use the following notation for collections of open covers of a space X whichare relevant for this paper.

O: the collection of open covers;Ω: the collection of ω-covers;K: the collection of k-covers;Γ: the collection of γ-covers;Γk: the collection of γk-covers.

An open cover U of X is:an ω-cover (a k-cover) if for each finite (compact) A ⊂ X, A ⊂ U for some

U ∈ U and X /∈ U ; thus we assume that considered spaces are not compact.a γ-cover (a γk-cover) if U is infinite and for each finite (compact) set A ⊂ X,

the set {U ∈ U : A � U} is finite.

We assume that γ-covers and γk-covers are countable.

A space X is said to be ω-Lindelof (k-Lindelof ) if each ω-cover (k-cover) U ofX contains a countable V ⊂ U which is an ω-cover (a k-cover) of X.

For a space X and a point x ∈ X we also use the notation:

Σx: the family of (nontrivial) sequences converging to x.

In [67], Tsaban classified αi-selection properties for A,B ∈ {O,Ω,Γ}. Weextended this classification considering also A,B ∈ {K,Γk}.

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It was established that for certain classes A and B some αi(A,B) selectionproperties either always hold or never hold, while some of them are nontrivialand either coincide with or are different from the corresponding S1(A,B) selectionprinciple. For example,

• α1(Γ,Γ) �= S1(Γ,Γ) [68].

Conjecture 2.1. α1(Γk,Γk) is strictly stronger than S1(Γk,Γk).

The following table summarizes the results about mentioned covering αi-selectionproperties (including some mild hypotheses on the considered spaces). The symbol+ denotes that a property always holds, - means that a property is not true, while� means that a property is new and different from classical selection properties.Finally, = S1 means that an αi-property coincides with the corresponding S1-typeproperty. The results involving O,Ω,Γ are from [67] (or are easily derived fromresults in that paper), and the results that involve K,Γk and = S1 are due to theauthor. Note that α1(Γ,Γ) is strictly stronger than S1(Γ,Γ).

α1 α2 α3 α4

(O,O) + - - +(O,Ω) - - - -(O,K) - - - -(O,Γ) - - - -(O,Γk) - - - -(Ω,O) + + + +(Ω,Ω) + + + +(Ω,K) - - - -(Ω,Γ) - = S1 = S1 = S1(Ω,Γk) - - - -(K,O) + + + +(K,Ω) + + + +(K,K) + + + +(K,Γ) - = S1 = S1 = S1(K,Γk) - = S1 = S1 = S1(Γ,O) + + + +(Γ,Ω) + + + +(Γ,K) - - - -(Γ,Γ) � = S1 = S1 = S1(Γ,Γk) - - - -(Γk,O) + + + +(Γk,Ω) + + + +(Γk,K) + + + +(Γk,Γ) + = S1 = S1 = S1(Γk,Γk) � = S1 = S1 = S1

Table 1

Let us mention a few results which complete information about Table 1.

Theorem 2.2. [42] For an ω-Lindelof space X the following are equivalent:

(1) X satisfies α2(Ω,Γ);(2) X satisfies α3(Ω,Γ);

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(3) X satisfies α4(Ω,Γ);(4) X satisfies S1(Ω,Γ).

Theorem 2.3. [42] For a k-Lindelof non-compact space X, the propertiesα2(K,Γ), α3(K,Γ), α4(K,Γ) and S1(K,Γ) are equivalent.

Theorem 2.4. [42] For a k-Lindelof non-compact space X, the propertiesα2(K,Γk), α3(K,Γk), α4(K,Γk) and S1(K,Γk) are equivalent.

Theorem 2.5. For a space X the properties α2(Γ,Γ), α3(Γ,Γ), α4(Γ,Γ) andS1(Γ,Γ) are equivalent.

In the next theorem we use the following selection principle introduced in [66].A space X satisfies

⋂∞(A,B) if for each sequence (An : n ∈ N) of elements of A

there is a sequence (Bn : n ∈ N) such that Bn is an infinite subset of An for eachn ∈ N and {

⋂Bn : n ∈ N} is an element of B.

Theorem 2.6. For a space X and B ∈ {Γ,Γk}, the following assertions areequivalent:

(1) X satisfies α2(Γk,B);(2) X satisfies α3(Γk,B);(3) X satisfies α4(Γk,B);(4) X satisfies S1(Γk,B);(5) X satisfies

⋂∞(Γk,B).

Proof. The equivalence of properties (1) - (4) was shown in [42]. We provethat (1) is equivalent to (5), showing only the case B = Γk, because the other caseB = Γ is similar; the proof is a modification of the proof of [66, Theorem 6].

(1) ⇒ (5): Let (Un : n ∈ N) be a sequence of γk-covers of X and assumethat Un ∩ Um = ∅ whenever n �= m; it is possible to assume because any infinitesubset of a γk-cover is also a γk-cover. By (2) there is a γk-cover V such that itsintersection with each Un is infinite. Let Vn = Un ∩V . We claim that the sequence(Vn : n ∈ N) witnesses for (Un : n ∈ N) that (5) holds. Suppose, to the contrary,that {

⋂Vn :∈ N} is not a γk-cover of X. There is a compact set K ⊂ X and an

infinite set M ⊂ N such that K is not contained in⋂Vn for each n ∈ M , i.e. for

each n ∈ M there is a Vn ∈ Vn such that K \ Vn �= ∅. Since Vn’s are disjointthis means that infinitely many elements Vn, n ∈ M , of V do not contain K. Acontradiction.

(5) ⇒ (1): Let (Un : n ∈ N) be a sequence of γk-covers of X. By (5) there isa sequence (Vn : n ∈ N) such that for each n, Vn is an infinite subset of Un and{⋂Vn : n ∈ N} ∈ Γk. Put V =

⋃{Vn : n ∈ N} and show that V witnesses that (1)

is true. Let K be a compact subset of X. Then there is a subset M of N such thatN \M is finite and K ⊂ ∩Vn for each n ∈ M . For each k ∈ N \M , Vk is a γk-coverof X being an infinite subset of Uk. So, K is not contained in only finitely manyelements of Vk for each k ∈ N \ M , i.e. K is contained in all but finitely manyelements of V . �

Recall that S1(K,Γ)-spaces have been first considered in [18] and called k-γ-sets, while S1(K,Γk)-spaces, introduced in [39], are called γk-sets (see also [13]where these two classes of spaces have been investigated).

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In [35] it was shown that the union of an increasing countable family of γ-setsis also a γ-set, while in [51] it was shown that a similar result holds for manyselection properties.

Using Theorem 2.6 and some methods from [35] and [51] we are going toshow that k-γ-sets and γk-sets have the same property. Recall that neither k-γ-setproperty nor γk-set property is finitely additive (for the first fact see [13], and forthe second [47]).

[Observe that the following holds: If X×Y is a k-γ-set, then X∪Y is a k-γ-set.Indeed, if U is a k-cover of X ∪ Y , then U2 = {U2 : U ∈ U} is a k-cover for X × Y ,since for a compact set K ⊂ X × Y the union of projections pX(K) and pY (K),being a compact set in X ∪Y , is contained in a member U ∈ U , hence K ⊂ U2. AsX × Y is a k-γ-set, there is a countable V ⊂ U such that V2 is a γ-cover of X × Y ;then V is a γ-cover of X ∪ Y witnessing for U that X ∪ Y is a k-γ-set.

Theorem 2.7. Let a k-Lindelof non-compact space X be the union of an in-creasing countable family X1 ⊂ X2 ⊂ · · · of k-γ-sets (γk-sets). Then X is also ak-γ-set (γk-set).

Proof. We prove only the γk-sets case. Let (Un : n ∈ N) be a sequence ofk-covers of X. Fix i ∈ N. As Xi is a γk-set each Un contains a countable subsetWn which is a γk-cover of Xi [13, Theorem 18], i.e. each compact set K ⊂ Xi iscontained in all but finitely many elements of Wn for each n. On the other hand,S1(K,Γk) implies S1(Γk,Γk), and thus, by Theorem 2.6,

⋂∞(Γk,Γk).

Claim 1: There are infinite sets Vn ⊂ Wn, n ∈ N, such that each compact K ⊂ Xi

is contained in⋂Vn for all but finitely many n.

Put Gn := {U ∈ Wn : Xi � U}. There are two possibilities:Case 1: The set M = {n ∈ N : |Gn| < ω} is finite.

In this case for each n let Vn = Gn. Clearly, each K ⊂ Xi is contained in ∩Vn

for all but finitely many n.

Case 2: The set M = {n ∈ N : |Gn| = ω} is infinite.For each n ∈ M the set Gn, being an infinite subset of Wn, is a γk-cover of Xi.

Apply S1(Γk,Γk) =⋂

∞(Γk,Γk) to Xi and the sequence (Gn : n ∈ N) and find asequence (Vn : n ∈ N) such that Vn is an infinite subset of Gn for each n ∈ N, andeach compact K ⊂ Xi is contained in ∩Vn for all but finitely many n. Then Vn’sare the desired sets.

This completes the proof of Claim 1.

Claim 2: There are infinite sets Vn ⊂ Un, n ∈ N, such that each compact K ⊂ Xis contained in ∩Vn for all but finitely many n.

For each i ∈ N, Ui is a k-cover of Xi, and since Xi is a γk-set there is an infiniteset Hi ⊂ Ui which is a γk-cover of Xi. By Claim 1 for each i ∈ N and each n ≥ ithere are infinite sets Gn ⊂ Hn ⊂ Un and ni ∈ N such that each compact subset ofXi belongs to all ∩Gn with n ≥ ni. For each i, let Vi = Gni

. The sets Vn, n ∈ N,are as required by Claim 2.

To finish the proof of the theorem, apply Claim 2, find sets Vn, n ∈ N, as inClaim 2, and for each n take Un ∈ Vn \ {U1, · · · , Un−1}. It is evident that the set{Un : n ∈ N} witnesses for (Un : n ∈ N) that X is a γk-set. �

Let us mention that some of results presented here were extended to Cechclosure spaces in [48].

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We end this section with some open problems1.

By known results on S1 selection principles (see [13], [55]) and Theorems 2.2,2.3, 2.5, we conclude that properties α2(Ω,Γ), α2(K,Γ) and α2(Γ,Γ) (under CH)are not finitely productive. According to Theorem 2.4 and [47, Lemma 4], theproperty α2(K,Γk) is finitely productive in the class of metric spaces.

Problem 2.8. Are properties α1(Γ,Γ) and α1(Γk,Γk) finitely productive? Arethese properties preserved by finite powers?

If the answer to this question is negative (our conjecture is that it is the case),then it is natural to ask the following.

Problem 2.9. Characterize the class S of spaces such that the product of anα1(Γ,Γ) space (an α1(Γk,Γk) space) X and a space Y ∈ S is again in the classα1(Γ,Γ) (α1(Γk,Γk)).

Theorem 2.7 suggests the following question.

Problem 2.10. Are properties α1(Γ,Γ) and α1(Γk,Γk) countably additive?Is an increasing union of countably many subspaces having property α1(Γ,Γ) orα1(Γk,Γk) a space with the same property?

Problem 2.11. Is α1(Γ,Γ) property hereditary for Fσ-subsets?

For a topological property P, the symbol non(P) denotes the minimal cardi-nality of a set X of reals such that X does not have P. The additivity number ofP, denoted add(P), is defined as

add(P) = min{|F| : ∀F ∈ F , F has P, and ∪F has no P}.In Section 4 we observe that non(α1(Γ,Γ)) = b.

Problem 2.12. Estimate or determine exactly non(α1(Γk,Γk))?

Problem 2.13. Estimate or determine exactly add(α1(Γ,Γ)) and add(α1(Γk,Γk))?

The following game Gα1(A,B) is naturally corresponded to α1(A,B). Two

players, ONE and TWO, play a round for each n ∈ N. In the n-th round ONEchooses an element An ∈ A, and TWO responds by choosing a cofinite set Bn

from An. TWO wins a play (A1, B1; · · · ;An, Bn; · · · ) if⋃

n∈NBn belongs to B;

otherwise, ONE wins.Clearly, if the player ONE does not have a winning strategy in the game

Gα1(A,B), then the selection property α1(A,B) is true.Problem 2.14. Is is true that a space X satisfies α1(Γ,Γ) (α1(Γk,Γk)) if and

only if ONE has no winning strategy in the corresponding Gα1game?

3. Topological groups

In this section we discuss some applications of general results on αi-selectionprinciples to topological groups. Selection principles in topological groups, initiatedindependently by Kocinac and Okunev in 1998, have been investigated in a seriesof papers, for example in [4], [5], [30], [31], [44], [46], [65].

Let (G, ·, τ ) be a topological group, e its neutral element, and Be a local baseat e. For U ∈ Be with U �= G we put

1After the paper was submitted B.Tsaban solved some problems; see Section 8.

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o(U) = {x · U : x ∈ G},ω(U) = {F · U : F ∈ F(G)},k(U) = {K · U : K ∈ K(G)},

and

O(e) = {o(U) : U ∈ Be};Ω(e) = {ω(U) : U ∈ Be and there is no F ∈ F(G) with F · U = G};K(e) = {k(U) : U ∈ Be and there is no K ∈ K(G) with K · U = G}

Observe that O(e) ⊂ O, Ω(e) ⊂ Ω, and K(e) ⊂ K.Similarly to the general situation, in topological groups some of the αi-selection

properties are always satisfied. Such selection principles are, for example, α1(O(e),O),α1(Ω(e),Ω) (thus α2(Ω(e),O) α2(Ω(e),Ω)), α1(K(e),O), α1(K(e),K), α1(K(e),Ω).On the other hand, some properties are never satisfied, for instance: αi(O(e),Ω),αi(O(e),K), αi(O(e),Γ), αi(O(e),Γk) for i = 1, 2.

We also have the following results.

Theorem 3.1. [42] For a topological group G the following statements areequivalent:

(1) G satisfies α2(Ω(e),Γ);(2) G satisfies α3(Ω(e),Γ);(3) G satisfies α4(Ω(e),Γ);(4) G satisfies S1(Ω(e),Γ);(5) G satisfies S1(K(e),Γ).

Recall that a topological group G is Hurewicz bounded if it satisfies: for eachsequence (Un : n ∈ N) from Be there is a sequence (An : n ∈ N) of finite subsets ofG such that each x ∈ X belongs to all but finitely many Fn · Un.

Combining the previous theorem and a result of Babinkostova [4] we have:α2(Ω(e),Γ) characterizes metrizable Hurewicz bounded (≡ σ-totally bounded) groups.

In [67], Tsaban proved that a topological group G satisfies α1(Ω(e),Γ) if andonly if it is totally bounded.

Theorem 3.2. [42] The following conditions are equivalent in any topologicalgroup G:

(1) G satisfies α2(Ω(e),Γk);(1) G satisfies α3(Ω(e),Γk);(3) G satisfies α4(Ω(e),Γk);(4) G satisfies S1(Ω(e),Γk);(5) G satisfies S1(K(e),Γk).

The proofs of Theorems 8 and 9 in [67] show that α1(K(e),Γk) holds if and onlyif for each neighborhood U of e there is a compact set K ⊂ G such that K ·U = G.

4. Function spaces

In this section all spaces are assumed to be Tychonoff. For a space X, Cp(X)denotes the space of all continuous real-valued functions on X endowed with thetopology of pointwise convergence. Since Cp(X) is a topological vector space, it ishomogeneous and so for local properties it is enough to consider the point 0 – theconstant function equal to 0.

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The situation with αi-selection principles in these spaces is interesting anddifferent from the situation in general topological spaces and topological groups.

To describe results in this connection we need some terminology.According to [56] a space X has the sequence selection property (SSP) if it

satisfies S1(Σx,Σx) for each x ∈ X.A sequence (fn)n∈N ⊂ RX converges quasi-normally to f ∈ RX if there exists

a (control) sequence (εn)n∈N of positive reals converging to 0 such that for eachx ∈ X, |fn(x) − f(x)| < εn for all but finitely many n [8]. (Note that this notionwas introduced in [14] under the name equal convergence.)

A space X is a QN-space (quasi-normal) [11] if every sequence (fn)n∈N inCp(X) converging pointwise to 0, converges also quasi-normally to 0.

X is a wQN-space [11] (see also [12]) if from every sequence (fn)n∈N in Cp(X)converging pointwise to 0, there is a subsequence (fnm

)m∈N converging to 0 quasi-normally.

In [28], Fremlin introduced the notion of s1-space. For a space Z, Σ(Z) denotessup{σ(A) : A ⊂ Z}, where σ(A) is the sequential order of A. Fremlin provedthat for any space X, Σ(Cp(X)) is either 0, 1 or ω1, and called X an s1-space ifΣ(Cp(X)) = 1.

The following theorem summarizes some results in connection with the Arhan-gel’skii αi-properties, i = 2, 3, 4, in function spaces.

Theorem 4.1. For a space X the following are equivalent:

(1) Cp(X) is an α2-space;(2) Cp(X) is an α3-space;(3) Cp(X) is an α4-space;(4) Cp(X) has SSP ;(5) Cp(X) is an s1-space;(6) X is a wQN-space.

The equivalence of (1), (2) and (3) was shown in [58], the equivalence of (1), (4)and (5) in [57]. The implication (1) ⇒ (6) has been proved in [57], and (6) ⇒ (1)in [29].

In [58] it was shown that if Cp(X) is α1-space, then X is a QN-space, and askedif the converse is true. The positive answer to this question was given independentlyby Bukovsky-Hales [10], and Sakai [53] (see also [9] for similar material).

In the same paper [53], Sakai gave characterizations of properties α1 and α2 ofCp(X) in terms of covering properties of X.

Let Γcl and Γcoz denote the family of γ-covers by clopen and cozero-sets of X,respectively.

Theorem 4.2. [53] For a Tychonoff space X the following hold:

(1) Cp(X) is an α1-space if and only if X satisfies the selection propertyα1(Γcoz,Γcoz);

(2) Cp(X) is an α2-space if and only if X satisfies both IndZ(X) = 0 andS1(Γcl,Γcl).

Here IndZ(X) = 0 means that for every pair (A,B) of disjoint zero-sets in X,there exists a clopen set U ⊂ X with A ⊂ U and B ∩ U = ∅.

Corollary 4.3. A perfectly normal space X satisfies α1(Γ,Γ) if and only ifCp(X) is an α1-space.

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10 LJ.D.R. KOCINAC

The previous theorem and corollary allow us to give a characterization of spacessatisfying the selection property α1(Γ,Γ) in terms of their continuous images intothe space Rω.

Theorem 4.4. For a perfectly normal space X the following are equivalent:

(1) X has the property α1(Γ,Γ);(2) For each continuous function f : X → Rω, the space f(X) satisfies

α1(Γ,Γ).

Proof. We prove only the non-trivial part (2) ⇒ (1). To prove this it isenough, according to Theorem 4.2 (1), to prove that the function space Cp(X) isan α1-space.

Let (sn : n ∈ N), sn = (fn,m)m∈N, be a sequence of sequences of Cp(X) suchthat each sn converges to 0. Let S = {fn,m : n ∈ N,m ∈ N} ∪ {0} and let g bethe diagonal product of mappings from S. Then g is a continuous mapping fromX onto the set Y = g(X) ⊂ Rω. Since by (2) Y satisfies α1(Γ,Γ), the space Cp(Y )is an α1-space. On the other hand, the set T = {h ◦ g : h ∈ Cp(Y )} is a subsetof Cp(X) which is homeomorphic to Cp(Y ) [3] and, as it is easily seen, containsS. Therefore, T is an α1-space. Thus there exists a sequence s = (ϕn)n∈N whichT -converges to 0 and such that for each n ∈ N, the sn \ s is finite. Clearly, s alsoCp(X)-converges to 0. This means that Cp(X) is an α1 space, which completes theproof of the theorem. �

In [58], it was observed that the minimal cardinality of a set X of reals suchthat Cp(X) does not have property α1 is b. So, we have

Corollary 4.5. The minimal cardinality of a set X of reals such that X doesnot have property α1(Γ,Γ) is b.

Let us emphasize the following nice result closely related to Theorems 4.2 and4.4.

Theorem 4.6. For a perfectly normal strongly zero-dimensional space X thefollowing conditions are equivalent:

(1) Cp(X) is an α1-space;(2) [68] each Borel image of X in the Baire space NN is bounded.

5. Hyperspaces

In this section we consider αi-selection properties in hyperspaces and show thatin some hyperspaces these properties coincide. For undefined notions in this sectionsee [41].

Let X be a Hausdorff topological space, A ⊂ X. We use the following notation.2X is the family of all closed subsets of X, while F(X) and K(X) denote the

family of all non-empty finite subsets of X and the collection of non-empty compactsubsets of X, respectively.

A+ = {F ∈ 2X : F ⊂ A}.The upper Fell topology τF+ on 2X is the topology whose base is the family

{(Kc)+ : K ∈ K(X)} ∪ {2X} (where Kc = X \K).

The sets of the form U+, U open in X, form a base for the upper Vietoris topologyτV+ .

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The following two theorems describe the status of αi-selection principles in thehyperspace (2X , τF+).

Theorem 5.1. [19] For a space X the following are equivalent:

(1) For each E ∈ 2X , (2X , τF+) satisfies α2(ΣE,ΣE);(2) For each E ∈ 2X , (2X , τF+) satisfies α3(ΣE,ΣE);(3) For each E ∈ 2X , (2X , τF+) satisfies α4(ΣE,ΣE);(4) For each E ∈ 2X , (2X , τF+) satisfies S1(ΣE,ΣE).

Theorem 5.2. [19] For a space X, (2X , τF+) is an α1-space if and only if eachopen Y ⊂ X satisfies α1(Γ,Γ).

We proceed with two theorems from [42] (see also [16]) which show that theclasses of γ-sets and γk-sets can be characterize by α2-type properties of hyper-spaces.

Theorem 5.3. [42] If X is an ω-Lindelof space, then the following are equiv-alent:

(1) (F(X), τV+) satisfies α2(Ω,Γ);(2) (F(X), τV+) satisfies α3(Ω,Γ);(3) (F(X), τV+) satisfies α4(Ω,Γ);(4) (F(X), τV+) satisfies S1(Ω,Γ);(5) X satisfies S1(Ω,Γ).

Theorem 5.4. [42] For a k-Lindelof space X, the following statements areequivalent:

(1) (K(X), τV+) satisfies α2(Ω,Γ);(2) (K(X), τV+) satisfies α3(Ω,Γ);(3) (K(X), τV+) satisfies α4(Ω,Γ);(4) (K(X), τV+) satisfies S1(Ω,Γ);(5) X satisfies S1(K,Γk).

For generalizations of some of the results from this section to Cech closurespaces see [48].

6. Real analysis and αi-selection principles

This section is devoted to some applications of αi-selection principles (andrelated games) to asymptotic analysis of divergent processes, mainly in so-calledKaramata’s theory of regular variation (Jovan Karamata, 1930s; [37]) and its spe-cial case, rapid variation (de Haan, 1970). For nice expositions of Karamata’stheory we refer the reader to the books [6] and [43]. In a series of papers (see[15], [20], [22], [23], [24], [25], and also the survey paper [21]) the author and hiscollaborators demonstrated nice relationships between selection principles theoryand the theory of regular and rapid variation.

Throughout this section the main object will be the set S of sequences of positivereal numbers, and A and B will be certain subsets of S. We shall often identify asequence and its range.

We begin with two results from [22] which show that for certain subclasses Aof S, the selection property α2(A,A) is satisfied. The proofs of these results (fordetails see [22]) bring some ideas which may be useful in more general situations.

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12 LJ.D.R. KOCINAC

Consider also the following game Gα2(A,B) (associated to the selection princi-

ple α2(A,B)). Two players, ONE and TWO, play a round for each n ∈ N. In then-th round ONE chooses a sequence sn ∈ A, and TWO responds by choosing aninfinite set Tn from sn. TWO wins a play (s1, T1; · · · ; sn, Tn; · · · ) if

⋃n∈N

Tn canbe arranged in a sequence from B; otherwise, ONE wins.

Clearly, if the player ONE does not have a winning strategy in the gameGα2

(A,B), then the selection property α2(A,B) is true.A sequence a = (an)n∈N ∈ S is negligible with respect to b = (bn)n∈N from S if

for every ε > 0, there is n0 = n0(ε) such that an ≤ ε · bn whenever n ≥ n0.

∇(a) denotes the set of all sequences b in S such that a is negligible with respectto b.

Theorem 6.1. [22] Let a = (an)n∈N ∈ S. The player TWO has a winningstrategy in the game Gα2

(∇(a),∇(a)).

Corollary 6.2. [22] Let a = (an)n∈N ∈ S. Then the selection principleα2(∇(a),∇(a)) holds.

Another similar situation is the following.

Let a = (an)n∈N ∈ S and μ > 0 be fixed. A sequence b = (bn)n∈N ∈ Sμ-dominates a if there is n0 = n0(μ) such that an ≤ μ · bn for all n ≥ n0.

The symbol {a}μ denotes the set of all sequences in S which μ-dominate a.

Theorem 6.3. [22] Let a = (an)n∈N ∈ S and μ > 0 be fixed. Then α2({a}μ, {a}μ)holds.

6.1. Rapid variation and αi-selection properties I. In [32], L. de Haanintroduced the notion of rapid variation, which is, in a sense, a special kind ofKaramata’s regular variation. In [20], a fundamental result regarding relationshipsbetween selection principles and rapid variation was shown (see Theorems 6.4 and6.5 below).

A sequence (an)n∈N in S is rapidly varying (of index of variability ∞) if

limn→∞

a[λn]

an= ∞

for each λ > 1.

R∞,s denotes the class of rapidly varying sequences.

Theorem 6.4. [20] The player TWO has a winning coding strategy in the gameG1(R∞,s,R∞,s).

From this theorem we obtain that the selection property S1(R∞,s,R∞,s) is sat-isfied. However, this property is equivalent to each of the properties αi(R∞,s,R∞,s),i = 2, 3, 4, so that we have:

Theorem 6.5. [20] The class R∞,s satisfies the following (equivalent) proper-ties:

(1) S1(R∞,s,R∞,s);(2) α2(R∞,s,R∞,s);(3) α3(R∞,s,R∞,s);(4) α4(R∞,s,R∞,s).

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These two results suggest the following two natural tasks:

Task 1. Improve Theorem 6.5 changing coordinates!

Task 2. What about regularly varying sequences?

6.2. Rapid variation and αi-selection properties II. Theorem 6.5 can besignificantly improved by changing both coordinates in it: the first coordinate witha bigger class, and the second coordinate by a smaller class of sequences.

A sequence x = (xn)n∈N ∈ S is in the class Tr(R∞,s) of translationally rapidlyvarying sequences [24] if for each λ ≥ 1, the following condition holds:

limn→∞

x[n+λ]

xn= ∞.

A sequence x = (xn)n∈N ∈ S is in the class ARVs if for each λ > 1 there aren0 ∈ N and c(λ) > 1 such that for each n ≥ n0 the following condition is satisfied:

x[λn] ≥ c(λ) · xn.

Let us observe [24] that

Tr(R∞,s) � R∞,s � ARVs.

Let x = (xn)n∈N be in S. For each k ∈ N define a new sequence V (k)(x) =

(V(k)n (x))n∈N inductively by

V (1)n (x) :=

xn+1

xn, n ∈ N;

V (k+1)n (x) :=

V(k)n+1(x)

V(k)n (x)

, n ∈ N.

The sequence V (k)(x) we call the quotient sequence of x of order k. We also putV (0)(x) = x.

The k-th quotient speed vk(x) of a sequence x = (xn)n∈N is ∞ if

limn→∞

V (k)n (x) = ∞.

PutTr(1)(R∞,s) = Tr(R∞,s),

for each k ≥ 2Tr(k)(R∞,s) = {x ∈ S : vk(x) = ∞},

and

Tr(∞)(R∞,s) =

∞⋂

k=1

Tr(k)(R∞,s).

According to a result from [25] it holds

Tr(1)(R∞,s) � Tr(2)(R∞,s) � · · · � Tr(k)(R∞,s) � · · ·The following theorem is an improvement of Theorem 6.5 we mentioned above

(and also of a theorem from [23]).

Theorem 6.6. [25] The following (equivalent) selection properties are all sat-isfied:

(1) S1(ARVs,Tr(2)(R∞,s));

(2) α2(ARVs,Tr(2)(R∞,s));

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14 LJ.D.R. KOCINAC

(3) α3(ARVs,Tr(2)(R∞,s));

(4) α4(ARVs,Tr(2)(R∞,s)).

The proof of this theorem is based on the fact that each sequence from ARVs

contains an unbounded subsequence belonging to ARVs (for a proof see [25], andalso [23]).

In fact, the previous theorem remains true if we take Tr(k)(R∞,s) instead of

Tr(2)(R∞,s), but the proof in this case is technically much more difficult.

6.3. Regular variation. We answer now the question posed at the end ofsubsection 6.1 (see Task 2 there).

A sequence x = (xn)n∈N ∈ S is regularly varying if for each λ > 0 it satisfies

kx(λ) := limn→∞

x[λn]

xn< ∞.

It is well known that kx(λ) = λρ for some ρ ∈ R called the index of variabilityof x (see [6]). If ρ = 0, then x is said to be slowly varying (in the sense of Karamata[37]).

RVs, RVρ,s, SVs denote the classes of regularly varying sequences, regularlyvarying sequences of index of variability ρ, and slowly varying sequences, respec-tively.

The following result shows that the situation with regularly varying sequencesin the second coordinate is quite different from the situations described in Theorem6.4 and Theorem 6.6.

Theorem 6.7. [24] ONE has a winning strategy in the game G1(ARVs,RVs)(in particular in the game G1(ARVs, SVs)).

A similar situation is with another class of sequences.A sequence x = (xn)n∈N ∈ S is in the class Tr(RVs) of translationally regularly

varying sequences if for each λ ∈ R,

rx(λ) := limn→∞

x[n+λ]

xn< ∞.

It was shown in [24] that rx(λ) = eρ[λ] for some ρ ∈ R, the index of variabilityof x.

Let us denote by Tr(RVs) the class of translationally regularly varying se-quences, and by Tr(RVρ,s) the class of translationally varying sequences of index ofvariability ρ.

Notice that for ρ > 0, Tr(RVρ,s) � R∞,s.Unlike the result from Theorem 6.4, and similarly to the result in Theorem 6.7,

we have:

Theorem 6.8. [24] For any ρ > 0 the player ONE has a winning strategy inG1(Tr(RVρ,s),Tr(RVs)).

However, we have a different situation in some special cases.The symbol G∗

fin(A,B) denotes the following infinitely long game for two play-ers ONE and TWO: in the n-th round, n ≥ 2, ONE chooses an element An ∈ A,and TWO responds by choosing a finite set Bn ⊂ An−1 ∪ An. Two wins a playA1, B1; · · · ;An, Bn; · · · if ∪n∈NBn ∈ B; otherwise ONE wins.

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Let IuTr(RV0,s) denote the class of strictly increasing, unbounded translation-ally regularly varying sequences of index of variability 0. Then the following issatisfied.

Theorem 6.9. [25] TWO has a winning strategy in the game G∗fin(IuTr(RV0,s), IuTr(RV0,s)).

7. Statistical convergence and αi’s

The idea of statistical convergence appeared (under the name almost conver-gence) in the first edition (Warsaw, 1935) of the celebrated monograph [69] ofZygmund. However, the notion of statistical convergence of sequences of real num-bers was explicitly introduced by H. Fast in [27] and H. Steinhaus in [63] (actually,in 1949 on a conference held at the Wroclaw University in Poland) and is based onthe notion of asymptotic (or natural) density of a set of natural numbers.

In this section we present several results and ideas of applications of statisticalconvergence to selection principles theory, in particular to αi-selection principles.For this kind of investigation see [15] and [17].

For a set A ⊂ N the asymptotic density of A, denoted δ(A) is defined as

δ(A) = limn→∞|{k∈A:k≤n}|

n .

In [17], using the idea of statistical convergence, we introduced the followingselection principles.

For families A and B of subsets of a set X the symbol st-αi(A,B), i = 1, 2, 3, 4,denotes that for each sequence (An = {an,m : m ∈ N} : n ∈ N) of countably infiniteelements of A there is a B ∈ B with:

st-α1(A,B): for each n ∈ N, δ({m ∈ N : an,m ∈ An \B}) = 0;st-α2(A,B): for each n ∈ N, δ({m ∈ N : an,m ∈ An ∩B}) = 1;st-α3(A,B): there is a set K ⊂ N such that δ(K) = 1 and for each k ∈ K,

δ({m ∈ N : ak,m ∈ Ak ∩B}) = 1;st-α4(A,B): there is K ⊂ N with δ(K) = 1 and for all k ∈ K, Ak ∩B �= ∅.

If in the previous definitions of st-αi properties, i = 2, 3, 4, we replace every-where asymptotic density 1 by positive asymptotic density, we obtain the definitionsof the properties st-α2, st-α3 and st-α4, respectively [15].

Evidently:

st-αi(A,B) ⇒ st-αi+1(A,B), (i = 1, 2, 3),α1(A,B) ⇒ st-α1(A,B),

st-α2(A,B) ⇒ st-α3(A,B) ⇒ st-α4(A,B),st-αi(A,B) ⇒ st-αi(A,B) ⇒ αi(A,B), (i = 2, 3, 4).

Investigation of these selection principles is at the very beginning and I believethat a systematic study in this direction can give good contributions to selectionprinciples theory.

We are going to mention a few results about sequences of positive real numbers.Our notation is as notation in Section 6; recall, especially, that S denotes the setof sequences of positive real numbers.

The following theorem is a statistical version of Theorem 6.3.

Theorem 7.1. [15] Let a = (an)n∈N ∈ S and μ > 0 be fixed. Then st-α2({a}μ, {a}μ) holds.

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16 LJ.D.R. KOCINAC

Let a = (an)n∈N ∈ S be fixed. A sequence b = (bn)n∈N in S is said to be stronglyasymptotically equivalent to a if for every μ > 1, b ∈ {a}μ and a ∈ {b}μ.

For a fixed a ∈ S, [a] is the set of all sequences from S which are stronglyasymptotically equivalent to a.

Let A and B be subsets of S. The symbol G(A,B) denotes the following game:two players, ONE and TWO, play a round for each n ∈ N; in the n-th round ONEchooses a sequence sn ∈ A, and TWO responds by choosing a subset Tn from sn ofpositive asymptotic density. TWO wins a play (s1, T1; · · · ; sn, Tn; · · · ) if

⋃n∈N

Tn

can be arranged in a sequence from B; otherwise, ONE wins.Evidently, if TWO has a winning strategy in G(A,B), then st-α2(A,B) is sat-

isfied.

Theorem 7.2. [15] Let a = (an)n∈N ∈ S be given. Then TWO has a winningstrategy in the game G([a], [a]).

Corollary 7.3. Let a ∈ S be given. Then st-α2([a], [a]) is true.

8. Closing remarks

After the paper was submitted Boaz Tsaban solved some problems from thisarticle. He proved the following:

1. (Problem 2.8) The property α1(Γ,Γ) is not productive.2. (Problem 2.10) Both classes α1(Γ,Γ) and α1(Γk,Γk) are b-additive.3. (Problem 2.11) The property α1(Γ,Γ) is hereditary for Fσ-subsets.4. (Problem 2.12) non(α1(Γk,Γk)) = b.5. (Problem 2.13) add(α1(Γ,Γ)) = b and add(α1(Γk,Γk)) = b.6. (Problem 2.14) The answer to Problem 2.14 is ”Yes”.

Acknowledgement. This survey is written on the basis of my invited lecture,delivered at the XVIII Boise Extravaganza in Set Theory (March, 27–29, 2009,Boise, Idaho). I thank the organizers for inviting me to write this survey and fortheir hospitality during the conference. I also thank the referee for a number ofuseful comments and suggestions.

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(1994), 371–382.29. D.H. Fremlin, SSP and wQN, Notes, January 14, 2003.30. C. Hernandez, Topological groups close to being σ-compact, Topology Appl. 102 (2000),

101–111.

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Topology Appl. 73 (1996), 241–266.37. J. Karamata, Sur un mode de croissance reguliere des fonctions, Mathematica (Cluj) 4

(1930), 38–53.38. Lj.D.R. Kocinac, Selected results on selection principles, Proc. Third Seminar on Geom-

etry and Topology, July 15–17, 2004, Tabriz, Iran, 71–104.39. Lj.D.R. Kocinac, γ-sets, γk-sets and hyperspaces, Math. Balkanica 19 (2005), 109–118.40. Lj.D.R. Kocinac, Some covering properties in topological and uniform spaces, Proc. Steklov

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(1990), 143–183.61. W. Sierpinski, Sur un probleme de M. Menger, Fund. Math. 8 (1926), 223–224.62. W. Sierpinski, Sur un ensemble nondenombrable, donc toute image continue est de mesure

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1979.

Faculty of Sciences and Mathematics, Visegradska 33, 18000 Nis, Serbia


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Jensen’s diamond principle and its relatives

Assaf Rinot

Abstract. We survey some recent results on the validity of Jensen’s diamondprinciple at successor cardinals. We also discuss weakening of this principlesuch as club guessing, and anti-diamond principles such as uniformization.

A collection of open problems is included.

Introduction. Cantor’s continuum hypothesis has many equivalent formula-tions in the context of ZFC. One of the standard formulations asserts the existenceof an enumeration {Aα | α < ω1} of the set P(ω). A non-standard, twisted, formu-lation of CH is as follows:

(∃)2 there exists a sequence, 〈Aα | α < ω1〉, such that for every subset Z ⊆ ω1,there exist two infinite ordinals α, β < ω1 such that Z ∩ β = Aα.

By omitting one of the two closing quantifiers in the above statement, we arriveto the following enumeration principle:

(∃)1 there exists a sequence, 〈Aα | α < ω1〉, such that for every subset Z ⊆ ω1,there exists an infinite ordinal α < ω1 such that Z ∩ α = Aα.

Jensen discovered this last principle during his analysis of Godel’s constructibleuniverse, and gave it the name of diamond, ♦. In [28], Jensen proved that ♦ holdsin the constructible universe, and introduced the very first ♦-based constructionof a complicated combinatorial object — a Souslin tree. Since then, this principleand generalizations of it became very popular among set theorists who utilized it tosettle open problems in fields including topology, measure theory and group theory.

In this paper, we shall be discussing a variety of diamond-like principles forsuccessor cardinals, including weak diamond, middle diamond, club guessing, sta-tionary hitting, and λ+-guessing, as well as, anti-diamond principles including theuniformization property and the saturation of the nonstationary ideal.

An effort has been put toward including a lot of material, while maintainingan healthy reading flow. In particular, this survey cannot cover all known resultson this topic. Let us now briefly describe the content of this survey’s sections, andcomment on the chosen focus of each section.

2000 Mathematics Subject Classification. Primary 03E05; Secondary 03E35, 03E50.Key words and phrases. Diamond, Uniformization, Club Guessing, Stationary Hitting,

Souslin Trees, Saturation, Square, Approachability, SAP.The author has been an invited speaker of 18th annual B.E.S.T. conference; I would like to

express my deep gratitude to the organizers for the invitation and the warm hospitality.


1



125



125

2 ASSAF RINOT

Organization of this paper. In Section 1, Jensen’s diamond principles, ♦S ,♦∗

S , ♦+S , are discussed. We address the question to which stationary sets S ⊆ λ+,

does 2λ = λ+ imply ♦S and ♦∗S , and describe the effect of square principles and

reflection principles on diamond. We discuss a GCH-free version of diamond, which

is called stationary hitting, and a reflection-free version of ♦∗S , denoted by ♦λ+

S . Inthis section, we only deal with the most fundamental variations of diamond, andhence we can outline the whole history.

Section 2 is dedicated to describing part of the set theory generated by White-head problem. We deal with the weak diamond, ΦS , and the uniformization prop-erty. Here, rather than including all known results in this direction, we decidedto focus on presenting the illuminating proofs of the characterization of weak dia-mond in cardinal-arithmetic terms, and the failure of instances of the uniformizationproperty at successor of singular cardinals.

In Section 3, we go back to the driving force to the study of diamond —the Souslin hypothesis. Here, we focus on aggregating old, as well as, new openproblems around the existence of higher souslin trees, and the existence of particularclub guessing sequences.

Section 4 deals with non-saturation of particular ideals — ideals of the formNSλ+ � S. Here, we describe the interplay between non-saturation, diamond andweak-diamond, and we focus on presenting the recent results in this line of research.

Notation and conventions. For ordinals α < β, we denote by (α, β) := {γ |α < γ < β}, the open interval induced by α and β. For a set of ordinals C, wedenote by acc(C) := {α < sup(C) | sup(C ∩ α) = α}, the set of all accumulationpoints of C. For a regular uncountable cardinal, κ, and a subset S ⊆ κ, let

Tr(S) := min{γ < κ | cf(γ) > ω, S ∩ γ is stationary in γ}.We say that S reflects iff Tr(S) �= ∅, is non-reflecting iff Tr(S) = ∅, and reflectsstationarily often iff Tr(S) is stationary.

For cardinals κ < λ, denote Eλκ := {α < λ | cf(α) = κ}, and [λ]κ := {X ⊆ λ |

|X| = κ}. Eλ>κ and [λ]<κ are defined analogously. Cohen’s notion of forcing for

adding κ many λ-Cohen sets is denoted by Add(λ, κ). To exemplify, the forcingnotion for adding a single Cohen real is denoted by Add(ω, 1).

Contents

1. Diamond2. Weak Diamond and the Uniformization Property3. The Souslin Hypothesis and Club Guessing4. Saturation of the Nonstationary IdealIndexReferences

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126

JENSEN’S DIAMOND PRINCIPLE AND ITS RELATIVES 3

1. Diamond

Recall Jensen’s notion of diamond in the context of successor cardinals.

Definition 1.1 (Jensen, [28]). For an infinite cardinal λ and stationary subsetS ⊆ λ+:

� ♦S asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ α;• if Z is a subset of λ+, then the following set is stationary:

{α ∈ S | Z ∩ α = Aα}.Jensen isolated the notion of diamond from his original construction of an ℵ1-

Souslin tree from V = L; in [28], he proved that ♦ω1witnesses the existence of

such a tree, and that:

Theorem 1.2 (Jensen, [28]). If V = L, then ♦S holds for every stationaryS ⊆ λ+ and every infinite cardinal λ.

In fact, Jensen established that V = L entails stronger versions of diamond,two of which are the following.

Definition 1.3 (Jensen, [28]). For an infinite cardinal λ and stationary subsetS ⊆ λ+:

� ♦∗S asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ P(α) and |Aα| ≤ λ;• if Z is a subset of λ+, then the there exists a club C ⊆ λ+ such that:

C ∩ S ⊆ {α ∈ S | Z ∩ α ∈ Aα}.� ♦+

S asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ P(α) and |Aα| ≤ λ;• if Z is a subset of λ+, then the there exists a club C ⊆ λ+ such that:

C ∩ S ⊆ {α ∈ S | Z ∩ α ∈ Aα & C ∩ α ∈ Aα}.Kunen [34] proved that ♦∗

S ⇒ ♦T for every stationary T ⊆ S ⊆ λ+, and that♦λ+ cannot be introduced by a λ+-c.c. notion of forcing.

Since, for a stationary subset S ⊆ λ+, ♦+S ⇒ ♦∗

S ⇒ ♦S ⇒ ♦λ+ ⇒ (2λ = λ+),it is natural to study which of these implications may be reversed.

Jensen (see [7]) established the consistency of ♦∗ω1

+ ¬♦+ω1, from the existence

of an inaccessible cardinal. In [46], it is observed that if λℵ0 = λ, then for everystationary S ⊆ λ+, ♦∗

S is equivalent to ♦+S . Devlin [8], starting with a model of

V |= GCH, showed that V Add(λ+,λ++) |= ¬♦∗λ+ + ♦λ+ .1 Jensen proved that, in

general, the implication ♦λ+ ⇒ (2λ = λ+), may not be reversed:

Theorem 1.4 (Jensen, see [10]). CH is consistent together with ¬♦ω1.

On the other hand, Gregory, in a paper that deals with higher Souslin trees,established the following surprising result.

1For this, he argued that if G is Add(λ+, 1)-generic over V , then

(1) V [G] |= ♦S for every stationary S ⊆ λ+ from V , and(2) every sequence 〈Aα | α < λ+〉 that witnesses ♦∗

λ+ in V , will cease to witness ♦∗λ+ in

V [G].

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4 ASSAF RINOT

Theorem 1.5 (Gregory, [25]). Suppose λ is an uncountable cardinal, 2λ = λ+.If σ < λ is an infinite cardinal such that λσ = λ, then ♦∗

Eλ+σ

holds.

In particular, GCH entails ♦∗Eλ+

<cf(λ)

for any cardinal λ of uncountable cofinality.

Unfortunately, it is impossible to infer ♦λ+ from GCH using Gregory’s theorem,in the case that λ > cf(λ) = ω. However, shortly afterwards, this missing case hasbeen settled by Shelah.

Theorem 1.6 (Shelah, [52]). Suppose λ is a singular cardinal, 2λ = λ+.If σ < λ is an infinite cardinal such that sup{μσ | μ < λ} = λ, and σ �= cf(λ),

then ♦∗Eλ+

σ

holds.

In particular, GCH entails ♦∗Eλ+

�=cf(λ)

for every uncountable cardinal, λ.

A closer look at the proof of Theorems 1.5, 1.6 reveals that moreover ♦+

Eλ+σ

may

be inferred from the same assumptions, and, more importantly, that the hypothesisinvolving σ may be weakened to: “sup{cf([μ]σ,⊇) | μ < λ} = λ”. However, itwas not clear to what extent this weakening indeed witnesses more instances ofdiamonds.

Then, twenty years after proving Theorem 1.6, Shelah established that theabove weakening is quite prevalent. In [63], he proved that the following conse-quence of GCH follows outright from ZFC.

Theorem 1.7 (Shelah, [63]). If θ is an uncountable strong limit cardinal, thenfor every cardinal λ ≥ θ, the set {σ < θ | cf([λ]σ,⊇) > λ} is bounded below θ.

In particular, for every cardinal λ ≥ �ω, the following are equivalent:

(1) 2λ = λ+;(2) ♦λ+ ;(3) ♦∗

Eλ+σ

for co-boundedly many σ < �ω.

Let CHλ denote the assertion that 2λ = λ+. By Theorems 1.4 and 1.7, CHλ

does not imply ♦λ+ for λ = ω, but does imply ♦λ+ for every cardinal λ ≥ �ω. Thisleft a mysterious gap between ω and �ω, which was only known to be closed in thepresence of the stronger cardinal arithmetic hypotheses, as in Theorem 1.5.

It then took ten additional years until this mysterious gap has been completelyclosed, where recently Shelah proved the following striking theorem.

Theorem 1.8 (Shelah, [68]). For an uncountable cardinal λ, and a stationary

subset S ⊆ Eλ+

�=cf(λ), the following are equivalent:

(1) CHλ;(2) ♦S .

Remark. In [31], Komjath provides a simplified presentation of Shelah’s proof.Also, in [44] the author presents a considerably shorter proof.2

Having Theorem 1.8 in hand, we now turn to studying the validity of ♦S for

sets of the form S ⊆ Eλ+

cf(λ). Relativizing Theorem 1.5 to the first interesting

case, the case λ = ℵ1, we infer that GCH entails ♦∗E

ω2ω. By Devlin’s theorem [8],

GCH �⇒ ♦∗ℵ2, and consequently, GCH does not imply ♦∗

Eℵ2ω1

. Now, what about the

2See the discussion after Theorem 1.19 below.

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unstarred version of diamond? It turns out that the behavior here is analogous tothe one of Theorem 1.4.

Theorem 1.9 (Shelah, see [29]). GCH is consistent with ¬♦S, for S = Eω2ω1.

The proof of Theorem 1.9 generalizes to successor of higher regular cardinals,suggesting that we should focus our attention on successors of singulars. Andindeed, a longstanding, still open, problem is the following question.

Question 1 (Shelah). Is it consistent that for some singular cardinal λ, CHλ

holds, while ♦Eλ+

cf(λ)

fails?

In [55, §3], Shelah established that a positive answer to the above question —in the case that λ is a strong limit — would entail the failure of weak square,3 andhence requires large cardinals. More specifically:

Theorem 1.10 (Shelah, [55]). Suppose λ is a strong limit singular cardinal,

and �∗λ holds. If S ⊆ Eλ+

cf(λ) reflects stationarily often, then CHλ ⇒ ♦S.

Applying ideas of the proof of Theorem 1.8 to the proof Theorem 1.10, Zemanestablished a “strong limit”-free version of the preceding.

Theorem 1.11 (Zeman, [75]). Suppose λ is a singular cardinal, and �∗λ holds.

If S ⊆ Eλ+

cf(λ) reflects stationarily often, then CHλ ⇒ ♦S .

The curious reader may wonder on the role of the reflection hypothesis in thepreceding two theorems; in [55, §2], Shelah established the following counterpart:

Theorem 1.12 (Shelah, [55]). Suppose CHλ holds for a strong limit singular

cardinal, λ. If S ⊆ Eλ+

cf(λ) is a non-reflecting stationary set, then there exists a

notion of forcing PS such that:

(1) PS is λ-distributive;(2) PS satisfies the λ++-c.c.;(3) S remains stationary in V PS ;(4) V PS |= ¬♦S.

In particular, it is consistent that GCH+�∗λ holds, while ♦S fails for some

non-reflecting stationary set S ⊆ Eλ+

cf(λ).

The next definition suggests a way of filtering out the behavior of diamond onnon-reflecting sets.

Definition 1.13 ([44]). For an infinite cardinal λ and stationary subsetsT, S ⊆ λ+:

� ♦TS asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ P(α) and |Aα| ≤ λ;• if Z is a subset of λ+, then the following set is non-stationary:

T ∩ Tr{α ∈ S | Z ∩ α �∈ Aα}.

3The weak square property at λ, denoted �∗λ, is the principle �λ,λ as in Definition 3.8.

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6 ASSAF RINOT

Notice that by Theorem 1.6, GCH entails ♦λ+

λ+ for every regular cardinal λ.

Now, if λ is singular, then GCH does not necessarily imply ♦λ+

λ+ ,4 however, if in

addition �∗λ holds, then GCH does entail ♦λ+

λ+ , as the following improvement oftheorem 1.10 shows.

Theorem 1.14 ([44]). For a strong limit singular cardinal, λ:

(1) if �∗λ holds, then CHλ ⇔ ♦λ+

λ+ ;

(2) if every stationary subset of Eλ+

cf(λ) reflects, then ♦λ+

λ+ ⇔ ♦∗λ+ .

Remark. An interesting consequence of the preceding theorem is that assum-ing GCH, for every singular cardinal, λ, �∗

λ implies that in the generic extensionby Add(λ+, 1), there exists a non-reflecting stationary subset of λ+. This is a rem-iniscent of the fact that �λ entails the existence non-reflecting stationary subset ofλ+.

Back to Question 1, it is natural to study to what extent can the weak squarehypothesis in Theorem 1.11 be weakened. We now turn to defining the axiom SAPλ

and describing its relation to weak square and diamond.

Definition 1.15 ([44]). For a singular cardinal λ and S ⊂ λ+, consider theideal I[S;λ]: a set T is in I[S;λ] iff T ⊆ Tr(S) and there exists a function d :[λ+]2 → cf(λ) such that:

• d is subadditive: α < β < γ < λ+ implies d(α, γ) ≤ max{d(α, β), d(β, γ)};• d is normal : for all i < cf(λ) and β < λ+, |{α < β | d(α, β) ≤ i}| < λ;

• key property : for some club C ⊆ λ+, for every γ ∈ T ∩C ∩Eλ+

>cf(λ), there

exists a stationary Sγ ⊆ S ∩ γ with sup(d“[Sγ ]2) < cf(λ).

Evidently, if I[S;λ] contains a stationary set, then S reflects stationarily often.The purpose of the next definition is to impose the converse implication.

Definition 1.16 ([44]). For a singular cardinal λ, the stationary approacha-bility property at λ, abbreviated SAPλ, asserts that I[S;λ] contains a stationary

set for every stationary S ⊆ Eλ+

cf(λ) that reflects stationarily often.

Our ideal I[S;λ] is a variation of Shelah’s approachability ideal I[λ+], andthe axiom SAPλ is a variation of the approachability property, APλ.

5 We shall becomparing these two principles later, but let us first compare SAPλ with �∗

λ.In [44], it is observed that for every singular cardinal λ, �∗

λ ⇒ SAPλ, andmoreover, �∗

λ entails the existence of a function, d : [λ+]2 → cf(λ), that servesas a unified witness to the fact for all S ⊆ λ+, Tr(S) ∈ I[S;λ]. Then, startingwith a supercompact cardinal, a model is constructed in which (1) GCH+SAPℵω

holds, (2) every stationary subset of Eℵω+1ω reflects stationarily often, and (3) for

4Start with a model of GCH and a supercompact cardinal κ. Use backward Easton supportiteration of length κ + 1, forcing with Add(α+ω+1, α+ω+2) for every inaccessible α ≤ κ. Now,work in the extension and let λ := κ+ω . Then the GCH holds, κ remains supercompact, and byDevlin’s argument [8], ♦∗

λ+ fails. Since cf(λ) < κ < λ, and κ is supercompact, we get that every

stationary subset of Eλ+

cf(λ)reflects, and so it follows from Theorem 1.14(2), that ♦λ+

λ+ fails in this

model of GCH.5For instance, if λ > cf(λ) > ω is a strong limit, then I[λ+] = P(Eλ+

ω ) ∪ I[Eλ+

ω ;λ]. For thedefinition of I[λ+] and APλ, see [15].

130130


every stationary S ⊆ Eℵω+1ω and any function d witnessing that I[S;ℵω] contains a

stationary set, there exists another stationary S′ ⊆ Eℵω+1ω such that this particular d

does not witness the fact that I[S′;ℵω] contains a stationary set. Thus, establishing:

Theorem 1.17 ([44]). It is relatively consistent with the existence of a super-compact cardinal, that SAPℵω

holds, while �∗ℵω

fails.

Once it is established that SAPλ is strictly weaker than �∗λ, the next task

would be proving that it is possible to replace �∗λ in Theorem 1.11 with SAPλ,

while obtaining the same conclusion. The proof of this fact goes through a certaincardinal-arithmetic-free version of diamond, which we now turn to define.

Definition 1.18 ([44]). For an infinite cardinal λ and stationary subsetsT, S ⊆ λ+, consider the following two principles:

� ♣−S asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ [α]<λ and |Aα| ≤ λ;• if Z is a cofinal subset of λ+, then the following set is stationary:

{α ∈ S | ∃A ∈ Aα(sup(Z ∩A) = α)} .� ♣−

S � T asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα ⊆ [α]<λ and |Aα| ≤ λ;• if Z is a stationary subset of T , then the following set is non-empty:

{α ∈ S | ∃A ∈ Aα(sup(Z ∩A) = α)} .

Notice that ♣−S makes sense only in the case that S ⊆ Eλ+

<λ. In [44], it is

established that the stationary hitting principle, ♣−S � λ+, is equivalent to ♣−

S , andthat these equivalent principles are the cardinal-arithmetic-free version of diamond:

Theorem 1.19 ([44]). For an uncountable cardinal λ, and a stationary subset

S ⊆ Eλ+

<λ, the following are equivalent:

(1) ♣−S +CHλ;

(2) ♦S .

It is worth mentioning that the proof of Theorem 1.19 is surprisingly short,and when combined with the easy argument that ZFC � ♣−

S for every stationary

subset S ⊆ Eλ+

�=cf(λ), one obtains a single-page proof of Theorem 1.8.

It is also worth mentioning the functional versions of these principles.

Fact 1.20. Let λ denote an infinite cardinal, and S denote a stationary subsetof λ+; then:

� ♦S is equivalent to the existence of a sequence 〈gα | α ∈ S〉 such that:• for all α ∈ S, gα : α → α is some function;• for every function f : λ+ → λ+, the following set is stationary:

{α ∈ S | f � α = gα}.� ♣−

S is equivalent to the existence of a sequence 〈Gα | α ∈ S〉 such that:• for all α ∈ S, Gα ⊆ [α× α]<λ and |Gα| ≤ λ;• for every function f : λ+ → λ+, the following set is stationary:

{α ∈ S | ∃G ∈ Gα sup{β < α | (β, f(β)) ∈ G} = α} .

131131

8 ASSAF RINOT

Finally, we are now in a position to formulate a theorem of local nature, fromwhich we derive a global corollary.

Theorem 1.21 ([44]). Suppose λ is a singular cardinal, and S ⊆ λ+ is astationary set. If I[S;λ] contains a stationary set, then ♣−

S holds.

Corollary 1.22 ([44]). Suppose SAPλ holds, for a given singular cardinal, λ.Then the following are equivalent:

(1) CHλ;(2) ♦S holds for every S ⊆ λ+ that reflects stationarily often.

Thus, the hypothesis �∗λ from Theorem 1.11 may indeed be weakened to SAPλ.

Having this positive result in mind, one may hope to improve the preceding, prov-ing that CHλ ⇒ ♦S for every S ⊆ λ+ that reflects stationarily often, without anyadditional assumptions. Clearly, this would have settle Question 1 (in the nega-tive!). However, a recent result by Gitik and the author shows that diamond mayfail on a set that reflects stationarily often, and even on an (ω1 + 1)-fat subset ofℵω+1:

Theorem 1.23 (Gitik-Rinot, [22]). It is relatively consistent with the exis-tence of a supercompact cardinal that the GCH holds above ω, while ♦S fails for a

stationary set S ⊆ Eℵω+1ω such that:

{γ < ℵω+1 | cf(γ) = ω1, S ∩ γ contains a club} is stationary.

In fact, the above theorem is just one application of the following general, ZFCresult.

Theorem 1.24 (Gitik-Rinot, [22]). Suppose CHλ holds for a strong limit sin-gular cardinal, λ. Then there exists a notion of forcing P, satisfying:

(1) P is λ+-directed closed;(2) P has the λ++-c.c.;(3) |P| = λ++;

(4) in V P, ♦S fails for some stationary S ⊆ Eλ+

cf(λ).

Note that unlike Theorem 1.14, here the stationary set on which diamond fails,is a generic one.

Utilizing the forcing notion from Theorem 1.24, Gitik and the author were ableto show that Corollary 1.22 is optimal: in [22], it is proved that replacing the SAPλ

hypothesis in Corollary 1.22 with APλ, or with the existence of a better scale forλ, or even with the existence of a very good scale for λ, is impossible, in the sensethat these alternative hypotheses do not entail diamond on all reflecting stationarysets.6 In particular:

Theorem 1.25 (Gitik-Rinot, [22]). It is relatively consistent with the existenceof a supercompact cardinal that APℵω

holds, while SAPℵωfails.

Moreover, in the model from Theorem 1.25, every stationary subset of Eℵω+1ω

reflects. Recalling that APℵωholds whenever every stationary subset of ℵω+1 re-

flects, we now arrive to the following nice question.

6The existence of a better scale at λ, as well as the approachability property at λ, are well-known consequences of �∗

λ. For definitions and proofs, see [15].

132132


Question 2. Is it consistent that every stationary subset of ℵω+1 reflects,while SAPℵω

fails to hold?

To summarize the effect of square-like principles on diamond, we now state

a corollary. Let Reflλ denote the assertion that every stationary subset of Eλ+

cf(λ)

reflects stationarily often. Then:

Corollary 1.26. For a singular cardinal, λ:

(1) GCH+�∗λ �⇒ ♦∗

λ+ ;(2) GCH+Reflλ +�∗

λ ⇒ ♦∗λ+ ;

(3) GCH+Reflλ +SAPλ �⇒ ♦∗λ+ ;

(4) GCH+Reflλ +SAPλ ⇒ ♦S for every stationary S ⊆ λ+;(5) GCH+Reflλ +APλ �⇒ ♦S for every stationary S ⊆ λ+.

Proof. (1) By Theorem 1.12. (2) By Theorem 1.14. (3) By the proof ofTheorem 1.17 in [44]. (4) By Corollary 1.25. (5) By the proof of Theorem 1.25 in[22]. �

The combination of Theorems 1.19 and 1.21 motivates the study of the idealI[S;λ]. For instance, a positive answer to the next question would supply an answerto Question 1.

Question 3. Must I[Eλ+

cf(λ);λ] contain a stationary set for every singular car-

dinal λ?

One of the ways of attacking the above question involves the following reflectionprinciples.

Definition 1.27 ([44]). Assume θ > κ are regular uncountable cardinals.R1(θ, κ) asserts that for every function f : Eθ

<κ → κ, there exists some j < κsuch that {δ ∈ Eθ

κ | f−1[j] ∩ δ is stationary} is stationary in θ.R2(θ, κ) asserts that for every function f : Eθ

<κ → κ, there exists some j < κsuch that {δ ∈ Eθ

κ | f−1[j] ∩ δ is non-stationary} is non-stationary.

It is not hard to see that R2(θ, κ) ⇒ R1(θ, κ), and thatMM implies R1(ℵ2,ℵ1)+¬R2(ℵ2,ℵ1). In [44], a fact from pcf theory is utilized to prove:

Theorem 1.28 ([44]). Suppose λ > cf(λ) = κ > ω are given cardinals.

The ideal I[Eλ+

cf(λ);λ] contains a stationary set whenever the following set isnon-empty:

{θ < λ | R1(θ, κ) holds}.

As a corollary, one gets a surprising result stating that a local instance ofreflection affects the validity of diamond on a proper class of cardinals.

Corollary 1.29 (implicit in [68]). Suppose κ is the successor of a cardinal

κ−, and that every stationary subset of Eκ+

κ− reflects.Then, CHλ ⇔ ♦

Eλ+

cf(λ)

for every singular cardinal λ of cofinality κ.

As the reader may expect, the principle R2 yields a stronger consequence.

Theorem 1.30 ([44]). Suppose θ > κ are cardinals such that R2(θ, κ) holds.Then:

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10 ASSAF RINOT

(1) For every singular cardinal λ of cofinality κ, and every S ⊆ λ+, we have

Tr(S) ∩ Eλ+

θ ∈ I[S;λ].

(2) if λ is a strong limit singular cardinal of cofinality κ, then CHλ ⇔ ♦Eλ+

θ

λ+ .

Unfortunately, there is no hope to settle Question 3 using these reflection prin-ciples, as they are independent of ZFC: by a theorem of Harrington and Shelah[26], R1(ℵ2,ℵ1) is equiconsistent with the existence of a Mahlo cardinal, whereas,by a theorem of Magidor [38], R2(ℵ2,ℵ1) is consistent modulo the existence of aweakly-compact cardinal. An alternative sufficient condition for I[S;λ] to containa stationary set will be described in Section 4 (See Fact 4.14 below).

2. Weak Diamond and the Uniformization Property

Suppose that G and H are abelian groups and π : H → G is a given epimor-phism. We say that π splits iff there exists an homomorphism φ : G → H such thatπ◦φ is the identity function on G. An abelian group G is free iff every epimorphismonto G, splits.

Whitehead problem reads as follows.

Question. Suppose that G is an abelian group such that every epimorphismπ onto G with the property that ker(π) � Z — splits;7

Must G be a free abelian group?

Thus, the question is whether to decide the freeness of an abelian group, itsuffices to verify that only a particular, narrow, class of epimorphism splits. Stein[71] solved Whitehead problem in the affirmative in the case that G is a countableabelian group. Then, in a result that was completely unexpected, Shelah [49]proved that Whitehead problem, restricted to groups of size ω1, is independentof ZFC. Roughly speaking, by generalizing Stein’s proof, substituting a counting-based diagonalization argument with a guessing-based diagonalization argument,Shelah proved that if ♦S holds for every stationary S ⊆ ω1, then every abeliangroup of size ω1 with the above property is indeed free. On the other hand, heproved that if MAω1

holds, then there exists a counterexample of size ω1.Since CH holds in the first model, and fails in the other, it was natural to

ask whether the existence of a counterexample to Whitehead problem is consistenttogether with CH. This led Shelah to introducing the uniformization property.

Definition 2.1 (Shelah, [50]). Suppose that S is a stationary subset of asuccessor cardinal, λ+.

• A ladder system on S is a sequence of sets of ordinals, 〈Lα | α ∈ S〉, suchthat sup(Lα) = α and otp(Lα) = cf(α) for all α ∈ S;

• A ladder system 〈Lα | α ∈ S〉 is said to have the uniformization propertyiff whenever 〈fα : Lα → 2 | α ∈ S〉 is a given sequence of local functions,then there exists a global function f : λ+ → 2 such that fα ⊆∗ f for alllimit α ∈ S. That is, sup{β ∈ Lα | fα(β) �= f(β)} < α for all limit α ∈ S.

Theorem 2.2 (Shelah, [53]; see also [17]). The following are equivalent:

• there exists a counterexample of size ω1 to Whitehead problem;

7Here, Z stands for the usual additive group of integers.

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• there exists a stationary S ⊆ ω1, and a ladder system on S that has theuniformization property.

Devlin and Shelah proved [11] that if MAω1holds, then every stationary S ⊆ ω1

and every ladder system on S, has the uniformization property. On the otherhand, it is not hard to see that if ♦S holds, then no ladder system on S has theuniformization property (See Fact 2.5, below). Note that altogether, this gives analternative proof to the independence result from [49].

Recalling that ¬♦ω1is consistent with CH (See Theorem 1.4), it seemed rea-

sonable to suspect that CH is moreover consistent with the existence of a laddersystem on ω1 that has the uniformization property. Such a model would also showthat the existence of a counterexample to Whitehead problem is indeed consistenttogether with CH, settling Shelah’s question.

However, a surprising theorem of Devlin states that CH implies that no laddersystem on ω1 has the uniformization property. Then, in a joint paper with Shelah,the essence of Devlin’s proof has been isolated, and a weakening of diamond whichis strong enough to rule out uniformization has been introduced.

Definition 2.3 (Devlin-Shelah, [11]). For an infinite cardinal λ and a station-ary set S ⊆ λ+, consider the principle of weak diamond.

� ΦS asserts that for every function F : <λ+

2 → 2, there exists a functiong : λ+ → 2, such that for all f : λ+ → 2, the following set is stationary:

{α ∈ S | F (f � α) = g(α)}.Note that by Fact 1.20, ♦S ⇒ ΦS . The difference between these principles is

as follows. In diamond, for each function f , we would like to guess f � α, while inweak diamond, we only aim at guessing the value of F (f � α), i.e., whether f � αsatisfies a certain property — is it black or white. A reader who is still dissatisfiedwith the definition of weak diamond, may prefer one of its alternative formulations.

Fact 2.4 (folklore). For an infinite cardinal λ and a stationary set S ⊆ λ+,the following principles are equivalent:

� ΦS;

� for every function F : <λ+

λ+ → 2, there exists a function g : S → 2, suchthat for all f : λ+ → λ+, the following set is stationary:

{α ∈ S | F (f � α) = g(α)}.� for every sequence of functions 〈Fα : P(α) → 2 | α ∈ S〉, there exists a

function g : S → 2, such that for every subset X ⊆ λ+, the following setis stationary:

{α ∈ S | Fα(X ∩ α) = g(α)}.Back to uniformization, we have:

Fact 2.5 (Devlin-Shelah, [11]). For every stationary set S, ΦS (and hence♦S) entails that no ladder system 〈Lα | α ∈ S〉 has the uniformization property.

Proof (sketch). For all α ∈ S and i < 2, let ciα : Lα → {i} denote the

constant function. Pick a function F : <λ+

2 → 2 such that for all α ∈ S and i < 2,if f : α → 2 and ciα ⊆∗ f , then F (f) = i. Now, let g : λ+ → 2 be given by applying

ΦS to F . Then, letting fα := c1−g(α)α for all α ∈ S, the sequence 〈fα | α ∈ S〉

cannot be uniformized. �

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12 ASSAF RINOT

Before we turn to showing that CH ⇒ Φω1, let us mention that since Φλ+ deals

with two-valued functions, its negation is an interesting statement of its own right:

Fact 2.6. Suppose that Φλ+ fails for a given infinite cardinal, λ.

Then there exists a function F : <λ+

(λ2) → λ2 such that for every g : λ+ → λ2,there exists a function f : λ+ → λ2, for which the following set contains a club:

{α < λ+ | F (f � α) = g(α)}.

Roughly speaking, the above states that there exists a decipher, F , such thatfor every function g, there exists a function f that F -ciphers the value of g(α) asf � α.

Since the (easy) proof of the preceding utilizes the fact that weak diamond dealswith two-valued functions, it is worth mentioning that Shelah also studied general-ization involving more colors. For instance, in [61], Shelah gets weak diamond formore colors provided that NSω1

is saturated (and Φω1holds).8

We now turn to showing that CH ⇒ Φω1. In fact, the next theorem shows that

weak diamond is a cardinal arithmetic statement in disguise. The proof given hereis somewhat lengthier than other available proofs, but, the value of this proof isthat its structure allows the reader to first neglect the technical details (by skippingthe proofs of Claims 2.7.1, 2.7.2), while still obtaining a good understanding of thekey ideas.

Theorem 2.7 (Devlin-Shelah, [11]). For every cardinal λ, Φλ+ ⇔ 2λ < 2λ+

.

Proof. ⇒ Assume Φλ+ . Given an arbitrary function ψ : λ+

2 → λ2, we now

define a function F : <λ+

2 → 2 such that by appealing to Φλ+ with F , we can showthat ψ is not injective.

Given f ∈ <λ+

2, we let F (f) := 0 iff there exists a function h ∈ λ+

2 such thath(dom(f)) = 0 and f ⊆ ψ(h) ∪ (h � [λ, λ+)).

Let g : λ+ → 2 be the oracle given by Φλ+ when applied to F , and let h : λ+ → 2be the function satisfying h(α) = 1− g(α) for all α < λ+.

Put f := ψ(h) ∪ (h � [λ, λ+)). Since f ∈ λ+

2, let us pick some α < λ+

with α > λ such that F (f � α) = g(α). Since f � α ⊆ ψ(h) ∪ (h � [λ, λ+)),the definition of F implies that F (f � α) = 0 whenever h(α) = 0. However,F (f � α) = g(α) �= h(α), and hence h(α) = 1. Since, F (f � α) = g(α) = 0, letus pick a function h′ such that h′(α) = 0 and f � α ⊆ ψ(h′) ∪ (h′ � [λ, λ+)). Bydefinition of f , we get that ψ(h) = f � λ = ψ(h′). By g(α) = 0, we also know thath(α) = 1 �= h′(α), and hence h �= h′, while ψ(h) = ψ(h′).

⇐ Given a function H : <λ+

(λ2) → <λ+

(λ2), let us say that a sequence〈(fn, Dn) | n < ω〉 is an H-prospective sequence iff:

(1) {Dn | n < ω} is a decreasing chain of club subsets of λ+;(2) for all n < ω, fn is a function from λ+ to λ2;(3) for all n < ω and α ∈ Dn+1, the following holds:

H(fn+1 � α) = fn � min(Dn \ α+ 1).

Note that the intuitive meaning of the third item is that there exists β > αsuch that the content of fn � β is coded by fn+1 � α.

8For the definition of “NSω1 is saturated” see Definition 4.1 below.

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Claim 2.7.1. Assume ¬Φλ+ .

Then there exists a function H : <λ+

(λ2) → <λ+

(λ2) such that for every func-tion f : λ+ → λ2, there exists an H-prospective sequence 〈(fn, Dn) | n < ω〉 withf0 = f .

Proof. Fix F as in Fact 2.6, and fix a bijection ϕ : λ2 → <λ+

(λ2). PutH := ϕ ◦ F . Now, given f : λ+ → λ2, we define the H-prospective sequence byrecursion on n < ω. Start with f0 := f and D0 := λ+. Suppose n < ω and fn andDn are defined. Define a function g : λ+ → λ2 by letting for all α < λ+:

g(α) := ϕ−1(fn � min(Dn \ α+ 1)).

By properties of F , there exists a function fn+1 and a club Dn+1 ⊆ Dn such thatfor all α ∈ Dn+1, we have F (fn+1 � α) = g(α). In particular,

H(fn+1 � α) = (ϕ ◦ F )(fn+1 � α) = (ϕ ◦ g)(α) = fn � min(Dn \ α+ 1). �

Claim 2.7.2. Given a function H : <λ+

(λ2) → <λ+

(λ2), there exists a function

H∗ : ω(<λ+

(λ2)) → ω(<λ+

(λ2)) with the following stepping-up property.For every H-prospective sequence, 〈(fn, Dn) | n < ω〉, and every α ∈

⋂n<ω Dn,

there exists some α∗ < λ+, such that:

(1) α∗ > α;(2) α∗ ∈

⋂n<ω Dn;

(3) H∗(〈fn � α | n < ω〉) = 〈fn � α∗ | n < ω〉.

Proof. Given H, we define functions Hm : ω(<λ+

(λ2)) → ω(<λ+

(λ2)) by

recursion on m < ω. For all σ : ω → <λ+

(λ2), let:

H0(σ) := σ,

and whenever m < ω is such that Hm is defined, let:

Hm+1(σ) := 〈H(Hm(σ)(n+ 1)) | n < ω〉.Finally, define H∗ by letting for all σ : ω → <λ+

(λ2):

H∗(σ) := 〈⋃

m<ω

Hm(σ)(n) | n < ω〉.

To see that H∗ works, fix an H-prospective sequence, 〈(fn, Dn) | n < ω〉, andsome α ∈

⋂n<ω Dn. Define 〈〈αm

n | n < ω〉 | m < ω〉 by letting α0n := α for all

n < ω. Then, given m < ω, for all n < ω, let:

αm+1n := min(Dn \ αm

n+1 + 1).

(1) Put α∗ := supm<ω αm0 . Then α∗ ≥ α1

0 > α01 = α.

(2) If n < ω, then Dn ⊇ Dn+1, and hence αm+1n+1 ≥ αm+1

n > αmn+1 for all m < ω.

This shows that supm<ω αmn = supm<ω αm

n+1 for all n < ω.For n < ω, since 〈αm

n | m < ω〉 is a strictly increasing sequence of ordinals fromDn that converges to α∗, we get that α∗ ∈ Dn.

(3) Let us prove by induction that for all m < ω:

Hm(〈fn � α | n < ω〉) = 〈fn � αmn | n < ω〉.

Induction Base: Trivial.Induction Step: Suppose m < ω is such that:

(�) Hm(〈fn � α | n < ω〉) = 〈fn � αmn | n < ω〉,

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14 ASSAF RINOT

and let us show that:

Hm+1(〈fn � α | n < ω〉) = 〈fn � αm+1n | n < ω〉.

By definition of Hm+1 and equation (�), this amounts to showing that:

〈H(fn+1 � αmn+1) | n < ω〉 = 〈fn � αm+1

n | n < ω〉.

Fix n < ω. Recalling the definition of αm+1n , we see that we need to prove that

H(fn+1 � αmn+1) = fn � min(Dn \αm

n+1 +1). But this follows immediately from thefacts that αm

n+1 ∈ Dn+1, and that 〈(fn, Dn) | n < ω〉 is an H-prospective sequence.Thus, it has been established that:

H∗(〈fn � α | n < ω〉) = 〈fn �⋃

m<ω

αmn | n < ω〉 = 〈fn � α∗ | n < ω〉. �

Now, assume ¬Φλ+ , and let us prove that 2λ+

= 2λ by introducing an injection

of the form ψ : λ+

(λ2) → ω(<λ+

2). Fix H as in Claim 2.7.1, and let H∗ be givenby Claim 2.7.2 when applied to this fixed function, H.

� Given a function f : λ+ → λ2, we pick an H-prospective sequence 〈(fn, Dn) |n < ω〉 with f0 = f and let ψ(f) := 〈fn � α | n < ω〉 for α := min(

⋂n<ω Dn).

To see that ψ is injective, we now define a function ϕ : ω(<λ+

2) → ≤λ+

(λ2)such that ϕ ◦ ψ is the identity function.

� Given a sequence σ : ω → <λ+

2, we first define an auxiliary sequence 〈στ |τ ≤ λ+〉 by recursion on τ . Let σ0 := σ, στ+1 := H∗(στ ), and στ (n) :=

⋃η<τ ση(n)

for limit τ ≤ λ+ and n < ω. Finally, let ϕ(σ) := σλ+(0).

Claim 2.7.3. ϕ(ψ(f)) = f for every f : λ+ → λ2.

Proof. Fix f : λ+ → λ2 and let σ := ψ(f). By definition of ψ, σ = 〈fn � α |n < ω〉 for some H-prospective sequence 〈(fn, Dn) | n < ω〉 and α ∈

⋂n<ω Dn. It

then follows from the choice of H∗, that there exists a strictly increasing sequence,〈ατ | τ < λ+〉, of ordinals from

⋂n<ω Dn, such that στ := 〈fn � ατ | n < ω〉 for all

τ < λ+, and then ϕ(ψ(f)) = ϕ(σ) = σλ+(0) = f0 � λ+ = f . �

This completes the proof. �

Evidently, Devlin’s pioneering theorem that CH excludes the existence of aladder system on ω1 with the uniformization property now follows from Fact 2.5and Theorem 2.7. It is interesting to note that if one considers the notion ofweak uniformization, in which the conclusion of Definition 2.1 is weakened fromsup{β ∈ Lα | fα(β) �= f(β)} < α to sup{β ∈ Lα | fα(β) = f(β)} = α, then we endup with an example of an anti-♦S principle, which is not an anti-ΦS principle:

Theorem 2.8 (Devlin, see [3]). It is consistent with GCH (and hence withΦω1

) that every ladder system on every stationary subset of ω1 has the weak uni-formization property.

Back to Whitehead problem, Shelah eventually established the consistency ofCH together with the existence of a counterexample:

Theorem 2.9 (Shelah, [50]). It is consistent with GCH+♦ω1that there exists

a stationary, co-stationary, set S ⊆ ω1 such that any ladder system on S has theuniformization property.

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It is worth mentioning that Shelah’s model was also the first example of a modelin which ♦ω1

holds, while for some stationary subset S ⊆ ω1, ♦S fails .We now turn to dealing with the uniformization property for successor of un-

countable cardinals. By Theorem 1.8 and Fact 2.5, there is no hope for getting a

model of GCH in which a subset of Eλ+

�=cf(λ) carries a ladder system that has the

uniformization property, so let us focus on sets of the critical cofinality. The firstcase that needs to be considered is Eω2

ω1, and the full content of Theorem 1.9 is now

revealed.

Theorem 2.10 (Shelah, [29],[51]). It is consistent with GCH that there existsa ladder system on Eω2

ω1with the uniformization property.

Knowing that 2ℵ1 = ℵ2 implies ♦Eω2ω

but not ♦Eω2ω1, and that 2ℵ1 < 2ℵ2 implies

Φω2but not ΦE

ω2ω1, one may hope to prove that 2ℵ1 < 2ℵ2 moreover implies ΦE

ω2ω.

However, a consistent counterexample to this conjecture is provided in [56].Note that Theorem 2.10 states that there exists a particular ladder system on

Eω2ω1

with the uniformization property, rather than stating that all ladder systems

on Eω2ω1

have this property.9 To see that Theorem 2.10 is indeed optimal, considerthe following theorem.

Theorem 2.11 (Shelah, [62]). Suppose that λ is a regular cardinal of the form

2θ for some cardinal θ, and that 〈Lα | α ∈ Eλ+

λ 〉 is a given ladder system.

If, moreover, Lα is a club subset of α for all α ∈ Eλ+

λ , and 2<λ = λ, then there

exists a coloring 〈fα : Lα → 2 | α ∈ Eλ+

λ 〉 such that for every function f : λ+ → 2,the following set is stationary:

{α ∈ Eλ+

λ | {β ∈ Lα | fα(β) �= f(β)} is stationary in α}.

In particular, CH entails the existence of a ladder system on Eω2ω1

that does notenjoy the uniformization property.

The proof of Theorem 2.10 generalizes to successor of higher regular cardinals,

showing that there may exist a ladder system on Eλ+

λ that enjoys the uniformizationproperty. Hence, we now turn to discuss the uniformization property at successorof singulars. We commence with revealing the richer content of Theorem 1.12.


cardinal, λ. If S ⊆ Eλ+

cf(λ) is a non-reflecting stationary set, then there exists a

notion of forcing PS such that:

(1) PS is λ-distributive;(2) PS satisfies the λ++-c.c.;(3) S remains stationary in V PS ;(4) in V PS , there exists a ladder system on S that has the uniformization

property.

By Theorem 1.23, it is consistent that diamond fails on a set that reflectsstationarily often. Now, what about the following strengthening:

9Compare with [11, Theorem 5.2], stating that under MAω1 , every ladder system on ω1 has

the uniformization property.

139139

16 ASSAF RINOT

Question 4. Is it consistent with GCH that for some singular cardinal, λ,

there exists a stationary set S ⊆ Eλ+

cf(λ) that reflects stationarily often, and a ladder

system on S that has the uniformization property?

Remark. By Corollary 1.22, SAPλ necessarily fails in such an hypotheticalmodel.

Now, what about the existence of ladder systems that do not enjoy the uni-formization property? Clearly, if λ is a strong limit singular cardinal, then Theorem2.11 does not apply. For this, consider the following.

Fact 2.13 (Shelah, [65]). Suppose CHλ holds for a strong limit singular cardi-nal, λ. Then, for every stationary S ⊆ λ+, there exists a ladder system on S thatdoes not enjoy the uniformization property.

Proof. Fix a stationary S ⊆ λ+. If S∩Eλ+

�=cf(λ) is stationary, then by Theorem

1.8, ♦S holds, and then by Fact 2.5, moreover, no ladder system on S has the

uniformization property. Next, suppose S ⊆ Eλ+

cf(λ) is a given stationary set. By the

upcoming Theorem 2.14, in this case, we may pick a ladder system 〈Lα | α ∈ S〉such that for every function f : λ+ → 2, there exists some α ∈ S such that if{αi | i < cf(λ)} denotes the increasing enumeration of Lα, then f(α2i) = f(α2i+1)for all i < cf(λ).

It follows that if for each α ∈ S, we pick fα : Lα → 2 satisfying for all β ∈ Lα:

fα(β) =

{0, ∃i < cf(λ)(β = α2i)

1, otherwise,

then the sequence 〈fα | α ∈ S〉 cannot be uniformized. �

Remark. Note that the sequence 〈fα | α ∈ S〉 that was derived in the preced-ing proof from the guessing principle of Theorem 2.14, is a sequence of non-constantfunctions that cannot be uniformized. To compare, the sequence that was derivedfrom weak diamond in the proof of Fact 2.5 is a sequence of constant functions. Inother words, weak diamond is stronger in the sense that it entails the existence ofa monochromatic coloring that cannot be uniformized.


cardinal, λ, S ⊆ Eλ+

cf(λ) is stationary and μ < λ is a given cardinal.

Then there exists a ladder system 〈Lα | α ∈ S〉 so that if {αi | i < cf(λ)}denotes the increasing enumeration of Lα, then for every function f : λ+ → μ, thefollowing set is stationary:

{α ∈ S | f(α2i) = f(α2i+1) for all i < cf(λ)}.

Proof. Without loss of generality, λ divides the order-type of α, for all α ∈ S.Put κ := cf(λ) and θ := 2κ. By 2λ = λ+, let {dγ | γ < λ+} be some enumerationof {d : θ × τ → μ | τ < λ+}.

Fix α ∈ S. Let 〈cαi | i < κ〉 be the increasing enumeration of some club subset ofα, such that (cαi , c

αi+1) has cardinality λ for all i < κ. Also, let {bαi | i < κ} ⊆ [α]<λ

be a continuous chain converging to α with bαi ⊆ cαi for all i < κ. Recall that wehave fixed α ∈ S; now, in addition, we also fix i < κ.

140140


For all j < κ, define a function ψj = ψα,i,j : (cαi , c

αi+1) → θ×bαj (μ+1) such that

for all ε ∈ (cαi , cαi+1) and (β, γ) ∈ θ × bαj :

ψj(ε)(β, γ) =

{dγ(β, ε), (β, ε) ∈ dom(dγ)

μ, otherwise.

For all j < κ, since |θ×bαj (μ+1)| < λ = |(cαi , cαi+1)| , let us pick two ordinals αji,0, α

ji,1

with cαi < αji,0 < αj

i,1 < cαi+1 such that ψj(αji,0) = ψj(α

ji,1).

For every function g ∈ κκ, consider the ladder system 〈Lgα | α ∈ S〉, where

Lgα := {αg(i)

i,0 , αg(i)i,1 | i < κ}.

Claim 2.14.1. There exists some g ∈ κκ such that 〈Lgα | α ∈ S〉 works.

Proof. Suppose not. Let {gβ | β < θ} be some enumeration of κκ. Then, forall β < θ, we may pick a function fβ : λ+ → μ and a club Eβ such that for allα ∈ S ∩ Eβ, there exists some i < κ such that

fβ(αgβ(i)i,0 ) �= fβ(α

gβ(i)i,1 ).

Now, let h : λ+ → λ+ be the function such that for all ε < λ+:

h(ε) = min{γ < λ+ | ∀(β, ε) ∈ θ × ε (dγ(β, ε) is defined and equals fβ(ε))}.Pick α ∈ S ∩

⋂β<θ Eβ such that h[α] ⊆ α.

Then we may define a function g : κ → κ by letting:

g(i) := min{j < κ | h(cαi+1) ∈ bαj }.Let β < θ be such that g = gβ and fix i < κ such that

fβ(αgβ(i)i,0 ) �= fβ(α

gβ(i)i,1 ).

Put j := g(i). By definition of αji,0

and αji,1

, we know that ψα,i,j(αji,0

) =

ψα,i,j(αji,1

) is a function from θ × bαj to μ+ 1.

Put γ := h(cαi+1); then (β, γ) ∈ θ × bαj , and hence:

ψα,i,j(αji,0

)(β, γ) = ψα,i,j(αji,1

)(β, γ).

It now follows from αji,0

< αji,1

< cαi+1 and γ = h(cαi+1), that:

fβ(αji,0

) = dγ(β, αji,0) = ψα,i,j(α

ji,0

)(β, γ) = ψα,i,j(αji,1

)(β, γ) = dγ(β, αji,1) = fβ(α

ji,1

)

Unrolling the notation, we must conclude that

fβ(αgβ(i)i,0 ) = fβ(α

ji,0

) = fβ(αji,1

) = fβ(αgβ(i)i,1 ),

thus, yielding a contradiction to α ∈ Eβ . �

Thus, it has been established that there exists a ladder system with the desiredproperties. �

In light of Theorem 1.24, the moral of Theorem 2.14 is that GCH entails someof the consequences of diamond, even in the case that diamond fails. Two naturalquestions concerning this theorem are as follows.

Question 5. Is it possible to eliminate the “strong limit” hypothesis fromTheorem 2.14, while maintaining the same conclusion?

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18 ASSAF RINOT

Question 6. Is Theorem 2.14 true also for the case that μ = λ?

Note that an affirmative answer to the last question follows from ♦S . In fact,even if 2λ > λ+, but Ostaszewski’s principle, ♣S , holds, then a ladder system as inTheorem 2.14 for the case μ = λ, exists.

Definition 2.15 (Ostaszewski, [42]). Let λ denote an infinite cardinal, and Sdenote a stationary subset of λ+. Consider the following principle.

� ♣S asserts that there exists a sequence 〈Aα | α ∈ S〉 such that:• for all α ∈ S, Aα is a cofinal subset of α;• if Z is a cofinal subset of λ+, then the following set is stationary:

{α ∈ S | Aα ⊆ Z} .

It is worth mentioning that unlike ♣−S , the principle ♣S makes sense also in

the case that S ⊆ Eλ+

λ . In particular, the missing case of Theorem 1.19 may becompensated by the observation that ♦S is equivalent to ♣S + CHλ. It is alsoworth mentioning that ♣λ+ + ¬CHλ is indeed consistent; for instance, in [53],Shelah introduces a model of ♣ω1

+ ¬Φω1.

Next, consider Theorem 2.14 for the case that μ = cf(λ). In this case, thetheorem yields a collection L ⊆ [λ+]cf(λ) of size λ+, such that for every functionf : λ+ → cf(λ), there exists some L ∈ L such that f � L is not injective (insome strong sense). Apparently, this fact led Shelah and Dzamonja to consider thefollowing dual question.

Question. Suppose λ is a strong limit singular cardinal.Must there exist a collection P ⊆ [λ+]cf(λ) of size λ+ such that for every function

f : λ+ → cf(λ) which is non-trivial in the sense that∧

β<cf(λ) |f−1{β}| = λ+, there

exists some a ∈ P such that f � a is injective?

We shall be concluding this section by describing the resolution of the abovequestion. To refine the question, consider the following two definitions.

Definition 2.16. For a function f : λ+ → cf(λ), let κf denote the minimal

cardinality of a family P ⊆ [λ+]cf(λ) with the property that whenever Z ⊆ λ+

satisfies∧

β<cf(λ) |Z ∩ f−1{β}| = λ+, then there exist some a ∈ P with sup(f [a ∩Z]) = cf(λ).10

Definition 2.17. For a singular cardinal λ, we say that λ+-guessing holds iff

κf ≤ λ+ for all f ∈ λ+

cf(λ).

Answering the above-mentioned question in the negative, Shelah and Dzamonjaestablished the consistency of the failure of λ+-guessing.

Theorem 2.18 (Dzamonja-Shelah, [13]). It is relatively consistent with theexistence of a supercompact cardinal that there exist a strong limit singular cardinalλ and a function f : λ+ → cf(λ) such that κf = 2λ > λ+.

Recently, we realized that the above-mentioned question is simply equivalentto the question of whether every strong limit singular cardinal λ satisfies CHλ.

10Note that if λ is a strong limit, then we may assume that P is closed under taking subsets.Thus, we may moreover demand the existence of a ∈ P such that a ⊆ Z and f � a is injective.

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Theorem 2.19 ([22]). Suppose λ is a strong limit singular cardinal. Then:

{κf | f ∈ λ+

cf(λ)} = {0, 2λ}.In particular, if λ is a strong limit singular cardinal, then λ+-guessing happens

to be equivalent to the, seemingly, much stronger principle, ♦+

Eλ+

�=cf(λ)

.

3. The Souslin Hypothesis and Club Guessing

Recall that a λ+-Aronszajn tree is a tree of height λ+, of width λ, and withoutchains of size λ+. A λ+-Souslin tree is a λ+-Aronszajn tree that has no antichainsof size λ+.

Jensen introduced the diamond principle and studied its relation to Souslintrees.

Theorem 3.1 (Jensen, [28]). If λ<λ = λ is a regular cardinal such that ♦Eλ+

λ

holds, then there exists a λ+-Souslin tree.In particular, ♦ω1

entails the existence of an ω1-Souslin tree.

Theorem 3.2 (Jensen, see [10]). GCH is consistent together with the non-existence of an ω1-Souslin tree.

Remark. This is how Jensen proves Theorem 1.4. For a more modern proofof Theorem 3.2, see [2] or [3].

Let V denote the model from Theorem 1.4/3.2, and let P := Add(ω, 1) denoteCohen’s notion of forcing for introducing a single Cohen real. Since V |= ¬♦ω1

and since P is c.c.c., the discussion after Definition 1.3 shows that V P |= ¬♦ω1.

By a theorem of Shelah from [54], adding a Cohen real introduces an ω1-Souslintree, and hence V P is a model of CH witnessing the fact that the existence of anω1-Souslin tree does not entail ♦ω1

.Now, one may wonder what is the role of the cardinal arithmetic assumption

in Theorem 3.1? the answer is that this hypothesis is necessary. To exemplify thecase λ = ℵ1, we mention that PFA implies ♦+

Eλ+

λ

, but it also implies that λ<λ �= λ

and the non-existence of λ+-Aronszajn trees.11

So, ♦Eω2ω1

per se does not impose the existence of an ω2-Souslin tree. Also,

starting with a weakly compact cardinal, Laver and Shelah [37] established thatCH is consistent together with the non-existence of an ℵ2-Souslin tree. This leadsus to the following tenacious question.

Question 7 (folklore). Does GCH imply the existence of an ω2-Souslin tree?

An even harder question is suggested by Shelah in [64].

Question 8 (Shelah). Is it consistent that the GCH holds while for someregular uncountable λ, there exists neither λ+-Souslin trees nor λ++-Souslin trees?

Gregory’s proof of Theorem 1.5 appears in the paper [25] that deals with Ques-tion 7, and in which this theorem is utilized to supply the following partial answer.

Theorem 3.3 (Gregory, [25]). Assume GCH (or just CHω +CHω1).

If there exists a non-reflecting stationary subset of Eω2ω , then there exists an

ω2-Souslin tree.

11For an introduction to the Proper Forcing Axiom (PFA), see [9].

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20 ASSAF RINOT

It follows that the consistency strength of a negative answer to Question 7 isat least that of the existence of a Mahlo cardinal. Recently, B. Koenig suggestedan approach to show that the strength is at least that of the existence of a weaklycompact cardinal. Let �(ω2) denote the assertion that there exists a sequence〈Cα | α < ω2〉 such that for all limit α < ω2: (1) Cα is a club subset of α, (2) if β isa limit point of α, then Cα ∩ β = Cβ , (3) there exists no “trivializing” club C ⊆ ω2

such that C ∩ β = Cβ for all limit points β of C.The principle �(ω2) is a consequence of �ω1

,12 but its consistency strength ishigher — it is that of the existence of a weakly compact cardinal. Thus, Koenig’squestion is as follows.

Question 9 (B. Koenig). Does GCH+�(ω2) imply the existence of an ω2-Souslin tree?

In light of Theorem 3.1, to answer Question 7 in the affirmative, one probablyneeds to find a certain consequence of ♦E

ω2ω1

that, from one hand, follows outright

from GCH and which is, on the other hand, strong enough to allow the constructionof an ℵ2-Souslin tree. An example of ZFC-provable consequences of diamond isShelah’s family of club guessing principles. The next theorem exemplifies only afew out of many results in this direction.

Theorem 3.4 (several authors). For infinite cardinals μ ≤ λ, and a stationary

set S ⊆ Eλ+

μ , there exists a sequence−→C = 〈Cα | α ∈ S〉 such that for all α ∈ S, Cα

is a club in α of order-type μ, and:

(1) if μ < λ, then−→C may be chosen such that for every club D ⊆ λ+, the

following set is stationary:

{α ∈ S | Cα ⊆ D}.

(2) if ω < μ = cf(λ) < λ, then−→C may be chosen such that for almost all

α ∈ S, 〈cf(β) | β ∈ nacc(Cα)〉 is a strictly increasing sequence cofinal inλ, and for every club D ⊆ λ+, the following set is stationary:

{α ∈ S | Cα ⊆ D}.

(3) if V = L, then−→C may be chosen such that for every club D ⊆ λ+, the

following set contains a club subset of S:

{α ∈ S | ∃β < α(Cα \ β ⊆ D)}.

(4) if ω < cf(μ) = λ, then−→C may be chosen such that for every club D ⊆ λ+,

the following set is stationary:

{α ∈ S | {β ∈ Cα | min(Cα \ β + 1) ∈ D} is stationary in α}.

� Theorem 3.4(1) is due to Shelah [59], and the principle appearing therereflects the most naive form of club guessing. Personally, we are curious whetherthe guessing may concentrate on a prescribed stationary set T :

Question 10. Suppose that S, T are given stationary subsets of a successorcardinal λ+. Must there exist a sequence 〈Cα | α ∈ S〉 with sup(Cα) = α for allα ∈ S, such that for every club D ⊆ λ+, {α ∈ S | Cα ⊆ D ∩ T ∩ α} is stationary?

12The square property at λ, denoted �λ, is the principle �λ,1 as in Definition 3.8.

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A positive answer follows from ♣−S , and a negative answer is consistent for

various cardinals λ and non-reflecting sets S ⊆ λ+, hence one should focus on sets

S ⊆ Eλ+

cf(λ) that reflect stationarily often, and, e.g., T = Tr(S) ∪ S.

� Theorem 3.4(2) is due to Shelah [59], but see also Eisworth and Shelah [16].Roughly speaking, the principle appearing there requires that, in addition to thenaive club guessing, the non-accumulation points of the local clubs to be of highcofinality. An hard open problem is whether their assertion is valid also in the caseof countable cofinality.

Question 11 (Eisworth-Shelah). Suppose that λ is a singular cardinal of

countable cofinality. Must there exist a ladder system 〈Lα | α ∈ Eλ+

cf(λ)〉 such

that for almost all α, 〈cf(β) | β ∈ Lα〉 is a strictly increasing ω-sequence cofinal in

λ, and for every club D ⊆ λ+, the set {α ∈ Eλ+

cf(λ) | Lα ⊆ D} is stationary?

While the above question remains open, Eisworth recently established the va-lidity of a principle named off-center club guessing [14], and demonstrated that thenew principle can serve as a useful substitute to the principle of Question 11.

� Theorem 3.4(3) is due to Ishiu [27], and the principle appearing there isnamed strong club guessing . The “strong” stands for the requirement that theguessing is done on almost all points rather that on just stationary many. Histori-cally, Foreman and Komjath first proved in [20] that strong club guessing may beintroduced by forcing (See Theorem 4.17 below), and later on, Ishiu proved thatthis follows from V = L. In his paper, Ishiu asks whether V = L may be reducedto a diamond-type hypothesis. Here is a variant of his question.

Question 12. Suppose that ♦+λ+ holds for a given uncountable cardinal λ.

Must there exist a regular cardinal μ < λ, a stationary set S ⊆ Eλ+

μ , and a ladder

system 〈Lα | α ∈ S〉 such that for every club D ⊆ λ+, for club many α ∈ S, thereexists β < α with Lα \ β ⊆ D?

We mention that ♦+ω1

is consistent together with the failure of strong clubguessing over ω1 (see [36]), while, for an uncountable regular cardinal λ, and a

stationary S ⊆ Eλ+

λ , ♦∗S suffices to yield strong club guessing over S.

� Theorem 3.4(4) is due to Shelah [60], and a nice presentation of the proofmay be found in [69]. The prototype of this principle is the existence of a sequence

of local clubs, 〈Cα | α ∈ Eλ+

λ 〉, such that for every club D ⊆ λ+, there exists some

α ∈ Eλ+

λ with sup(nacc(Cα) ∩D) = α. Now, if {αi | i < λ} denotes the increasingenumeration of Cα, then Theorem 3.4(3) states that for every club D ⊆ λ+, thereexists stationarily many α ∈ S, for which not only that sup(nacc(Cα) ∩ D) = α,but moreover, {i < λ | αi+1 ∈ D} is stationary in λ. According to Shelah [64],to answer Question 7 in the affirmative, it suffices to find a proof of the followingnatural improvement.

Question 13 (Shelah). For a regular uncountable cardinal, λ, must there exist

a sequence 〈Cα | α ∈ Eλ+

λ 〉 with each Cα a club in α whose increasing enumerationis {αi | i < λ}, such that for every club D ⊆ λ+, there exists stationarily many α,for which {i < λ | αi+1 ∈ D and αi+2 ∈ D} is stationary in λ?

To exemplify the tight relation between higher Souslin trees and the precedingtype of club guessing, we mention the next principle.

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22 ASSAF RINOT

Definition 3.5 ([46]). Suppose λ is a regular uncountable cardinal, T is a

stationary subset of λ, and S is a stationary subset of Eλ+

λ .〈T 〉S asserts the existence of sequences 〈Cα | α ∈ S〉 and 〈Aα

i | α ∈ S, i < λ〉such that:

(1) for all α ∈ S, Cα is a club subset of α of order-type λ;(2) if for all α ∈ S, {αi | i < λ} denotes the increasing enumeration of

Cα, then for every club D ⊆ λ+ and every subset A ⊆ λ+, there existstationarily many α ∈ S for which:

{i ∈ T | αi+1 ∈ D & A ∩ αi+1 = Aαi+1} is stationary in λ.

It is obvious that ♦S ⇒ 〈T 〉S. It is also not hard to see that 〈T 〉S ⇒ ♦S

whenever NSλ � T is saturated.13 A strengthening of Theorem 3.1 is the following.

Theorem 3.6 (implicit in [30]). If λ<λ = λ is a regular uncountable cardinaland 〈λ〉

Eλ+

λholds, then there exists a λ+-Souslin tree.

We now turn to discuss Souslin trees at the of successor of singulars. By Magi-dor and Shelah [39], if λ is a singular cardinal which is a limit of strongly compactcardinals, then there are no λ+-Aronszajn trees. In particular, it is consistent withGCH that for some singular cardinal λ, there are no λ+-Souslin trees. On the otherhand, Jensen proved the following.

Theorem 3.7 (Jensen). For a singular cardinal λ, CHλ +�λ entails the exis-tence of a λ+-Souslin tree.

Since �λ ⇒ �∗λ and the latter still witnesses the existence of a λ+-Aronszajn

tree, the question which appears to be the agreed analogue of Question 7 is thefollowing.

Question 14 (folklore). For a singular cardinal λ, does GCH+�∗λ imply the

existence of a λ+-Souslin tree?

A minor modification of Jensen’s proof of Theorem 3.7 entails a positive an-swer to Question 14 provided that there exists a non-reflecting stationary subset of

Eλ+

�=cf(λ). However, by Magidor and Ben-David [4], it is relatively consistent with

the existence of a supercompact cardinal that the GCH holds, �∗ℵω

holds, and every

stationary subset of Eℵω+1

�=ω reflects.

A few years ago, Schimmerling [48] suggested that the community should per-haps try to attack a softer version of Question 14, which is related to the followinghierarchy of square principles.

Definition 3.8 (Schimmerling, [47]). For cardinals, μ, λ, �λ,<μ asserts theexistence of a sequence 〈Cα | α < λ+〉 such that for all limit α < λ+:

• 0 < |Cα| < μ;• C is a club subset of α for all C ∈ Cα;• if cf(α) < λ, then otp(C) < λ for all C ∈ Cα;• if C ∈ Cα and β ∈ acc(C), then C ∩ β ∈ Cβ .

We also write �λ,μ for �λ,<μ+ .

Question 15 (Schimmerling). Does GCH+�ℵω ,ω imply the existence of anℵω+1-Souslin tree?

13See Definition 4.1 below.

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In [1], Abraham, Shelah and Solovay showed that if CHλ +�λ holds for a givenstrong limit singular cardinal, λ, then a principle which is called square with built-in diamond may be inferred. Then, they continued to show how to construct aλ+-Souslin tree with a certain special property, based on this principle.

There are several variations of square-with-built-in-diamond principles (the firstinstance appearing in [24]), and several constructions of peculiar trees that utilizesprinciples of this flavor (see [5], [6], [30], [72]). Recalling the work of Abraham-Shelah-Solovay in [1], it seems reasonable to seek for a principle that ramifies thehypothesis of Question 15. Here is our humble suggestion.

Definition 3.9 ([43]). For cardinals, μ, λ, ♦λ,<μ asserts the existence of twosequences, 〈Cα | α < λ+〉 and 〈ϕθ | θ ∈ Γ〉, such that all of the following holds:

• 〈Cα | α < λ+〉 is a �λ,<μ-sequence;• ∅ �= Γ ⊆ {θ < λ+ | cf(θ) = θ};• ϕθ : P(λ+) → P(λ+) is a function, for all θ ∈ Γ;• for every subset A ⊆ λ+, every club D ⊆ λ+, and every cardinal θ ∈ Γ,

there exists some α ∈ Eλ+

θ such that for all C ∈ Cα:sup{β ∈ nacc(acc(C)) ∩D | ϕθ(C ∩ β) = A ∩ β} = α.

We write ♦λ,μ for ♦λ,<μ+ .

Notice that the above principle combines square, diamond and club guessing.The value of this definition is witnessed by the following.

Theorem 3.10 ([43]). Suppose that λ is an uncountable cardinal.If ♦λ,λ holds, then there exists a λ+-Souslin tree.

Remark. An interesting feature of the (easy) proof of the preceding theoremis that the construction does not depend on whether λ is a regular cardinal or asingular one.

It follows that if GCH+�ℵω,ω entails ♦ℵω ,ℵω, then this would supply an affir-

mative answer to Question 15. However, so far, a ramification is available only forthe case μ ≤ cf(λ).

Theorem 3.11 ([43]). For cardinals λ ≥ ℵ2, and μ ≤ cf(λ), the following areequivalent:

(a) �λ,<μ +CHλ;(b) ♦λ,<μ.

Remark. In the proof of (a)⇒(b), we obtain a ♦λ,<μ-sequence as in Definition3.9 for which, moreover, Γ is a non-empty final segment of {θ < λ | cf(θ) = θ}.

Clearly, in the presence of a non-reflecting stationary set, one can push Theorem3.11 much further (Cf. [43]). Thus, to see the difficulty of dealing with the caseμ = cf(λ)+, consider the following variation of club guessing.

Question 16. Suppose that λ is a singular cardinal, �λ,cf(λ) holds, and every

stationary subset of λ+ reflects.Must there exist a regular cardinal θ with cf(λ) < θ < λ and a �λ,cf(λ)-

sequence, 〈Cα | α < λ+〉, such that for every club D ⊆ λ+, there exists some

α ∈ Eλ+

θ satisfying sup(nacc(C) ∩D) = α for all C ∈ Cα?

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24 ASSAF RINOT

To conclude this section, let us mention two questions that suggests an alter-native generalizations of Theorems 3.1 and 3.7.

Question 17 (Juhasz). Does ♣ω1entail the existence of an ℵ1-Souslin tree?

Question 18 (Magidor). For a singular cardinal λ, does �λ entail the existenceof a λ+-Souslin tree?

Juhasz’s question is well-known and a description of its surrounding resultsdeserves a survey paper of its own. Here, we just mention that most of theseresults may be formulated in terms of the parameterized diamond principles of[41]. For instance, see [40].

To answer Magidor’s question, one needs to find a yet another GCH-free versionof diamond which suggests some non-trivial guessing features. In [66], Shelahintroduced a principle of this flavor, named Middle Diamond , and a corollary tothe results of [67, §4] reads as follows (compare with Definitions 1.1 and 2.15.)

Theorem 3.12 (Shelah, [67]). For every cardinal λ ≥ �ω1, there exist a finite

set d ⊆ �ω1, and a sequence 〈(Cα, Aα) | α < λ+〉 such that:

• for all limit α, Cα is a club in α, and Aα ⊆ Cα;• if Z is a subset of λ+, then for every regular cardinal κ ∈ �ω1

\ d, thefollowing set is stationary:

{α ∈ Eλ+

κ | Z ∩ Cα = Aα}.

For more information on the middle diamond, consult [45].

4. Saturation of the Nonstationary Ideal

Definition 4.1 (folklore). Suppose that S is a stationary subset of a cardinal,λ+. We say that NSλ+ � S is saturated iff for any family F of λ++ many stationarysubsets of S, there exists two distinct sets S1, S2 ∈ F such that S1∩S2 is stationary.

Of course, we say that NSλ+ is saturated iff NSλ+ � λ+ is saturated.

Now, suppose that ♦S holds, as witnessed by 〈Aα | α ∈ S〉. For every subsetZ ⊆ λ+, consider the set GZ := {α ∈ S | Z ∩ α = Aα}. Then GZ is stationaryand |GZ1

∩ GZ2| < λ+ for all distinct Z1, Z2 ∈ P(λ+). Thus, ♦S entails that

NSλ+ � S is non-saturated. For stationary subsets of Eλ+

<λ, an indirect proof ofthis last observation follows from Theorem 4.3 below. For this, we first remind ourreader that a set X ⊆ P(λ+) is said to be stationary (in the generalized sense) ifffor any function f : [λ+]<ω → λ+ there exists some X ∈ X with f“[X]<ω ⊆ X.

Definition 4.2 (Gitik-Rinot, [22]). For an infinite cardinal λ and a stationaryset S ⊆ λ+, consider the following two principles.

� (1)S asserts that there exists a stationary X ⊆ [λ+]<λ such that:• the sup-function on X is 1-to-1;• {sup(X) | X ∈ X} ⊆ S.

� (λ)S asserts that there exists a stationary X ⊆ [λ+]<λ such that:• the sup-function on X is (≤ λ)-to-1;• {sup(X) | X ∈ X} ⊆ S.

Theorem 4.3. For an uncountable cardinal λ, and a stationary set S ⊆ Eλ+

<λ,the implication (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) holds:

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(1) ♦S ;(2) (1)S;(3) (λ)S;(4) ♣−

S ;(5) NSλ+ � S is non-saturated.

Proof. For a proof of the implication (1) ⇒ (2) ⇒ (3) ⇒ (4), see [22]. Theproof of the last implication appears in [44], building on the arguments of [12]. �

Note that by Theorem 1.19, the first four items of the preceding theorem coin-cide assuming CHλ. In particular, the next question happens to be the contrapos-itive version of Question 1.

Question 19. Suppose that λ is a singular cardinal. Does CHλ entail theexistence of a stationary X ⊆ [λ+]<λ on which X �→ sup(X) is an injective map

from X to Eλ+

cf(λ)?

Back to non-saturation, since ZFC � ♣−S for every stationary subset S ⊆

Eλ+

�=cf(λ), one obtains the following analogue of Theorem 1.8.

Corollary 4.4 (Shelah, [59]). If λ is an uncountable cardinal, and S is a

stationary subset of Eλ+

�=cf(λ), then NSλ+ � S is non-saturated.

Thus, as in diamond, we are led to focus our attention on the saturation ofNSλ+ � S for stationary sets S which concentrates on the set of critical cofinality.

Kunen [33] was the first to establish the consistency of an abstract saturatedideal on ω1. As for the saturation of the ideal NSω1

, this has been obtained first bySteel and Van Wesep by forcing over a model of determinacy.

Theorem 4.5 (Steel-VanWesep, [70]). Suppose that V is a model of “ZF+ADR +Θis regular”. Then, there is a forcing extension satisfying ZFC+NSω1

is saturated.

Woodin [73] obtained the same conclusion while weakening the hypothesis tothe assumption “V = L(R) + AD”. Several years later, in [18], Foreman, Magidorand Shelah introduced Martin’s Maximum, MM, established its consistency from asupercompact cardinal, and proved that MM entails that NSω1

is saturated, andremains as such in any c.c.c. extension of the universe.

Then, in [58], Shelah established the consistency of the saturation of NSω1from

just a Woodin cardinal. Finally, recent work of Jensen and Steel on the existenceof the core model below a Woodin cardinal yields the following definite resolution.

Theorem 4.6 (Shelah, Jensen-Steel). The following are equiconsistent:

(1) ZFC+“there exist a woodin cardinal”;(2) ZFC+“NSω1

is saturated”.

However, none of these results serves as a complete analogue of Theorem 1.4in the sense that the following is still open.

Question 20 (folklore). Is CH consistent with NSω1being saturated?

Remark. By [43], “CH+NSω1is saturated” entails ♦ω1,ω1

.

Recalling that CH ⇒ Φω1, it is worth pointing out that while the saturation of

NSω1is indeed an anti-♦ω1

principle, it is not an anti-Φω1principle. To exemplify

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26 ASSAF RINOT

this, start with a model of MM and add ℵω1many Cohen reals over this model;

then as a consequence of Theorem 2.7 and the fact that Cohen forcing is c.c.c., oneobtains a model in which NSω1

is still saturated, while Φω1holds.

Let us consider a strengthening of saturation which does serve as an anti-Φλ+

principle.

Definition 4.7 (folklore). Suppose that S is a stationary subset of a cardinal,λ+. We say that NSλ+ � S is dense iff there exists a family F of λ+ many stationarysubsets of S, such that for any stationary subset S1 ⊆ S, there exists some S2 ∈ Fsuch that S2 \ S1 is non-stationary.

Of course, we say that NSλ+ is dense iff NSλ+ � λ+ is dense.

It is not hard to see that if NSλ+ � S is dense, then it is also saturated. Theabove discussion and the next theorem entails that these principles do not coincide.

Theorem 4.8 (Shelah, [57]). If Φω1holds, then NSω1

is not dense.

Improving Theorem 4.5, Woodin proved:

Theorem 4.9 (Woodin, [74]). Suppose that V is a model of “V = L(R)+AD”.Then there is a forcing extension of ZFC in which NSω1

is dense.

The best approximation for a positive answer to Question 20 is, as well, due toWoodin, who proved that CH is consistent together with NSω1

� S being dense forsome stationary S ⊆ ω1. Woodin also obtained an approximation for a negativeanswer to the very same question. By [74], if NSω1

is saturated and there exists ameasurable cardinal, then CH must fail.

As for an analogue of Theorem 1.9 — the following is completely open:

Question 21 (folklore). Is it consistent that NSω2� Eω2

ω1is saturated?

A major, related, result is the following unpublished theorem of Woodin (for aproof, see [19, §8].)

Theorem 4.10 (Woodin). Suppose that λ is an uncountable regular cardinaland κ is a huge cardinal above it. Then there exists a < λ-closed notion of forcingP, such that in V P the following holds:

(1) κ = λ+;

(2) there exists a stationary S ⊆ Eλ+

λ such that NSλ+ � S is saturated.

Moreover, if GCH holds in the ground model, then GCH holds in the extension.

Foreman, elaborating on Woodin’s proof, established the consistency of the

saturation of NSλ+ � S for some stationary set S ⊆ Eλ+

λ and a supercompactcardinal, λ, and showed that it is then possible to collapse λ to ℵω, while preservingsaturation. Thus, yielding:

Theorem 4.11 (Foreman). It is relatively consistent with the existence of asupercompact cardinal and an almost huge cardinal above it, that the GCH holds,

and NSℵω+1� S is saturated for some stationary S ⊆ E

ℵω+1ω .

Since the stationary set S was originally a subset of Eλ+

λ , it is a non-reflectingstationary set. This raises the following question.

Question 22 (folklore). Suppose that λ is a singular cardinal, and S ⊆ Eλ+

cf(λ)

reflects stationarily often, must NSλ+ � S be non-saturated?

150150


Recently, the author [44] found several partial answers to Question 22. To startwith, as a consequence of Theorem 1.21 and Theorem 4.3, we have:

Theorem 4.12 ([44]). Suppose S ⊆ λ+ is a stationary set, for a singularcardinal λ. If I[S;λ] contains a stationary set, then NSλ+ � S is non-saturated.

In particular, SAPλ (and hence �∗λ) impose a positive answer to Question 22.

Recalling Theorem 1.30, we also obtain the following.

Theorem 4.13 ([44]). If λ is a singular cardinal of uncountable cofinality andS ⊆ λ+ is a stationary set such that NSλ+ � S is saturated, then for every regularcardinal θ with cf(λ) < θ < λ, at least one of the two holds:

(1) R2(θ, cf(λ)) fails;

(2) Tr(S) ∩Eλ+

θ is nonstationary.

Next, to describe an additional aspect of Question 22, we remind our readerthat a set T ⊆ λ+ is said to carry a weak square sequence iff there exists sequence〈Cα | α ∈ T 〉 such that:

(1) Cα is a club subset of α of order-type ≤ λ, for all limit α ∈ T ;(2) |{Cα ∩ γ | α ∈ T}| ≤ λ for all γ < λ+.

Fact 4.14 ([44]). Suppose λ is a singular cardinal, and S ⊆ λ+ is a givenstationary set. If some stationary subset of Tr(S) carries a weak square sequence,then I[S;λ] contains a stationary set, and in particular, NSλ+ � S is non-saturated.

The consistency of the existence of a stationary set that does not carry a weaksquare sequence is well-known, and goes back to Shelah’s paper [52]. However, thefollowing question is still open.

Question 23. Suppose that λ is a singular cardinal. Must there exist a sta-

tionary subset of Eλ+

>cf(λ) that carries a partial weak square sequence?

Remark. The last question is closely related to a conjecture of Foreman andTodorcevic from [21, §6]. Note that by Fact 4.14 and Theorems 1.19, 1.21, a positiveanswer imposes a negative answer on Question 1.

Back to Question 22, still, there are a few ZFC results; the first being:

Theorem 4.15 (Gitik-Shelah, [23]). If λ is a singular cardinal, then NSλ+ �Eλ+

cf(λ) is non-saturated.

Gitik and Shelah’s proof utilizes the ZFC fact that a certain weakening of theclub guessing principle from Theorem 3.4(2) holds for all singular cardinal, λ. Then,

they show that if NSλ+ � Eλ+

cf(λ) were saturated, then their club guessing principle

may be strengthened to a principle that combines their variation of 3.4(2), togetherwith 3.4(3). However, as they show, this strong combination is already inconsistent.

In [32], Krueger pushed further the above argument, yielding the followinggeneralization.

Theorem 4.16 (Krueger, [32]). If λ is a singular cardinal and S ⊆ λ+ is astationary set such that NSλ+ � S is saturated, then S is co-fat.14

14Here, a set T ⊆ λ+ is fat iff for every cardinal κ < λ and every club D ⊆ λ+, T ∩ Dcontains some closed subset of order-type κ.

151151

28 ASSAF RINOT

To conclude this section, we mention two complementary results to the Gitik-Shelah argument.

Theorem 4.17 (Foreman-Komjath, [20]). Suppose that λ is an uncountableregular cardinal and κ is an almost huge cardinal above it. Then there exists anotion of forcing P, such that in V P the following holds:

(1) κ = λ+;

(2) there exists a stationary S ⊆ Eλ+

λ such that NSλ+ � S is saturated;

(3) Eλ+

μ carries a strong club guessing sequence for any regular μ ≤ λ.

Remark. By strong club guessing, we refer to the principle appearing in The-orem 3.4(3).

Theorem 4.18 (Woodin, [74]). Assuming ADL(R), there exists a forcing ex-tension of L(R) in which:

(1) NSω1is saturated;

(2) there exists a strong club guessing sequence on Eω1ω .

For interesting variations of Woodin’s theorem, we refer the reader to [35].

152152

Index

CH , 128λ

Ladder system, 134

Anti-♦ PrinciplesNSλ+ � S is dense, 150

NSλ+ � S is saturated, 148The uniformization property, 134

Guessing Principlesλ+-guessing, 142

(1) , 148S

(λ) , 148S

〈T 〉S , 146Φ , 135S

♦∗S , 127

♦+S , 127

♦TS , 129

♦S

Functional version, 131

Set version, 127

♣−SFunctional version, 131

Set version, 131

♣S , 142Club guessing, 144

Stationary hitting, 131Strong club guessing, 145

Middle Diamond, 148

Reflection Principles

R1(θ, κ), 133

R2(θ, κ), 133

Square PrinciplesI[S; λ], 130

SAP , 130λ

�(ω2), 144�∗

λ, 129

�λ, 144

�λ,<μ, 146♦λ,<μ, 147

Open Problems

Question 01, 129

Question 02, 133Question 03, 133

Question 04, 140



Question 09, 144


Question 12, 145


Question 15, 146Question 16, 147Question 17, 148Question 18, 148Question 19,Question 20, 149Question 21, 150Question 22, 150Question 23,

153153

149

151

30 ASSAF RINOT

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[37] Richard Laver and Saharon Shelah. The ℵ2-Souslin hypothesis. Trans. Amer. Math. Soc.,264(2):411–417, 1981.

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(2), 14(3):505–516, 1976.[43] Assaf Rinot. A unified approach to higher souslin trees constructions. work in progress.[44] Assaf Rinot. A relative of the approachability ideal, diamond and non-saturation. J. Symbolic

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1987.

[59] Saharon Shelah. Cardinal arithmetic, volume 29 of Oxford Logic Guides. The Clarendon PressOxford University Press, New York, 1994. Oxford Science Publications.

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[61] Saharon Shelah. More on weak diamond. preprint of paper #638, arXiv:math.LO/9807180,1998.

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[63] Saharon Shelah. The generalized continuum hypothesis revisited. Israel J. Math., 116:285–321, 2000.

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School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

E-mail address: [email protected]: http://www.assafrinot.com

156156


Paracompactness and normality in box products:old and new

Judith Roitman

Abstract. Old results on the paracompactness and normality of boxproducts are reviewed. New results include a combinatorial principlewhich implies the paracompactness of �(ω+1)ω and extensions of recentwork on subspaces.

1. Outline

The problem of the paracompactness and normality of box products is(arguably) the last of the problems from M.E. Rudin’s 1974 CBNS lectureswhich still has major unsolved chunks. The problem seems to have circu-lated orally for decades before solutions began to appear. Its first form iscredited to Tietze in the 1940’s: Is �Rω normal? A second version is attrib-uted to Arthur Stone in the 1950’s: Is the box product of countably manyseparable metrizable spaces normal? A positive answer to Stone’s questionunder CH for compact metrizable spaces was found by M.E. Rudin [14]in 1972; she actually found that the box product was paracompact. Thusparacompactness entered the picture.

Previous surveys on this problem were written by van Douwen [3],Williams [17], and Lawrence [10] — the most recent of these appearedin 1993. The purpose of this paper is to summarize past results from a(semi)-coherent combinatorial point of view, present old proofs from thatpoint of view, present some new results which build on this point of view,and then to present a new result which seems to ignore it.

Section 2 gives basic definitions and basic set-theoretic propositions; sec-tion 3 states old positive results; section 4 states old negative results (forcontext only; none of them are proved here); section 5 gives basic topologicalfacts; section 6 proves old positive consistency results incorporating some

1991 Mathematics Subject Classification. Primary 54-02, 54D20, 03E75; Secondary54B99, 54G99, 03E35.

Key words and phrases. box product, paracompact, ∇-product.

1


157


157

2 JUDITH ROITMAN

new methods; section 7 includes recent and new work on real paracompactsubspaces; and section 8 gives a simple principle proving the paracompact-ness of �(ω+1)ω which holds in all the models in which we know �(ω+1)ω

is paracompact. Section 9 lists some open questions.

2. Definitions

In this paper, all spaces are assumed to be regular Hausdorff.Let’s recall the definition of box product:

Definition 2.1. Let I be an arbitrary index set, and for each i ∈ Ilet Xi be a topological space. �i∈IXi is the set Πi∈IXi under the topologywhose base is all Πi∈Iui where each ui is open in Xi. Sets of the form Πi∈IYi

where each Yi ⊆ Xi will be called boxes.1

We will also need some cardinal invariants on ωω. Here the notation=∗, <∗,≤∗ etc. mean “mod finite.”

Definition 2.2. (a) F ⊆ ωω is bounded iff ∃g ∈ ωω ∀f ∈ F g >∗ f ;otherwise F is unbounded.

(b) F ⊆ ωω is dominating iff ∀g ∈ ωω∃f ∈ F g <∗ f .(c) b is the minimal cardinality of an unbounded set.(d) d is the minimal cardinality of a dominating set.

Note that ω1 ≤ b ≤ d ≤ c, but no other relations among ω1, b, d, and c

can be proved in ZFC. Also, b is regular, but neither d nor c need be regular.

Proposition 2.1. 1. If F ⊆ ωω and |F | < b then F is bounded.2. If F ⊆ ωω, A ⊆ P(ω) and |F |, |A| < d then ∃f ∈ ωω ∀g ∈ F ∀a ∈

A {n ∈ A : f(n) > g(n)} is infinite.3. b = d iff there is a dominating family {fα : α < b} ⊆ ωω so that if

α < β then fα <∗ fβ.

The proof of proposition 2.1.1 and proposition 2.1.3 is trivial; the proofof proposition 2.1.2 can be found in [11]. A family as in proposition 2.1.3 iscalled a scale.

While many consistency results about the paracompactness of box prod-ucts involve cardinal invariants, some of them use a combinatorial statementwhich is independent of any particular cardinal invariant equality or inequal-ity:

Definition 2.3. The Model Hypothesis (MH) is the following state-ment: For some κ, H(ω1) =

⋃α<κHα where each Hα is an elementary

submodel of (H(ω1),∈) and each Hα ∩ ωω is not dominating.

Here H(κ) is the collection of all sets whose transitive closures have sizeless than κ. In particular, both ωω and P(ω) ⊆ H(ω1), and a space ofcountable weight or countable character be coded as (hence is isomorphicto) a subset of H(ω1).

1These are sometimes called cylinders.

158158

PARACOMPACTNESS AND NORMALITY IN BOX PRODUCTS: OLD AND NEW 3

MH follows from d = c, hence from Martin’s axiom (hence from CH),and holds in any forcing extension by uncountably many Cohen reals.

Definition 2.4. Let P = {possibly partial functions from ω to ω}. Qis a good enough submodel of P iff Q is closed under Boolean operations;if f, g ∈ Q and a = domain f ∩ domain g is nonempty then f |a ∈ Q; andif ϕ is a formula with f as a parameter for some f ∈ Q, and g = {(n,m) :ϕ(n,m, f)}, then g ∈ Q.

Note that if f ∈ Q and Q is good enough then there is a total functionx ∈ ωω ∩ Pα with x(n) > f(n) for all but finitely many n ∈ domain f .

A possibly weaker hypothesis related to MH is

Definition 2.5. The Weak Model Hypothesis (WMH) is the followingstatement: P =

⋃α<κ Pα where each Pα is a good enough submodel of P ,

∀α ∃y ∈ ωω ∩ Pα+1 ∀f ∈ Pα y(n) > f(n) for all but finitely many n ∈domain f .

WMH is consistent with ω1 < b < d < c. IfH is an elementary submodelof H(ω1) then H ∩ P is a good enough submodel of P .

Proposition 2.2. If Q is a good enough submodel of P and Q ∩ ωω isnot dominating, then there is f ∈ ωω \Q so that for all g ∈ Q ∩ ωω and alla ∈ Q ∩ [ω]ω, {n ∈ a : f(n) > g(n)} is infinite.

Proof. Let a ∈ [ω]ω. For n ∈ ω we define n+a be the least element in a

greater than n, i.e., n+a = a \ n+ 1.

For g ∈ Q ∩ ωω and a ∈ Q ∩ [ω]ω we define hg,a : ω → ω by hg,a(n) =1 + sup{g(m) : m ≤ g(n+

a )}. Then each g(n+a ) < hg,a(n). Since g, a ∈

Q, hg,a ∈ Q.Suppose the conclusion fails. Let f ∈ ωω be increasing. ∃g, a ∈ Q so a

infinite and∀∞m ∈ a f(m) ≤ g(m). But then ∀∞n ∈ ω f(n) ≤ f(n+a ) ≤

g(n+a ) < hg,a(n). I.e., Q ∩ ωω is dominating, a contradiction.

�Both definition 2.3 and proposition 2.2 are implicit in [11], which uses

combinatorial consequences of MH for compact first countable spaces. WMHis sufficient to prove paracompactness of �(ω + 1)ω.

3. Positive results

In this section we list the strongest (so far) positive results known. Laterwe will prove several results from this section.

Theorem 3.1. (Roitman) The box product of countably many compactfirst countable spaces is paracompact if d = c or if MH holds.

Theorem 3.2. (Williams) The box product of countably many compactspaces of weight ≤ ω1 is paracompact if d = ω1.

2

2In the paper in which this result first appeared, [17], there is a crucial misprint: thecorrect “d = ω1” is written as “b = ω1.”

PARACOMPACTNESS AND NORMALITY IN BOX PRODUCTS 159159

4 JUDITH ROITMAN

Section 5.2 will present a slight generalization of theorem 3.2.

Theorem 3.3. (van Douwen) The box product of countably many com-pact metrizable spaces is paracompact if b = d.

Note that theorem 3.3 subsumes M.E. Rudin’s pioneering result.

Theorem 3.4. (Kunen) The box product of countably many compactscattered spaces is paracompact if CH holds.

All of these results are about compact spaces. 3 How far can we getaway from compactness?

Theorem 3.5. (Lawrence) The box product of countably many countablemetrizable spaces is paracompact if b = d or d = c.

I.e., “compact” isn’t necessary. The weakest generalization is to σ-compact.

Theorem 3.6. (Wingers) The box product of countably many σ-compact0-dimensional first countable spaces of cardinality ≤ c is paracompact ifd = c.

Theorem 3.6 subsumes theorem 3.5 under d = c.Theorem 3.1 appeared in [11]; theorem 3.2 in [17]; theorem 3.3 in [3]

(and was anticipated in the case of �(ω + 1)ω in [16]); theorem 3.4 in [6];theorem 3.5 in [8]; and theorem 3.6 in [19].

Theorem 3.4 is the only theorem in which cardinality, weight and char-acter are unlimited. All the other positive results limit at least one of theseinvariants. All the positive results involve some approximation of compact-ness. The reasons will be become clear in the next section, when we meetvarious non-normal box products involving spaces which violate one or theother of these restrictions.

And all of these theorems involve box products of countably manyspaces. The reason for this will also become clear in the next section.

4. Not normal

The previous section listed results in which various box products areproved to be paracompact. None of them were ZFC results. Here we listresults in which various box products are proved to be not normal. Almostall of these are ZFC results. We include these results for completeness, andwill prove none of them.

Theorem 4.1. (van Douwen) P×�(ω+1)ω is not normal; hence �(ω+1)ω is not hereditarily normal.

Here P is the space of irrationals.

3By [6] we can extend most of them — the ones that use the nabla product in theirproofs (see below) — to locally compact spaces.

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Theorem 4.2. (Kunen) �(2c+)ω is not normal.

Theorem 4.3. (Kunen) d×�(ω + 1)ω is not normal if b = d.

Theorem 4.4. (Kunen; Erdos and Rudin) �(2ω1+1LEX )ω is not normal.

Theorem 4.5. (van Douwen) �(2ω2)ω is not normal.

The next theorem requires the negation of a generalization of compact-ness.

Definition 4.1. X is Hurewicz iff given any countable collection of opencovers {Un : n < ω} of X there are finite Vn ⊆ Un so X =

⋃n<ω

⋃Vn.

4

In particular, Hurewicz spaces are Lindelof and countable spaces areHurewicz.

Theorem 4.6. (Wingers) If X is Lindelof, not Hurewicz, with a denseHurewicz subspace, then X ×�(ω + 1)ω is not normal.

Note that theorem 4.6 implies theorem 4.1, since P is not menger buthas a countable (hence Menger) dense subspace.

Theorem 4.7. (Wingers) Assume MA. ∃X ⊆ R, X Hurewicz, X ×�(ω + 1)ω is paracompact but X2 ×�(ω + 1)ω is not normal.

These results can be seen as delineating the parameters for paracom-pactness: theorems 4.2, 4.4, 4.5 tell us that large weight, size, or characterare problematic; theorem 4.6 tells us that X can’t be too far from compact;theorem 4.7 tells us that the borderline is subtle.

And, finally, one of the most beautiful and difficult theorems in the sub-ject, the theorem that tells us that there’s no point in looking at uncountablymany factors:

Theorem 4.8. (Lawrence) �(ω + 1)ω1 is not normal.

Theorem 4.1 appeared in [1]; theorem 4.2 first appeared in a preprint of[6] but was left out of the final paper, and its statement and proof appearedin [15]; theorem 4.3 appeared in [5] for the case d = ω1, and in [4] forthe case d > ω1; theorem 4.4 appeared in [7]; theorem 4.5 appeared in [2];theorems 4.6 and 4.7 in [20]; and theorem 4.8 in [9].

5. Basic facts

Results in this section are well-known and not hard to prove, so proofswill be minimal and frequently left out.

Definition 5.1. ∇n<ωXn is the quotient space �n<ωXn/ =∗, and isknown as the nabla product. σ is the quotient map from �n<ωXn to∇n<ωXn.

4The property defined here is often called the Menger property — there is a strongerproperty also known as the Hurewicz property. I am using this notation because it is theone used in the papers I am discussing.


6 JUDITH ROITMAN

When it doesn’t cause confusion we will conflate x ∈ �n<ωXn with itsimage in ∇n<ωXn and write x(n) for the nth component of x. We will attimes conflate u = �n<ωun with u = ∇n<ωun (and call the latter a box aswell). When we need to distinguish box from nabla products, we’ll write x∇

for σ(x);Y ∇for{y∇ : y ∈ Y }; and Y∇ for {Y ∇ : Y ∈ Y}. If x ∈ ∇n<ωXn

we write E(x) = σ−1{x} and if Y ⊆ ∇n<ωXn we write E(Y ) =⋃

y∈Y E(y).

If A ⊆ �n<ωXn we’ll write A|[m,ω) for the projection of A onto �n≥mXn,and so on.

Definition 5.2. A space is κ-open iff the intersection of fewer than κmany open sets is open.5

Definition 5.3. A space is κ-Lindelof iff every open cover has a subcoverof size < κ.

Proposition 5.1. 1. Each ∇n<ωXn is ω1-open.2. If each Xn is first countable, then ∇n<ωXn is b-open.3. If each Xn is first countable, β < d and {Hα : α < β} is a collection

of closed boxes in the nabla product, then⋃

α<β Hα is closed.

Proof. Here’s a brief sketch:1. uses the fact that every countable collection of functions in ωω is

bounded.2. uses the fact that every set of fewer than b function in ωω is bounded.

Something like first countability is needed to have a uniform way to dealwith an uncountable sequence of open sets, hence the restriction.

3. uses proposition 2.1.2. �Definition 5.4. A space is ultraparacompact iff every open cover has

a pairwise disjoint covering refinement.

Many of our positive proofs will in fact prove that either the box productor the nabla product is ultraparacompact.

Proposition 5.2. 1. If each Xn is regular, then ∇n<ωXn is 0-dimensional.2. If X is 0-dimensional, κ-open and κ-Lindelof then X is ultrapara-

compact.3. If each Xn is first countable, D is a family of boxes in ∇ = ∇n<ωXn,

and |D| < d then⋃

D∈D cl D is closed.4. If each Xn is first countable, then each point in ∇n<ωXn has a neigh-

borhood base of size d.

Proof. For 1: Let ∇ = ∇n<ωXn, x ∈ u open in ∇. There are openboxes ui, i < ω where for each i, x ∈ ui+1 ⊆ cl ui+1 ⊆ ui ⊆ u. By proposition5.1.1,

⋂i<ω ui is clopen.

For 2: Given an open cover U we may assume it’s by clopen sets. Let{uα : α < κ} be a subcover of U . Then {uα \

⋃β<α uβ : α < κ} is a pairwise

disjoint open cover of X.

5This is sometimes called a Pκ space.

162162


For 3: This uses proposition 5.1.3, and the observation that for any boxD = ∇n<ωDn, cl D = ∇n<ωcl Dn.

For 4: For each s ∈ Xn let {us,n : n < ω} be a decreasing open base. Letx ∈ ∇n<ωXn. Let F be a dominating family in ωω. Then {∇n<ωux(n),f(n) :f ∈ F} is a neighborhood base for x. �

Remark 5.1. The conclusion of proposition 5.2.3 remains valid if eachD ∈ D is a closed intersection of countably many boxes.

Proposition 5.3. If each Xn is first countable and Y ∈ [∇n<ωXn]≤d

then Y is ultraparacompact.

Proof. Let Y = {xα : α < δ} where δ ≤ d. If U is an open cover ofY , we can inductively construct a pairwise disjoint clopen refinement of Uusing remark 5.1. �

An immediate corollary of proposition 5.3 is:

Corollary 5.1. If d = c and ∇ is a nabla product of countably manycompact first countable spaces, then ∇ is ultraparacompact.

We are interested in the nabla product because of the following theoremof Kunen:

Theorem 5.1. If each Xn is compact, then �n<ωXn is paracompact iff∇n<ωXn is paracompact.

An immediate corollary of theorem 5.1 and corollary 5.1 is

Corollary 5.2. If d = c and � is a box product of countably manycompact first countable spaces, then � is ultraparacompact.

Theorem 5.1 is proved by a general theorem whose application is thefollowing:

Proposition 5.4. Consider a box product of compact spaces. If {un :n < ω} is a cover of some E(x) then {z : E(z) ⊆

⋃n<ω un} is open.

Proposition 5.4 is in turn proved by showing that the quotient map from�n<ωXn to ∇n<ωXn is closed. Using the characterization of paracompact-ness via σ-locally finite refinements, it’s easily seen by proposition 5.4 that�n<ωXn is paracompact if ∇n<ωXn is paracompact; the other directionfollows directly from the quotient map being closed.

When we are unable to use theorem 5.1 we need a closer analysis ofsmall subsets and small families. The analysis we give is due to Wingers.

Definition 5.5. A family of subsets of a space X is discrete iff no pointin X is in the closure of more than one element of the family.

Remark 5.2. A family of sets if discrete iff the family is pairwise disjointand the union of any subset of its closures is closed.


8 JUDITH ROITMAN

A partial analog to proposition 5.1.3 for the box product (and an imme-diate corollary of it) is:

Proposition 5.5. Let � be a box product of countably many first count-ably spaces. If Y ∈ �<d then {E(y) : y ∈ Y } is a discrete collection, infact we can find a discrete collection of open sets {uy : y ∈ Y } with eachE(y) ⊆ uy.

A full analog of proposition 5.1.3 for the box product would be useful, butsuch an analog needs a stronger hypothesis — not just any small collectionof closed sets will do. Why? Let � = �i<ωXi and suppose s ∈ X0 is alimit point of a sequence {ti : i < ω}. Fix some x with x(0) = s and definethe point xi by xi|[1,ω) = x|[1,ω), xi(0) = ti. Clearly each {xi} is closed, but{xi : i < ω} is not.

Definition 5.6. Let D be a family of boxes, x ∈ � = �n<ωXn.1. ∀n < ω D(x, n) = {D ∈ D : x|[n,ω) ∈ D|[n,ω)}.2. D(x) =

⋃n<ω D(x, n).

3. D is simple iff ∀x ∈ � if there is some n < ω with D(x, n) infinitethen E(x, n) = {y : y|[n,ω) = x|[n,ω) ⊆

⋃D.

Remark 5.3. 1. Note that if D(x, n) is infinite, so is D(x,m) for allm ≥ n, hence if D is simple and D(x, n) is infinite for some n, then E(x) ⊆⋃

D.2. Note that x /∈

⋃D iff D(x, 0) = ∅.

The full box product analog to theorem 5.1.3 is:

Proposition 5.6. If H is a simple collection of closed boxes with |H| < d

then⋃

H is closed.

Proof. Let x /∈⋃H. By the proof of proposition 5.1.3, there is an

open box w with x ∈ w and w ∩⋃(H \H(x)) = ∅. H(x) =

⋃n<ω(H(x, n+

1) \ H(x, n)) and H(x, 0) = ∅. Each H(x, n) is finite, so ∀n ∃vn open in Xn

with x(n) ∈ vn and vn ∩ H(n) = ∅ for all H ∈ H(x, n + 1) \ H(x, n). Letv = �n<ωvn. v ∩

⋃H(x) = ∅. Letting u = v ∩ w completes the proof. �

Finally, we formalize the property in proposition 5.5 and show how touse it to get paracompactness.6

Definition 5.7. Y is a strongly separated subspace of X iff there is adiscrete open collection U = {uy : y ∈ Y } with each y ∈ uy and if y �= y′

then uy �= uy′ . We say that U separates Y .

Proposition 5.7. If X if κ-open, 0-dimensional, X =⋃

α<κXα whereeach

⋃β<αXβ is closed, and each Xα is strongly separated in X \

⋃β<αXβ,

then X is ultraparacompact.

6This approach was made explicit in [12] and [13] but is implicit in much of Lawrence’sand Wingers’ work.

164164


Proof. Let Vα be a discrete clopen collection separating Xα in X \⋃β<αXβ. Given an open cover U of X we let W0 be a pairwise disjoint

clopen refinement of V0 refining U and covering X0. At stage α > 0 welet Wα be a pairwise disjoint clopen refinement of {v ∈ Vα : v ∩ Xα ∩⋃

β<α

⋃Wβ = ∅} where

⋃Wα ∩

⋃β<α

⋃Wβ = ∅,

⋃β<αXβ ⊂

⋃β<α

⋃Wβ.

W =⋃

α<κWα is a pairwise disjoint refinement of U .�

The following remark will be useful in proving theorem 3.1:

Remark 5.4. In proposition 5.7 we don’t need κ a cardinal, or X isκ-open. All we really need is κ a limit ordinal, and a way to pick discreteclopen covers Uα of each Xα so if β < κ then

⋃α<β Uα is closed.

6. Proofs of old theorems

In this section we give proofs of old theorems. With the exception ofsection 6.1 and the first proof in section 6.2, the proofs are streamlined fromthe originals, mostly by using the results of section 5. The basic theme,after section 6.1, is stratification. Because we are presenting a semi-unifiedpoint of view, notation here and in the next section will often differ fromthe original papers.

6.1. Compact scattered factors. We begin with theorem 3.4: UnderCH, the box product of countably many compact scattered spaces is para-compact. Its proof is quite different from the ones that will follow, in factfrom the proofs of all the other theorems in section 3.7

Definition 6.1. A space is scattered iff every subspace has an isolatedpoint.

Basic facts about scattered spaces include:

Proposition 6.1. 1. Subspaces of scattered spaces are scattered.2. If X is scattered there’s a height function htX : X → ON . The

range of this function is called ht(X) and {x : htX(x) < α} is open for allα ≤ ht(X).

3. If X is compact scattered then ht(X) = α+1 for some ordinal α and{x ∈ X : htX(x) = α} is finite.

4. If Y ⊆ X scattered then for all y ∈ Y htY (y) ≤ htX(y).

The function in proposition 6.1.2 is known as the Cantor-Bendixsondecomposition; {{x : htX(x) = α} : α < ht(X)} is known as the Cantor-Bendixson hierarchy.

Now for the proof of theorem 3.4:

7Which is why we include it, as a sort of cautionary tale.


10 JUDITH ROITMAN

Proof. Let Xn be compact scattered for all n < ω, � = �n<ωXn. Byproposition 5.1.1 and proposition 5.2.2, it suffices to show that ∇ = ∇n<ωXn

is ω1-Lindelof. Hence it suffices to show that � = �n<ωXn is ω1-Lindelof.Let U be an open cover of �. It suffices to find a family K of closed

boxes refining U and covering �, |K| = ω1.Let T = ω<ω1

1 . By CH, |T | = ω1. If σ ∈ T we write σ�β for thesuccessor of σ in T whose last element takes on the value β. We will define{Kσ : σ ∈ T} a collection of closed boxes; K will be a particular subset of{Kσ : σ ∈ T}.

K∅ = �. Suppose we know K = Kσ = Πn<ωKn. Let βn + 1 = ht(Kn),and let Yn = {s ∈ Kn : ht(s) = βn}. Yn is finite. Y = Πn<ωYn has size ω1

so there is V a family of open boxes covering Y with {cl V : V ∈ V} refiningU and |V| = ω1. Define J to be the set of J = Πn<ωJn where each Jn iscompact and one of the following conditions holds:

(i) for some v ∈ V J = cl v ∩ K; or(ii) there is some n < ω so ∀m �= n Jm = Km and there is a finite set

R ⊆ V so Jn = Kn \⋃

c∈R c(n) and Yn ⊆⋃

c∈R c(n)Note that |J | = ω1. If (i) fails for some J ∈ J and n is as in (ii), then

ht(Jn) < ht(Kn).We claim that K =

⋃J : Let x ∈ K. If ∀J ∈ J with J satisfying (ii),

x /∈ J , then for each n < ω there is yn ∈ Yn so that ∀v = Πn<ωvn ∈ V ifyn ∈ vn then x(n) ∈ v(n). Define y ∈ Y by y(n) = yn for all n. ∀v ∈ V ify ∈ v then x ∈ v. Since there is some v ∈ V with y ∈ v, x ∈ cl v ∩K ∈ J .

Let {Kσ�α : α < ω1} enumerate J . Let K = {Kσ : σ ∈ T and Kσ

satisfies (i)}.For each x ∈ � there is f : ω1 → ω1 so x ∈

⋂α<ω1

Kf |α . There is β < ω1

so if γ > β, J = Kf |β and J ′ = Kf |γ then ht(Jn) = ht(J ′n) for all n. Hence

J satisfies (i). Hence x ∈ J ∈ K, as desired. �6.2. Compact factors with small weight. Most proofs of paracom-

pactness involve some kind of stratification. The proofs in this section strat-ify a base.

We prove van Douwen’s theorem 3.3: If b = d then the box product ofcountably many compact metrizable spaces is paracompact.8

First we define

Definition 6.2. X is κ-metrizable iff it has an open base B = {ux,α :α < κ, x ∈ X} so that {ux,α : α < κ} is a neighborhood base at x, andgiven two points x, y and two ordinals α ≤ β < κ then (i) if y ∈ ux,α thenuy,β ⊆ ux,α; and(ii) if y /∈ ux,α then uy,β ∩ ux,α = ∅.

Proposition 6.2. If X is κ-metrizable witnessed by B, and each Bα ={ux,α : x ∈ X} then each Bα is pairwise disjoint.9

8Van Douwen also proved [1] that if a nabla product of countably many compactmetrizable spaces is κ-metrizable, then κ = b = d.

9I.e., ω-metrizable �= metrizable.

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Proof. Suppose y ∈ ux,α. By (ii), x ∈ uy,α, so by (i), ux,α ⊆ uy,α ⊆ux,α. �

The proof of theorem 3.3 is immediate from:

Proposition 6.3. (a) If b = d then the nabla product of countably manycompact metrizable spaces is b-metrizable.

(b) A κ-metrizable space is paracompact.

Proof. For (a): Let ∇ = ∇n<ωXn where each Xn is compact metriz-able. Let {fα : α < b} be a scale, and for x ∈ ∇ define ux,α =

⋂n<ω ∇k<ωB

(x(k), 2−(n+fα(k))),where B(x(n), r) is the ball around x(n) ∈ Xn of radius r.For (b): Let B be as in proposition 6.2, and given an open cover U and

a point x ∈ X let ux = ux,α where α is the least ordinal so ∃u ∈ U ux,α ⊆ v.{ux : x ∈ X} is easily seen to be a partition of X. �

We prove Williams’ theorem 3.2: If d = ω1 then the box product ofcountably many compact spaces of weight ≤ ω1 is paracompact.10

Proof. Suppose d = ω1 (and hence b = d), each Xn has weight ≤ ω1,each Xn is compact. The claim is that ∇ = ∇n<ωXn is ω1-metrizable.

Let Cn be an open base for Xn, |Cn| ≤ ω1. We can consider Cn to be apartial order on ω1.

Let {Mα : α ≤ ω1} be an increasing continuous sequence of countableelementary submodels of H(ω2) such that each Cn is an element of each Mα

and each Cn ∈⋃

α<ω1Mα.

11

We define Xn,α to be the compact space with Cn ∩Mα as its base. EachXn is a limit of {Xn,α : α < ω1}.

We define ∇α = ∇n<ωXn,α. Each Xn,α is metrizable, so by b = d = ω1,each∇α is ω1-metrizable, with a base

⋃γ<ω1

Bγ,α where each Bγ,α is pairwise

disjoint and if γ < γ′ then Bγ′,α refines Bγ,α

Let Bα = {⋂

β,γ<α uγ,β : each uγ,β ∈ Bγ,β}. Each Bα is pairwise disjoint,

if β < α then Bα refines Bβ , and⋃

α<ω1Bα is a base for ∇. So ∇ is ω1-

metrizable. �

We strengthen theorem 3.2 as follows:

Theorem 6.1. If b = d < ℵω and each Xn has weight ≤ d then ∇n<ωXn

is b-metrizable, hence paracompact.

First a proposition:

Proposition 6.4. Let κ < b = d < ℵω. If the nabla product of countablymany spaces of weight κ is b-metrizable, then the nabla product of countablymany spaces of weight κ+ is b-metrizable.

10Williams’ original proof is presented in the context of uniformities,11The use of elementary submodels is a convenient way of closing things off at count-

able stages. For example, if u, v ∈ Cn ∩Mα, then so is u ∩ v.


12 JUDITH ROITMAN

Proof. Suppose each Xn has weight κ+. We consider a base Cn for thetopology on each Xn to be a partial order on κ+. Let {Mα : α < κ+} bean increasing continuous sequence of elementary submodels of H(κ++) suchthat each Cn is an element of each Mα and each Cn ∈

⋃α<κ+ Mα. As before,

define Xn,α to be the compact space with Cn ∩Mα as its base. As before,define ∇α = ∇n<ωXn,α. By hypothesis, each ∇α is b-metrizable. For α < b

let Bα = {⋂

β,γ<α uγ,β : each uγ,β ∈ Bγ,β}. Each element of Bα is open by

proposition 5.1.2, each Bα is pairwise disjoint, and if α < α′, Bα′ refines Bα.⋃α<bBα is a base for ∇n<ωXn, so ∇n<ωXn is b-metrizable. �Theorem 6.1 follows from proposition 6.4 by finite induction.

6.3. First countable compact factors. Recall theorem 3.1: If d = c

or MH then the box product of countably many compact first countablespaces is paracompact. Corollary 5.1 took care of the case d = c. In thissection we give a proof under MH in which the space is stratified. Again,because of compact factors, we only have to consider the nabla product.

Proposition 6.5. Assume MH, where H(c) =⋃

α<κHα as in definition2.3, and let ∇ be the nabla product of countably many compact first countablespaces. Then, for each α, ∇ ∩ Hα is strongly separated and closed in ∇ \⋃

β<αHα.

Proof. First note that, by MH, , since each first countable space hassize ≤ c, and since in a compact first countable space points can be codedby countable open neighborhood bases, if ∇ = �n<ωXn where each Xn iscompact first countable, then each Xn ⊂ H(c) and ∇ ⊂ H(c). For s ∈ Xn

let {un,s : n < ω} be a neighborhood base for s. For x, y ∈ ∇ ∩Hα there isfx,y ∈ Hα ∩ ωω and ax,y ∈ Hα ∩ [ωω] so ∀i ∈ ax,y uf(i),x(i) ∩ uf(i),y(i) = ∅.There is g ∈ ωω so ∀x, y ∈ ∇ ∩Hα {i ∈ ax,y : g(i) > fx,y(i)} is infinite. Forx ∈ ∇ ∩Hα, define ux = ∇i<ωug(i),x(i). If x �= y then ux ∩ uy = ∅.

For x ∈ ∇ ∩Hα let vn,x = ∇i<ωug(i)+n,x(i). Let vx =⋂

n<ω vn,x. Eachvx is clopen and if x �= y then vx ∩ vy = ∅. Let Vα = {vx : x ∈ ∇ ∩Hα}.

Each Vα is pairwise disjoint. We show that Vα is discrete.Suppose z ∈ ∇ \

⋃Vα. There is β > α with z ∈ Hβ. For x ∈ ∇ ∩Hα

there is fx ∈ Hβ ∩ [ω]ω so if wx = ∇i<ωufx(i),z(i) then wx ∩ vx = ∅. I.e.,there is ax ∈ [ω]ω ∩Hβ and gx ∈ Hβ so ∀i ∈ ax (wx)i ∩ vg(i),x(i) = ∅. Thereis f ∈ ωω so that for all x ∈ ∇ ∩Hα {i ∈ ax : f(i) > fx(i)} is infinite. I.e.,∇i<ωuf(i),z(i) ∩

⋃Vα = ∅.

A simpler version of the preceding paragraphs shows that ∇ ∩ Hα isclosed in ∇ \

⋃β<αHα.

�Proposition 6.5 lets us prove theorem 3.1:

Proof. Assume the hypothesis of proposition 6.5. Let U be an opencover of ∇. In the construction of proposition 6.5, require that each v ∈ Vα

refine some element of U . Let Dα =⋃

Vα. By the remark after proposition

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5.2, each⋃

α<β Dα is closed. So we construct W a pairwise disjoint clopen

refinement of U covering ∇ as follows: Wβ = {v \⋃

α<β Dβ : v ∈ Vβ};W =⋃

β<δ Wβ. �

6.4. σ-compact first countable 0-dimensional factors. In this sec-tion we prove Wingers’ theorem: If d = c then the box product of countablymany σ-compact first countable 0-dimensional spaces of size ≤ c is paracom-pact. In this proof we stratify the space by well-ordering the equivalenceclasses E(x). But we cannot rely on the ∇-product, which makes the proofmore difficult.

By propositions 5.5 and 5.6 our first task is to find and extend simpleopen families.

Assume � = �n<ωXn. How do you construct a simple open family?

Proposition 6.6. Suppose each Xn is σ-compact where each Xn =⋃i<ω Ci,n, each Ci,n compact and Ci,n ⊆ Ci+1,n. If x ∈ � then there is

a countable simple open family U with E(x) ⊆ U , and every element of U isa finite union of open boxes.

Proof. Fix x. Let k = {kn : n < ω} ⊂ ω be a strictly increasingsequence, each kn > n so ∀m < ω if m ≤ n then x(m) ∈ Ckn,m. Defineσn ∈ ωn by: each σn(i) = kn. Note that if τ ∈ ωm then there is σn withn ≥ m and τ(i) ≤ σn(i) for all i < m.

Define Kn = Πm≤nCkn,m. For each n there is a finite collection ofopen boxes {uj,n : j ≤ mn} with Kn × {x|(n,ω)} ⊆

⋃j≤mn

uj,n. We may

assume that if j, j′ ≤ mn and i > mn then uj,n(i) = uj′,n(i); and if n < n′

and i > mn′ then each uj,n(i) = uj′,n′(i). Let un =⋃

j≤mnuj,n and let

U = {un : n < ω}. U is simple. Why? Suppose U(y, s) is infinite, and letz ∈ E(y, s). Then there are infinitely many n with un ∈ U(y, s). For all butfinitely many of these n, z|[0,s) ∈ Πi<sCσ(n),i. For such n, z|(s,ω) = y|(s,ω) ∈Πi>s

⋂j≤mi

uj,i. So z ∈ un. �

In the construction above, we say that U is constructed via k. Note thatwe could in fact do this so {uj,n : j ≤ mn, n < ω} refines a given open coverof �.

So we’d be done if we knew that if U simple and x ∈⋃U then E(x) ⊆ U .

Unfortunately that is false. The complications come from working aroundthis.

The next step is to see how to construct simple families whose unionsare simple. Before doing this we need the following

Definition 6.3. Let S = {Sn : n < ω} be a family of finite unions ofboxes, each Sn =

⋃i≤mn

Si,n. Then S is tapered iff ∀n ∀i > n ∀j ≤ mn ∀j′ ≤mn+1 Sj,n(i) = Sj′,n+1(i).

Note that in the construction of proposition 6.6, we required that {un :n < ω} was tapered.


14 JUDITH ROITMAN

Proposition 6.7. Suppose ξ < d and consider a family {Uη : η < ξ}where each Uη = {un,η : n < ω} is simple, tapered, and constructed viakη = {kn,η : n < ω} so that the following property holds:

(∗) ∀η < ρ < ξ ∀n < ω if un,η ∩ un,ρ �= ∅ then kn,η < kn,ρ.

Then V =⋃

η<ξ Uη is simple.

Proof. Suppose V(x, n) is infinite. We need to show that E(x, n) ⊆⋃V.If some Uη(x, n) is infinite, we are done. So we may assume there is

an increasing sequence {ηm : m < ω} and a sequence {im : m < ω} wherex|[n,ω) ∈ ukim ,ηm |[n,ω) for all m.

Suppose {im : m < ω} is unbounded. If y ∈ E(x, n) then there isim,m > n with y|[0,n) ∈ Kkim,ηm

|[0,n) and we are done.Suppose {im : m < ω} is bounded. Then there is some i so, without loss

of generality, each ui,ηm ∈ Uηm(x, n). Suppose⋂

m<ω ui,ηm �= ∅. Then, by(*), {ki,ηm : m < ω} is unbounded.

If i ≥ n then {ki,ηm : m < ω} unbounded completes the proof.If i < n then, since each ki,η < ki+1,η and{ki,ηm : m < ω} is unbounded,

{kn,ηm : m < ω} is unbounded, which completes the proof.So we need to show that

⋂m<ω ui,ηm �= ∅.

If i ≥ n, it is immediate that⋂

m<ω ui,ηm ⊇ Πj<iC0,j × x|[i,ω) �= ∅. Ifi < n then, by tapered, if i ≤ j < n then x(j) ∈ ukim ,ηm(j) for all m, so,again,

⋂m<ω ui,ηm ⊇ Πj<iC0,j × x|[i,ω) �= ∅.

�

The next step is to construct a collection of simple tapered familiessatisfying (∗).

Proposition 6.8. Suppose ξ ≤ d and U =⋃

η<ξ Uη is a simple collection

of clopen families, where each Uη = {un,η : n < ω} is simple, tapered and

constucted via kη. Suppose x /∈⋃

U . Then there is k and a simple tapered

family V = {vn : n < ω} constructed via k with E(x) ⊆⋃

V so that ∀η ∀j ifuj,η ∩ vj �= ∅ then kj,η < kj.

Proof. We construct V as in 6.6 adding two extra requirements.First, using proposition 5.1.3, we require the following: (†) if x /∈ un,η(i)

for infinitely many i then v0(i) ∩ un,η(i) = ∅ for infinitely many i.By (†), if x /∈ un,η(i) for infinitely many i then vj ∩ un,η = ∅ for all j.Now we invoke U simple: If x /∈

⋃U then ∀n U(x, n) is finite. So our

second requirement is: if uj,η ∈ U(x, n) then kn > kj,η. �

Finally, we prove a theorem which has theorem 3.6 as a corollary.

Theorem 6.2. Let � = �n<ωXn where each Xn is first countable σ-compact 0-dimensional, and suppose Y ∈ [�]≤d. Then E(Y ) is paracompact.

170170


Proof. Let Y = {yξ : ξ < d}, let W be an open cover of Y , andby induction, using proposition 6.8, construct {Uξ : ξ < d} a collection ofsimple tapered families whose elements are finite unions of clopen boxesrefining W satisfying (∗), where if E(xξ) �⊆

⋃η<ξ Uη then E(xξ) ⊆ Uξ;

otherwise Uξ = ∅. If un,ξ ∈ Uξ let un,ξ =⋃

j<mn,ξuj,n,ξ as in proposition

6.6, and define vj,n,ξ = uj,n,ξ \ (⋃

η<ξ

⋃Uξ ∪

⋃i<n ui,ξ ∪

⋃i<j ui,n,ξ). Then

{vj,n,ξ : j < mn,ξ, n < ω, ξ < d} is a clopen pairwise disjoint refinement ofW covering Y . �

Theorem 3.6 follows since the hypothesis that each Xn is first countableσ-compact gives that each |Xn| ≤ c, and hence |�| ≤ c.

6.5. Countable metrizable factors. We prove theorem 3.5: If b = d

then the box product of countably many countable metric spaces is para-compact.

The proof under b = d involves stratifying the space with reference toa tree. To do this, Lawrence introduced the order hypothesis (OH). OHis about the box product. In the next section we will also be interestedin stratifying the nabla product in a similar way, so we present a modifiedorder hypothesis (MOH).

Definition 6.4. Let X be space, Y ⊆ X. MOH(Y ) is the followingstatement: there is a partial order � on Y so that (Y,�) is a tree, and∀y ∈ Y uy = {z ∈ Y : y � z} is open.

(Recall that a tree is a partial order so that the set of predecessors ofany element is well-ordered.)

Two immediate corollaries of the definition are: z � y iff uz ⊇ uy; if z, yare incomparable then uz ∩ uy = ∅.

It’s also useful to introduce a cardinal invariant we have implicitly usedbefore:12

Definition 6.5. If B is a clopen base for X the closure number cn(B) =sup{κ : if C ∈ [B]<κ then cl

⋃C =

⋃C. cn(X) = inf{cn(B) : B a clopen

base for X}.

In particular, a nabla product of countable many first countable spaceshas closure number d.

Proposition 6.9. If X is 0-dimensional, Y ⊆ X and MOH(Y ) whereht(Y,�) ≤ cn(X), then Y is ultraparacompact.

Proof. Let B be a clopen base so cn(B) = cn(X). Define Yβ = {y ∈Y : {z ∈ Y : z ≺ y} has order type β}. Each

⋃γ<β Yγ is closed in Y .

12I don’t know if this invariant has appeared elsewhere; it seems to have rather narrowapplication.


16 JUDITH ROITMAN

Since ∀β {uy : y ∈ Yβ} is pairwise disjoint, each Yβ is strongly separated in

Y \⋃

γ<β Yγ .13

Given an open cover U of Y , we construct by induction subspaces Zβ ⊆Yβ and, for z ∈ Zβ , vz ∈ B, vz ⊆ uz, so

⋃β<κ

⋃z∈Zβ

vz is a pairwise disjoint

covering refinement of U , as follows:Given Zγ for all γ < β, and vz for all z ∈

⋃γ<β Zγ let Zβ = Yβ \

⋃γ<β

⋃z∈Zγ

vz. Since Zβ is strongly separated in Y \⋃

γ<β

⋃z∈Zγ

vz, and

cn(X) > β, we can easily find vz as required.�

By theorem 5.1, if all factors of the space are compact, to show the boxproduct is compact, proposition 6.9 for the nabla product is sufficient. Buttheorem 3.5 is about spaces which need not be compact, for which theorem5.1 might not hold.

Definition 6.6. Let X be a box product of countably many spaces,Y ⊆ X. SMOH(Y ) is the following statement: there is a partial order � onY so that (Y,�) is a tree with height≤ d, and ∀y ∈ Y if uy = {z ∈ Y : y � z}then E(uy) is open.

An immediate corollary of the definition is that z � y iff E(uz) ⊇ E(uy);if z, y are incomparable then E(uz) ∩ E(uy) = ∅.

Lawrence’s principle OH is the statement that if � is the box productof countably many countable metrizable spaces, and Y is an =∗ transversalof �, then SMOH(Y ).

Proposition 6.10. Assume b = d. Let � be the box product of countablymany countable metrizable spaces. Let Y be an =∗ transversal of �. ThenSMOH(Y ).

Proof. . Since a disjoint union of metrizable spaces is metrizable, with-out loss of generality, � = �Mω for some countable metric space M whosepoint-set is ω with distance function d. I.e., as a set, � = ωω.

Let {hα : α < b} be a scale. For each α we want a sort of quasi-distancefunction δα : �2 → (R+)ω so that the sets ux,α = {y ∈ � : δα(x, y) convergesto 0} give rise to the ux’s of SMOH as follows: ux = ux,αx where αx is theleast α with x ≤∗ hα.

Define γα : ω → R+ as follows: γα(n) = inf{d(i, j) : 0 ≤ i < j ≤hα(n) + 1}. For x, y ∈ � we define

δα(x, y)(n) =d(x(n), y(n))

γα(n).

Note that if y ∈ E(x) then ux,α = uy,α for all α. Hence E(x) ⊆ ux,α forall x, α.

13At this point we might wish to invoke proposition 5.7, but we don’t know that Xis κ-open.

172172


Subclaim 6.10.1. If αx, αy ≤ α and x �=∗ y then ux,α ∩ uy,α = ∅.

Proof. If αx, αy ≤ α then ∀∞n x(n), y(n) < hα(n). So ∀∞n d(x(n), y(n)) ≥γα(n) and δα(x, y)(n) ≥ 1. �

Subclaim 6.10.2. If αx < αy then either ux ⊇ uy or ux ∩ uy = ∅.

Proof. First note that ux,α ⊇ ux,β for all x, for all α < β. So uy,αx ⊇uy,αy . Either uy,αx = ux,αx or uy,αx ∩ ux,αx = ∅. �

To complete the proof of the proposition, define the order x � y iffux ⊇ uy.

�

Propositions 6.9 and 6.10 tell us that a nabla product of countably manymetrizable spaces is paracompact if b = d. What about the box product?

To move from ∇ to �, Lawrence, writing several years before Wingers,refined the tree of SMOH very carefully level by level. The method is rel-atively long and technical, but Wingers’ method provides a way to shortenit.

Theorem 6.3. If b = d then the box product of countably many countablemetrizable spaces is paracompact.

Proof. Let W be an open cover of �, where � is a box product ofcountably many countable metrizable spaces.. We let Y be an =∗-transversalwith partial order � as in SMOH, T = (Y,�), and {uy : y ∈ Y } be the familygiven by SMOH. For α < ht(T ) we construct Vα =

⋃ht(y)=α Vy refining W

where Vα is simple, each Vy is a simple tapered collection of finite unions of

clopen boxes {vn,y : n < ω} via ky, vn,y =⋃

j<mnvj,n,y, each vj,n,y refines W

with E(y) ⊆⋃

Vy, each v∇n,y ⊆ u∇y , and if x ≺ y and vn,x ∩ vn,y �= ∅ thenkn,x < kn,y. Defining wj,n,y = vj,n,y \ (

⋃x≺y

⋃Vx ∪

⋃i<n vi,y ∪

⋃i<j vi,n,y)

gives a pairwise disjoint open refinement of W .�

7. Subspaces

In this section we are concerned with ZFC results about subspaces.Proposition 5.1 says that if Y is a subspace of a nabla product of countablymany first countable spaces, ∇, and |Y | ≤ d, then Y is an ultraparacompactsubspace of ∇. In a different language, it says that if Y is a partial transver-sal of the =∗ relation in a box product of countably many first countablespaces, �, and |Y | ≤ d, then Y is an ultraparacompact subspace of �. Whatabout larger subspaces?

In this and the next subsection we are interested not only in provingthat subspaces of nabla products are paracompact in their own right, butthat they are (ultra)paracompact subspaces in the following sense, strongerthan “is a subspace and is (ultra)paracompact:”


18 JUDITH ROITMAN

Definition 7.1. Y is a(n) (ultra)paracompact subspace of X iff everyopen cover of Y has a locally finite (in X) (discrete (in X)) open (in X)refinement covering Y .14

The basic technique is similar to the use of proposition 6.9: a well-founded pre-order of the nabla product is constructed so that elements whoseinitial segments are of order type strictly less than the maximum order typeform a strongly separated family. The pre-order determines an equivalencerelation; taking a transversal of this equivalence relation gives us a treeunder the pre-order, hence an ultraparacompact subspace.

Moving from an ultraparacompact subspace in a ∇-product to an ul-traparacompact subspace in a box product in general can be done by thefollowing:

Proposition 7.1. Let U be an open family in � = �n<ωXn and supposeV is a pairwise disjoint open refinement of U∇ covering Y ⊆ ∇. Then thereis {xy : y ∈ Y } ⊆ � and {wv : v ∈ V} so each (xy)

∇ = y, each w∇v = v and

{wv : v ∈ V} is a pairwise disjoint open refinement of U .

Proof. First, for each v ∈ V there is uv ∈ U with v ⊆ u∇v . So pickwv with w∇

v = v and wv ⊆ uv. Next, for each y ∈ v pick xy ∈ wv withx∇y = y. �

Here, moving back to the box product is post hoc; it depends on whichrepresentation we choose for each element in the ∇ pairwise disjoint openfamily. However, much of the time we can use Wingers’ technique to movefrom Y ⊂ ∇ to E(Y ) (as a subset of �) being ultraparacompact.

7.1. Subspaces of �(ω+1)ω under �⊥. The results in this subsectionand the next can be found in [13]. The machinery of MOH tightens theoriginal combinatorics, hence shortens the proofs.

Definition 7.2. 1. ω⊆ω = {x : dom x, range x ⊆ ω}.2. If x ∈ ω⊆ω we define x : ω → ω + 1 as: x ⊇ x and if n /∈ dom x, then

x(n) = ω.3. If x ∈ ω⊆ω and f ∈ ωω then N(x, f) = {y : y ⊇∗ x and y(n) > f(n)

for all but finitely many n /∈ dom x}.4. Given x ∈ ω⊆ω we define ⊥(x) = {n ∈ dom x : ∀m > n with m ∈

dom x x(m) ≥ x(n)};x⊥ = x|⊥(x).

5. Given x ∈ ω⊆ω we define x0 = x⊥; xn = (x \⋃

i<n xi)⊥.

6. For x ∈ (ω + 1)ω, ht(x) = sup{n : xn is infinite}.

Note that the N(x, f)’s form a base.

Definition 7.3. Let x, y ∈ (ω + 1)ω. x �⊥ y iff ht(x) ≤ ht(y) and∀i ≤ ht(x)xi =∗ yi; x ≈⊥ y iff x �⊥ y and y �⊥ x.

14For example, the discrete space ω is paracompact in the convergent sequence ω+1,but no cover of ω is locally finite in ω + 1.

174174


Definition 7.4. For x ∈ (ω + 1)ω, ux,≈⊥ = {y : x � y}.We need to show that we can apply proposition 6.9 to a ≈⊥ transversal.First we need a definition and another proposition:

Definition 7.5. INC = {x ∈ ω⊆ω so ∀n,m ∈ dom x, if n < m thenx(n) ≤ x(m).

Proposition 7.2. Let Y be an ≈ transversal of INC. Then Y isstrongly separated in �(ω + 1)ω.

Proof. For x ∈ INC define fx(n) = 1 + x(mn) where mn = inf{m ∈dom x : m > n}. Let vx = {y : y ⊇ x and if (n, k) ∈ y \ x then k > fx(n)}.If Y is a maximal ≈⊥ transversal of INC then {vx : x ∈ Y } is a discreteclopen separating family. �

Proposition 7.3. Each ux,≈⊥ is open in the nabla product, and x �⊥ yiff ux,≈⊥ ∩ uy,≈⊥ �= ∅ iff ux,≈⊥ ⊇ uy,≈⊥.

Proof. For z ∈ INC let fz, vz be as in proposition 7.2. For eachx ∈ ω⊆ω consider fxn , vxn where each xn as in definition 7.2. Let Gx = {g ∈ωω : g ≥∗ fxn for all n < ht(x)}. ux,≈⊥ ⊆ Ux =

⋃g∈G

⋃x�y N(y, g). We

need to show that ux,≈⊥ = Ux.Suppose z �� x. Then either ht(z) < ht(x( or there is some i ≤ ht(x)

and and infinite set S so ∀n ∈ S xi(n) �= zi(n) (note that one of xi(n) orzi(n) might not be defined). If ht(z) < ht(x) then z /∈ Ux. If S exists andS ⊆∗ dom x then if x � y then z �⊇ y, so z /∈ Ux. If S\ dom x is infinite andg ∈ G then z \ x �>∗ g. In all cases, z /∈ Ux. So ux.≈⊥ = Ux.

If x �≈⊥ y, x ��≈⊥ y and y ��≈⊥ x then there is some n with xn �=∗ yn.So there is no z with z ⊇∗ x, y.

�From proposition 7.3 it is easy to prove

Proposition 7.4. If Y is a ≈⊥ transversal, then Y satisfies MOH,where the tree has height ω + 1.

Hence

Theorem 7.1. If Y is an ≈⊥ transversal of (ω + 1)ω, then Y is anultraparacompact subspace of ∇(ω + 1)ω.

Because the tree has height ω + 1 < d, Wingers’ technique gives

Theorem 7.2. If Y is an ≈⊥ transveral of (ω + 1)ω then E(Y ) is anultraparacompact subspace of �(ω + 1)ω.

7.2. Subspaces of �(ω + 1)ω under � h. Fix h = {hα : α < b} an

unbounded family, where each hα is strictly increasing, and if α < β thenhα ≤∗ hβ.

We construct a partial order � hand an associated equivalence relation

≈ hso if Y is an ≈ h

transversal then E(Y ) is an ultraparacompact subspaceof �(ω + 1)ω.


20 JUDITH ROITMAN

Definition 7.6. For x ∈ ω⊆ω and β < b we define ax,β = {n : x(n) <hβ(n)}.

Definition 7.7. For x an infinite partial function from ω to ω we definea countable sequence βx,α, α < δx < ω1 and, for each β = βx,α, a function

xβ as follows:Given βx,γ and xβx,γ , γ < α let x∗,α = x\

⋃γ<α x

βx,γ . If x∗,α is finite, let

xβx,α = ∅ and stop. Otherwise define βx,α = the least β with ax∗,β infinite;

xβx,α = x|ax∗,β

. Let δx = sup{α+ 1 : xβx,α �= ∅}.For x ∈ ω⊆ω we define I(x) = {βx,α : α < δx}.15

Definition 7.8. For x, y ∈ ω⊆ω, x � hy iff I(y) is an end-extension of

I(x) and ∀β ∈ I(x) xβ =∗ yβ. x ≈ hy iff x � h

y and y � hx. x � h

y iffx � h

y; x ≈ hy iff x ≈ h

y.

Remark 7.1. � his transitive and well-founded.

Definition 7.9. For x ∈ ω⊆ω, ux,≈h= {y : x � h

y}.

The next proposition is the combinatorial lemma that serves the role ofproposition 7.2:

Proposition 7.5. Fix h ∈ ωω. If Y is an =∗ transversal of {x : ∀n ∈dom x x(n) ≤ h(n)} then Y is a strongly separated subset of �(ω + 1)ω.

Proof. For each y ∈ Y define uy = {z ∈ (ω+1)ω : z ⊇ y and ∀(n, k) ∈z \ y z(n) > h(n)}. {uy : y ∈ Y } is a discrete clopen separating family.

�

From proposition 7.5 we can prove:

Proposition 7.6. (a) Each ux,≈his open

(b) ux,≈h∩ uy,≈h

= ∅ iff x, y are � hincomparable.

(c) ux,≈h⊇ uy,≈h

iff x � hy

From propositions 7.1 and 7.6, MOH holds. Hence

Theorem 7.3. If Y is an ≈ htransversal of (ω + 1)ω, then Y is an

ultraparacompact subspace of ∇(ω + 1)ω.

Note that the height of this tree is is ω1. So again, using Wingers’technique,

Theorem 7.4. If Y is an ≈ htransversal of (ω + 1)ω, then E(Y ) is an

ultraparacompact subspace of �(ω + 1)ω.

15In [13] I(x) is denoted E(x).

176176


7.3. Countable metrizable factors. In this section we modify thetechniques of the previous two subsections to the situation of countablemetrizable factors.

Fix ∇ = ∇n<ωXn where each Xn is countable metrizable. If b = d weknow that ∇ (in fact �) is paracompact, so we suppose b < d and find largesubspaces which are ultraparacompact.

Fix h = {hα : α < b} an unbounded family where, if α < β thenhα <∗ hβ. Define δα(x, y), ux,α as in the proof of proposition 6.10. We nolonger have a base, but we can still do a lot.

Proposition 7.7. ∀x, y, α either ux,α = uy,α or ux,α ∩ uy,α = ∅.Proof. Suppose z ∈ ux,α ∩ uy,α. ∀n < ω δα(x, y)(n) ≤ δα(x, z)(n) +

δα(z, y)(n). So y ∈ ux,α. Simlarly, if y, z ∈ ux,α then z ∈ uy,α. I.e.,ux,α \ uy,α = ∅ for all y with uy,α ∩ ux,α �= ∅. �

Proposition 7.8. For all x and all α < β, ux,α ⊇ ux,β.

Proof. This is because each δα(x, y)(n) ≤ δβ(x, y)(n). �For each α < b define Tα = {ux,α : x ∈ ∇} then each Tα is a pairwise

disjoint clopen refinement of each Tβ with β < α. Let T =⋃

α<bTα.Definition 7.10. x ≈T y iff ∀α < b ux,α = uy,α.

Theorem 7.5. Let ϕ : T → ∇ so that Y =range ϕ is an ≈T transversalwith ϕ(u) ∈ u for all u ∈ T . Then Y is an ultraparacompact subspace of ∇.

Proof. Let W be an open cover of ∇. If y = ϕ(u) we write ht(y) =htT (u) and y � z iff y, z ∈ Y, ht(y) = ht(z) and y ≈T z. At stage α < b,assume we have a pairwise disjoint cover by clopen boxes Vα covering {y ∈Y : ht(y) < α} and refining both W and {ux,β : β < α}. Given y ∈ Y withht(y) = α and y /∈

⋃Vα we only need to find a clopen box vy with y ∈ vy

and vy refining W . Since y has fewer than b �-predecessors, we can do thisvia proposition 5.2.3. �

Again using Wingers’ technique, since the height of T = b,

Theorem 7.6. Let � = �n<ωXn be a box product of countably manycountable metrizable spaces, and ϕ : T → ∇n<ωXn so that Y =range ϕ is an≈T transversal with ϕ(u) ∈ u for all u. Then E(Y ) is an ultraparacompactsubspace of �.

7.4. Countable first countable. In this section we see what we canget for countable first countable factors. The ∇ result is easy, but there isn’tan easy box product result.

Let ∇ = ∇n<ωXn where each Xn is countable first countable (hence0-dimensional). We assume that, as a set, each Xn = ω (with a topologythat need not be discrete).

Definition 7.11. For f ∈ ωω let ∇f = {x ∈ ∇ : ∀∞n x ≤∗ f}.


22 JUDITH ROITMAN

Definition 7.12. For each k, n let Vk,n be a pairwise disjoint clopen (inXn) cover of {0, ...k} so if v ∈ Vk,n then there is exactly one i ≤ k with i ∈ v.For f ∈ ωω let Vf = {∇n<ωvn : vn ∈ Vf(n),n}.

Note that each Vf is a pairwise disjoint clopen cover of ∇f .

Proposition 7.9. Each ∇f is strongly separated.

Proof. We need to show that Vf is discrete. So suppose x /∈⋃

Vf . Ifx(n) ∈

⋃Vf(n),n define un,f,x to be the unique set in Vf(n),n with x(n) ∈

un,f,x. If x(n) /∈⋃Vf(n),n let un,f,x = ω \

⋃Vf(n),n. Then uf,x = ∇n<ωun,f,x

is a clopen neighborhood of x disjoint from⋃

Vf . �Proposition 7.10. If F ∈ [ωω]<d then ∇F =

⋃f∈F ∇f is closed.

Proof. Suppose x /∈ ∇F . For each f ∈ F let af = {n : x(n) > f(n)}.By proposition 2.1.2 there is u a clopen box neighborhood of x so ∀f ∈F {n ∈ af : u(n) ∩ (1 + f(n)) = ∅} is infinite. Hence u ∩ ∇F = ∅. �

Hence by proposition 5.7 we have

Theorem 7.7. If F ∈ [ωω]≤d then ∇F is an ultraparacompact subspaceof ∇.

Without an well-founded pre-order on points of∇, we can’t applyWingers’technique to get analogs of theorems 7.2, 7.4, 7.6.

8. A combinatorial principle for �(ω + 1)ω

In this section we give a combinatorial principle which implies that�(ω+1)ω is paracompact, and show that it holds in the models we have beenconsidering. It suffices to consider only ∇ = ∇(ω + 1)ω; for the rest of thissection that is the space we are working in. This principle does not explicitlyrefer to any stratification, but the proofs of its consistency do.

Definition 8.1. ω⊂ω = {x ∈ ω⊆ω : x �=∗ x}.Definition 8.2. Δ is the following statement: there is F : ω⊂ω → ωω

so that if x, y ∈ ω⊂ω, x\y and y \x are both infinite and ¬∃∞n x(n) �= y(n),then (x \ y) �>∗ F (y) or (y \ x) �>∗ F (x). We refer to F (x) as fx.

Note that if x =∗ x then any f would have the property of fx in Δ,vacuously. So there’s no point in discussing such points.

Definition 8.3. ∇∗ = {x ∈ ∇ : x ∈ ω⊂ω}.Remark 8.1. Since ∇ \ ∇∗ consists of a one non-isolated point and all

isolated points in ∇, ∇ is paracompact iff ∇∗ is.

Recall definition 7.2.3. Using the notation of this section,

Definition 8.4. For x ∈ ω⊂ω, Nx,f = {y : y ⊇∗ x and ∀∞n ∈ dom y\dom x y(n) > f(n)}.

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Theorem 8.1. If Δ holds then ∇∗ is ultraparacompact.

Proof. Let {xα : α < c} ⊂ ∇∗ so ∀x ∈ ∇∗ ∃α x ⊆∗ xα. Define∇α = {x ∈ ∇∗ : α is least with x ⊆∗ xα}. Let U be an open cover of ∇. Foreach x ∈ ω⊂ω let fx be as in Δ. If x ∈ ∇α define gx > fxα , gx > xα so Nx,gx

refines some u ∈ U . By the proof of proposition 7.5, Wα = {Nx,gx : x ∈ Xα}is a discrete family.

Suppose α < β, x ∈ ∇α, y ∈ ∇β. Then y �⊆∗ x. If x �⊆∗ y then, by Δ,Nx,gx ∩Ny,gy = ∅.

Subclaim 8.0.1. For x, y ∈ ω⊂ω and f ∈ ωω if x ⊆∗ y, x �=∗ y, theneither y ∈ Nx,f or ∀g ∈ ωωNx,f ∩Ny,g = ∅.

Proof. If ∀∞n ∈ dom y\ dom x y(n) > f(n) then y ∈ Nx,f . Otherwise∀z ⊇∗ y ∃∞n ∈ dom z\ dom x z(n) ≤ f(n). �

For α < c we define Dα ⊆ ∇α and a clopen family Vα covering Dα soV =

⋃α<cVα covers ∇∗ and V is a pairwise disjoint refinement of U , as

follows:Let D0 = ∇0,V0 = {Nx,gx : x ∈ X0}. Given Dβ,Vβ for all β < α we

define Dα = ∇α \⋃

β<α

⋃Vβ . Vα = {Nx,gx : x ∈ Dα}.

To show that V is pairwise disjoint, suppose Nx,gx , Ny,gy ∈ V, x �=∗ y. Ifthere is α with x, y ∈ Dα, we’re done, since Wα is a discrete family. Supposex ∈ Dβ, y ∈ Dα, β < α. If y ⊇∗ x then since y /∈ Nx,gx , by the subclaimNx,gx ∩Ny,gy = ∅. The remaining case is x \ y, y \ x both infinite, in whichcase we’re done by Δ. �

Where does Δ hold? Not surprisingly

Theorem 8.2. If b = d or d = c or MH holds, then Δ holds.

Proof. For b = d: Let {hα : α < b} be a scale in ωω and let Xα ={x ∈ ∇∗ : α is least with x <∗ hα}. For x ∈ Xα define fx = hα. Thenif x ∈ Xα, y ∈ Xβ, α ≤ β, if x \ y is infinite and ¬∃∞n x(n) �= y(n) then(x \ y) �>∗ fy.

For d = c: Let ∇∗ = {xα : α < d}. Suppose we’ve assigned fxα foreach α < β so if γ < α < β and (xγ \ xα) �=∗ ∅ and (xα \ xγ) �=∗ ∅ and¬∃∞n xγ(n) �= xγ(n)then either (xγ \ xα) �>∗ fxα or (xα \ xγ) �>∗ fxγ . Letaα = dom xα\ dom xβ. Let A = {aα : aα is infinite}. By proposition 2.1.2,there is g ∈ ωω so for all aα ∈ A g|aα �<∗ xα|aα and let fxβ

= g.For MH: Let Hα be as in the definition of MH, let fα be as in proposition

2.2 and if α is least with x ∈ Δ∗ ∩Hα, let fx = fα.�

9. Some questions

The main question is, of course:

Question 9.1. Are any of the following �n<ωXn really paracompact(i.e., without extra assumptions): each Xn is compact scattered, each Xn is


24 JUDITH ROITMAN

σ-compact first countable; each Xn is compact first countable; each Xn iscompact metrizable; each Xn is countable metrizable; each Xn = ω + 1?

Other questions include:

Question 9.2. What other subspaces of box products are really para-compact?

Question 9.3. Do theorems which hold under one of the hypothesesused in this paper (CH, b = d, d = c, d = ω + 1, MH) hold under any of theothers?

Question 9.4. If a nabla product of countably many compact firstcountable spaces is paracompact, is it hereditarily paracompact? Is it ultra-paracompact?16

Question 9.5. Can Δ be used to prove that any other box productsare paracompact?

References

1. E.K. van Douwen, The box product of countably many metrizable spaces need not benormal, Fund. Math. 88 (1975), pp. 127–132

2. E.K. van Douwen, Another nonnormal box product, General Topology and its Appli-cations 7 (1977), pp. 71–76

3. E. K. van Douwen, Covering and separation properties of box products, Surveys inGeneral Topology, ed. G.M. Reed, Academic Press (New York) 1980, pp. 55–129

4. P. Erdos and M.E. Rudin, A non-normal box product, Infinite and finite sets (Colloq.Math. Soc. Janos Bolyai 1973) 10 North-Holland, Amsterdam (1973) pp.629-631

5. K. Kunen, Some comments on box products, Infinite and finite sets (Colloq. Math.Soc. Janos Bolyai 1973) 10 North-Holland, Amsterdam (1973) pp.1011-1016

6. K. Kunen, Paracompactness of box products of compact spaces, TAMS 240 (1978),pp. 307–316

7. K. Kunen, Box products of ordered spaces, Topology and its Applications 20 (1985)no. 3, pp. 245-250

8. L.B. Lawrence, The box product of countably many copies of the rationals is consis-tently paracompact, Trans. Amer. Math. Soc. 309 (1988) no. 2, pp. 787-796

9. L.B. Lawrence, Failure of normality in the box product of uncountably many real lines,TAMS 348 (1996) no. 1, 187–203

10. L.B. Lawrence, Towards a Theory of Normality and Paracompactness in Box Products,Ann. New York Acad. Sci., 75 (1993), pp. 78–91

11. J. Roitman, More paracompact box products, PAMS 74 (1979), pp. 71–7612. J. Roitman, Paracompactness of box products and their subspaces, Top. Proc. 31 (2007)

no. 1, pp. 265 – 28113. J. Roitman, More paracompact subspaces of �(ω+1)ω, Top. Proc. 34 (2009) pp. 53–7614. M.E. Rudin, The box product of countably many compact metric spaces, General

Topology and its Applications 2 (1972), 293 - 29815. M.E. Rudin, Lectures on set-theoretic topology, Regional Conf. Series in Math., 23,

AMS, Providence, 197516. S. W. Williams, Is �ω(ω + 1) paracompact?, Topology Proc., 2 (1976), pp. 141 – 146

16At one point I thought Kunen had given a positive answer to the second part ofthis question, but this was a misunderstanding.

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17. S. W. Williams, Box products, Handbook of Set-Theoretic Topology, K. Kunen andJ.E. Vaughan ed., Elsevier, Amsterdam, 1984, pp. 169–200

18. S. W. Williams, Box products twenty-five years later, preprint19. L. Wingers, Box products of σ-compact spaces, Topology and its Applications 56, no.

2, (1994), 185 - 19720. L. Wingers, Box products and Hurewicz spaces, Topology and its Applications 64,

no. 1, (1995), pp. 9–21

Department of Mathematics, University of Kansas, Lawrence, Kansas 66045





Some problems and techniques in set-theoretic topology

Franklin D. Tall1

Abstract. I survey some problems and techniques that have interested meover the years, e.g. normality vs. collectionwise normality, reflection, preserva-tion by forcing, forcing with Souslin trees, and Lindelof problems.

Introduction

In 2008–9 I gave a graduate course in set-theoretic topology at the Universityof Toronto. With retirement on the horizon, I wanted to pass on to the next gen-eration some of the topological problems I had found interesting and some of theset-theoretic techniques I had found useful over my 40 years of research. MarionScheepers — one of the editors of this volume — asked me to send him copies ofthe lecture notes my students prepared. He then suggested I make them into anarticle for this volume. I was initially dubious, since most of the material had beenpublished elsewhere. I concluded, however, that I had indeed added enough valueto the extracted material to make the project worthwhile. In particular, I had in-cluded lots of “what’s important” and “what’s really going on” commentary oftenmissing from journal articles. I also realized that these notes could serve as an up-date of my 1984 survey Normality versus collectionwise normality [Tal84], as wellas call attention to some interesting work over the past few years on forcing withSouslin trees, the publication of which has been indefinitely delayed. The intendedaudience, then, is composed of graduate students interested in set-theoretic topol-ogy, topologists interested in learning some useful set-theoretic techniques scatteredthrough the literature, and set theorists interested in applications of set theory totopological problems. Prerequisites include a basic knowledge of general topology,and a first graduate course in set theory. For the former, I recommend Engelking[Eng89]; for the latter, Kunen [Kun80].

2010 Mathematics Subject Classification. Primary 54A35, 54D10, 54D20, 54D45; Secondary54D50.

Key words and phrases. Normal, collectionwise Hausdorff, collectionwise normal, first count-able, locally compact, Cohen real, random real, forcing, σ-centered, supercompact, reflection,preservation under forcing, elementary submodel, irrationals, forcing with a Souslin tree, Lin-delof.

1 The author was supported in part by NSERC Grant A-7354.


1



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2 FRANKLIN D. TALL

1. Normality versus collectionwise Hausdorffness

The relationship between normality and collectionwise Hausdorffness (everyclosed discrete subspace can be separated, i.e. expanded to a disjoint collection ofopen sets, each containing only one of the points of the subspace) was one of thefirst areas in general topology to feel the impact of set-theoretic methods [Tal77].[Tal84] exhaustively surveys the subject up to the time of its writing. Since then,there have been several interesting developments.

The question of whether locally compact normal metacompact spaces wereparacompact had been raised by Arhangel’skiı and by the author [Arh72], [Tal74].[Wat82] showed that V = L yielded an affirmative answer. By a difficult forcing ar-gument, a consistent counterexample was constructed by Gruenhage and Koszmider[GK96a]. It had been known since the 1960’s that collectionwise Hausdorffness suf-ficed to make such spaces paracompact; in [GK96b], Gruenhage and Koszmidershowed that ℵ1-collectionwise Hausdorffness sufficed in the course of their surprisingproof that:

Proposition 1.1. MAω1implies locally compact normal metalindelof spaces

are paracompact.

It had been known for a long time that Katetov’s problem: are compact spaceswith hereditarily normal squares metrizable? (“Yes” for cubes [Kat48]) could besolved if one combined “S & L” consequences of MAω1

with the non-existenceof separable normal first countable spaces which are not collectionwise Hausdorff.The solution of this problem by Larson and Todorcevic [LTo02] opened a newfront in the study of normality versus collectionwise normality. The question be-came, could other long-standing open problems be solved by consistently combiningconsequences of MAω1

or PFA with “normality implies collectionwise Hausdorff”consequences of V = L? There has been one notable success here, which we shallget into shortly, and a number of interesting open problems. We will start off witha result that no one had even conjectured before [LTaa], but in retrospect seems aplausible variation on the following classical theorem [Tal77], [Tal84], [Tal88].

Proposition 1.2. Adjoin ℵ2 Cohen subsets of ω1. Then normal spaces ofcharacter ≤ ℵ1 are ℵ1-collectionwise Hausdorff.

The newer result is:

Theorem 1.3 ([LTaa]). Force with a Souslin tree. Then normal first countablespaces are ℵ1-collectionwise Hausdorff.

The proof uses ideas from both Fleissner’s♦-for-stationary-systems proof [Fle74]and from the Cohen-subsets-of-ω1 proof. These are both included in my 1984 sur-vey. From the former, construct a partition that ruins all normalizations, and fromthe latter, arrange to successively decide initial segments of potential normaliza-tions. First countability, the countable chain condition, and ω-distributivity ensurethat closed unboundedly often, one can decide f | α, for f ∈ ω1ω and α ∈ ω1. Itis not known whether Theorem 1.3 can be improved to “character ≤ ℵ1”. Now forthe details.

Lemma 1.4. After forcing with a Souslin tree, the following holds. Supposethat {N(α, i) : i < ω, α < ω1} are sets such that for all α, i, N(α, i+ 1) ⊆ N(α, i).

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SOME PROBLEMS AND TECHNIQUES IN SET-THEORETIC TOPOLOGY 3

Suppose further that for all A ⊆ ω1, there is an f : ω1 → ω such that:⋃

{N(α, f(α)) : α ∈ A} ∩⋃

{N(β, f(β)) : β ∈ ω1 \A} = ∅.

Then there is a g : ω1 → ω and a closed unbounded C ⊆ ω1, such that:

whenever α < β and C ∩ (α, β] = ∅, then N(α, g(α)) ∩N(β, g(β)) = ∅.

It should be clear that Lemma 1.4 yields ℵ1-collectionwise Hausdorffness in firstcountable normal spaces: let the N(α, i)’s be a descending neighborhood base at α,where we have labeled the points of a discrete closed subspace of the space X withthe countable ordinals. Let g and C be as in the statement of the Lemma. Definec : ω1 → ω1 by letting c(α) = sup((C \ {0})∩ (α+1)), and let α ∼ β if c(α) = c(β).The ∼-classes are countable and normality implies ℵ0-collectionwise Hausdorffness,so there is a q : ω1 → ω such that c(α) = c(β) implies N(α, q(α))∩N(β, q(β)) = ∅.Let r(α) = max(g(α), q(α)). Then {N(α, r(α)}α<ω1

is the required separation. �

Proof of Lemma 1.4. Let S be a Souslin tree. Let {N(α, i) : i < ω, α < ω1}be S-names as in the hypothesis. For s ∈ S, let �(s) be the length of s. Since Shas countable levels and its corresponding forcing poset is ω-distributive, we canconstruct an increasing function h : ω1 → ω1 such that for all α < ω1 and all s ∈ Swith �(s) = h(α), s decides all statements of the form “N(β, j) ∩ N(α, i) = ∅”, forall i, j < ω and β < α.

Let A be an S-name for a subset of ω1 such that for no α < ω1 does any s ∈ Swith �(s) = h(α) decide whether α ∈ A. To define such an A, for each α < ω1 pick

two successors of each s ∈ S with �(s) = h(α) and let one force α ∈ A and let the

other force α ∈ A.Let f be an S-name for a function f : ω1 → ω as in the hypothesis of the

lemma, with respect to A. Let C be a closed unbounded subset of ω1 in V suchthat for each s ∈ S with �(s) ∈ C, s decides f |�(s) and A|�(s), and such thatfor all α < β < ω1, if β ∈ C then h(α) < β. We will define an S-name g for afunction from ω1 to ω such that whenever α < β < ω1, if (α, β] ∩ C = ∅, thenN(α, g(α)) ∩N(β, g(β)) = ∅.

Let c : ω1 → ω1 be defined as above. Then for all β < α < ω1, c(α) = c(β) if

and only if C∩ (α\β) = ∅. Fix β < ω1. Each s ∈ S with �(s) = h(β) decides f |c(β)and A|c(β) and “N(α, f(α))∩ N(β, i) = ∅” for all i < ω, α < c(β), but not whetherβ ∈ A. Fix s ∈ S with �(s) = h(β). Since s does not decide whether β ∈ A, we

claim that there is an i0 < ω such that for all α < c(β) such that s � α ∈ A,

s � N(α, f(α)) ∩ N(β, i0) = ∅.To see this, extend s to t ∈ S forcing that β ∈ A and deciding f(β). Let i0 be

the value of f(β) as decided by t. Then for each α < c(β) such that s � α ∈ A,t forces that N(α, f(α)) ∩ N(β, i0) = ∅, but these facts were already decided by

s. Similarly, there is an i1 < ω such that for all α < c(β) such that s � α ∈ A,

s � N(α, f(α)) ∩ N(β, i1) = ∅.Since s decidesA|c(β), letting i = max{i0, i1}, for all α < c(β), s � N(α, f(α))∩

N(β, i)) = ∅.We have such an is for each s in the c(β)-th level of the tree, so we can construct

a name g such that s � g(β) = max{is, f(β)} for each s ∈ S with �(s) = c(β). Theng is as required. �

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4 FRANKLIN D. TALL

It is not clear how Theorem 1.3 fits into the panoply of “normal implies collec-tionwise normal” results. Because the proof doesn’t obviously generalize to charac-ter ≤ ℵ1, the standard technique [Wat82] for obtaining “locally compact normalimplies ℵ1-collectionwise Hausdorff” is unavailable. However, one desired extensiondoes go through without difficulty, Although Souslin tree forcing does not make nor-mal first countable spaces collectionwise Hausdorff [Tal11c], one can arrange forthat statement to hold in the models one is likely to be interested in, simply bystarting with a nice enough ground model. In particular, one can start with amodel of GCH, adjoin λ+ Cohen subsets of λ for each regular λ via an Eastonextension [Eas70], [Tal88], establishing ♦ for stationary systems for all such λ.Forcing over this model with an ℵ2-chain condition partial order P of size ℵ2 willpreserve ♦ for stationary systems at ℵ2 and above. Thus, if, say, Q = P×S, whereS is a Souslin tree, in the resulting model normal first countable spaces will becollectionwise Hausdorff.

This observation is quite interesting from the point of view of obtaining para-compactness. It is well-known that paracompactness is equivalent to subparacom-pactness (every open cover has a σ-discrete closed refinement) plus collectionwisenormality: expand a σ-discrete closed refinement to a σ-discrete open refinement. Ifthe space is e.g. locally compact, one can get away with collectionwise normality forcompact sets or even (this is not obvious) collectionwise Hausdorffness. With this inmind, models in which one can easily obtain σ-discrete closed refinements as wellas obtaining “normal implies collectionwise Hausdorff” results are quite promis-ing. In 1983, Balogh [Bal83] proved under MAω1

that locally countable subsetsof size ℵ1 in a compact countably tight space are σ-discrete. Under suitable hy-potheses, this can be improved to σ-closed-discrete, and the paracompactness ideamentioned above comes into play, if one assumes or proves collectionwise Haus-dorffness. Balogh assumed collectionwise Hausdorffness in [Bal83] and [Bal02], asdid Nyikos in a number of papers on hereditarily normal manifolds, e.g. [Nyi02].What one would like to do is obtain Balogh’s MAω1

result in one of the modelsobtained by forcing with a Souslin a tree. The assumption of “collectionwise Haus-dorff” could then likely be removed, since normality was assumed anyway. We willcomment further on Balogh’s result shortly, but we now see the interest in retainingconsequences of MAω1

(or PFA, etc.) when forcing with a Souslin tree.

Definition 1.5. A space is ℵ1-collectionwise Hausdorff if each closed dis-crete subspace D of size ℵ1 can be separated by disjoint open sets. It is weaklyℵ1-collectionwise Hausdorff if ℵ1 of the points of D can always be separated.It is stationarily ℵ1-collectionwise Hausdorff if stationarily many can be sep-arated for any well-ordering of D. It is very stationarily ℵ1-collectionwiseHausdorff if stationarily many of the points of any stationary subset of D can beseparated, for any well-ordering of D. closed unboundedly ℵ1-collectionwiseHausdorff is defined analogously to “stationarily ℵ1-collectionwise Hausdorff”.

Proposition 1.6 ([Tay81]). Closed unboundedly ℵ1-collectionwise Hausdorffspaces are ℵ1-collectionwise Hausdorff.

Proposition 1.7. 2ℵ0 < 2ℵ1 implies that normal spaces of character ≤ 2ℵ0 arestationarily ℵ1-collectionwise Hausdorff [Tay81] and hence locally compact normalspaces are stationarily ℵ1-collectionwise Hausdorff [Tal84].

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There is an example consistent with GCH of a first countable, locally com-pact normal space which is stationarily ℵ1-collectionwise Hausdorff, but not verystationarily ℵ1-collectionwise Hausdorff [DS79], [Tal84].

Problem 1.8. Is every locally compact, normal, very stationarily ℵ1-collectionwiseHausdorff space ℵ1-collectionwise Hausdorff?

The proof of Gruenhage and Koszmider actually establishes that:

Proposition 1.9. Locally compact, normal, metalindelof, very stationarily ℵ1-collectionwise Hausdorff spaces are paracompact.

It is an interesting and largely unexplored question as to what one can get byjust forcing with a Souslin tree. See [LTo01], [LTo02], [Tod]. It turns out thatby forcing with a particular kind of Souslin tree, over particular models, one canget quite a lot.

2. Forcing with coherent Souslin trees

Definition 2.1. A coherent tree is a downward closed subtree S of <ω1ωwith the property that {

⇀

ξ ∈ dom s ∩ dom t : s(⇀

ξ) = t(⇀

ξ)} is finite for all s, t ∈ S.A coherent Souslin tree is a Souslin tree given by a coherent family of functionsin <ω1ω closed under finite modifications.

Coherent Souslin trees can be constructed from ♦ [Lar99].

Theorem 2.2 ([LTo02]). Let S be a coherent Souslin tree. Iterate countablechain condition posets that preserve S to obtain MAω1

restricted to such posets.Then force with S. In the resulting model, compact spaces with hereditarily normalsquares are metrizable.

The most interesting models to force over with coherent Souslin trees are modelsof iteration axioms such as MAω1

, PFA, and Martin’s Maximum, restricted toposets that preserve the “Souslinity” of the tree. A good many of the consequencesof these axioms survive the additional forcing, but whether some others do are hardopen questions we shall mention later.

Definition 2.3. Let S be a coherent Souslin tree. PFA(S) is PFA restrictedto proper posets that preserve S. Let Φ be a proposition. PFA(S)[S] � Φ is anabbreviation for “Φ holds in any model obtained by forcing with S over a model ofPFA(S)”. MAω1

(S)[S], MM(S)[S] are similarly defined.

PFA(S)[S] is not an axiom, but is convenient to act as if it were. Proofsinvolving PFA(S)[S] are similar to, but more difficult than, those for PFA, sinceone tries to apply PFA(S) to a partial order involving S-names. The intuitionbehind the use of coherence is that when forcing with a coherent Souslin tree, twodifferent interpretations of an S-name will essentially be isomorphic, since, up toan automorphism of S, S has only one generic branch. It is unfortunate that thefundamental work of Todorcevic concerning forcing with a coherent Souslin treeover a model of PFA(S) has not been published. It is hoped that [Tod] willeventually appear.

Of course if PFA(S)[S] only provided harder proofs of PFA consequences,that would not be very interesting. As we shall see, that trick is to combine thoseconsequences with consequences of forcing with S.

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6 FRANKLIN D. TALL

The most useful consequence of PFA(S)[S] from the point of view of obtainingparacompactness is Balogh’s

∑∑∑: Suppose X is a compact, countably tight space and Y is a

subset of X of size ℵ1 such that for each y ∈ Y there exist opensets Uy, Vy such that y ∈ Vy ⊆ V y ⊆ Uy, and |Uy∩Y | is countable.Then Y =

⋃n<ω Yn, where each Yn is closed discrete in

⋃{Vy : y ∈

Y }.It is not hard to show that

∑∑∑implies there are no compact S-spaces.

∑∑∑was

obtained from MAω1in [Bal83]. If the space has Lindelof number ≤ ℵ1, we may

assume the Yn’s are closed discrete in the whole space. Todorcevic [unpublished]proved this from PFA(S)[S] in 2002. A detailed proof of the special case in whichfinite products of S are sequential is supposed to appear in [Fis]. An essentialingredient in Todorcevic’s proof is:

Theorem 2.4. PFA(S)[S] implies compact countably tight spaces are sequen-tial.

The compact S-space proof and the proof of Theorem 2.4 were sketched byTodorcevic in his talk at Erice in 2008 [Tod08].

The reader may have some idea as to how normality plus collectionwise Haus-dorffness plus

∑∑∑could yield a σ-discrete open refinement of an open cover of size

ℵ1, but what about open covers of other sizes? The answer is that in a major caseof interest, those are all one needs to be concerned about.

Definition 2.5. Axiom R [Fle86] is the assertion that if S is a stationarysubset of [κ]ω and C is a subset of [κ]ω1 closed under unions of strictly increasingω1-chains, then there is a Y ∈ C such that S ∩ [Y ]ω is stationary.

Lemma 2.6 ([Bal02]). Axiom R implies that if X is a locally Lindelof, regular,countably tight space such that every Lindelof Y ⊆ X has Lindelof closure, thenif X is not paracompact, it has a clopen non-paracompact subspace with Lindelofnumber ℵ1.

Although Axiom R does not follow from PFA(S)[S] [Tal11c], it holds in thenatural iteration model for PFA(S)[S] described above [LTab].

Reducing the question of paracompactness to subspaces with Lindelof numberℵ1 in countably tight spaces in which Lindelof spaces have Lindelof closures allowsa further simplification.

Definition 2.7 ([Nyi84]). A space X is of Type I if X =⋃

α<ω1Xα, where

each Xα is open, α < β implies Xα ⊆ Xβ , and each Xα is Lindelof. {Xα : α < ω1}is canonical if for limit α, Xα =

⋃β<α Xβ.

Lemma 2.8 ([EN09]). If X is locally compact with Lindelof number ≤ ℵ1,and each Lindelof subspace of X has Lindelof closure, then X is of Type I, with acanonical sequence.

Lemma 2.9 ([Nyi84]). If X is of Type I, then X is paracompact if and only if{α : Xα \Xα = 0} is non-stationary.

Theorem 2.10. There is a model of PFA(S)[S] in which every ℵ1-collectionwiseHausdorff, locally compact normal space in which closures of Lindelof subspaces areLindelof, and which does not include a perfect preimage of ω1, is paracompact.

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Proof. We may confine ourselves to Type I spaces. If X is not paracompact,we may pick a canonical sequence and a stationary S ⊆ ω1 and xα ∈ Xα \Xα, foreach α ∈ S. S is σ-closed-discrete because of:

Lemma 2.11 ([Bal83]). The one-point compactification of a locally compactspace X is countably tight if and only if X does not include a perfect pre-image ofω1.

By normality and collectionwise Hausdorffness, we can find a stationary setS′ ⊆ S of limit ordinals such that {xα : α ∈ S′} is closed discrete. By normality andcollectionwise Hausdorffness, we can expand the xα’s to a discrete collection of openUα’s. Each Uα intersects some Xβ, β < α; pressing down yields an uncountable

closed discrete subspace of some Xβ, contradiction. �

I believe the “ℵ1-collectionwise Hausdorff” hypothesis is superfluous, but I havenot yet finished proving that.

Corollary 2.12 ([LTaa]). There is a model of PFA(S)[S] in which everylocally compact, perfectly normal space is paracompact.

Proof. It is easy to see that a perfectly normal space cannot include a perfectpreimage of ω1. A locally compact, perfectly normal space is first countable. Itthus suffices to show that Lindelof subspaces have Lindelof closures. We use an ideafrom [Nyi03]. In fact, it suffices to show that open Lindelof subspaces do, sinceby local compactness, such a subspace Y can be covered by countably many openLindelof sets. The closure of that union is then Lindelof and includes Y . Let thenB be a right-separated subspace of the boundary of an open Lindelof subspace Y .We claim B is countable, whence the boundary of Y is hereditarily Lindelof. If Bis uncountable, it has an uncountable closed discrete subspace. Via normality andℵ1-collectionwise Hausdorffness, we can expand this to a discrete collection of opensets, giving a contradiction. �

It is very much an open question what other consequences of PFA hold underPFA(S)[S] (orMM(S)[S], orMAω1

(S)[S]). Three of the most intriguing problemsare:

(1) Does PFA(S)[S] imply every hereditarily separable T3 space is hereditar-ily Lindelof?

(2) Does MM(S)[S] imply first countable perfect pre-images of ω1 must in-clude a copy of ω1?

(3) Is there a model of PFA(S)[S] in which locally compact normal spacesare collectionwise Hausdorff?

The first is very natural, considering that PFA implies the conclusion, and thatPFA(S)[S] implies there are no such compact spaces. The answer is not knowneven if the additional assumption of first countability is made.

The second would solve a longstanding open problem of Nyikos [Nyi83]: I haveshown that i) the conclusion of (2) plus ii) a form of P -ideal dichotomy [AT97]plus iii)

∑∑∑plus iv) normal first countable spaces are ℵ1-collectionwise Hausdorff

implies hereditarily normal manifolds of dimension > 1 are metrizable. All ex-cept i) are known to follow from PFA(S)[S] [Tod], [Fis], [LTaa]. Todorcevicproposed MM(S)[S] rather than PFA(S)[S] since a proof may involve shooting aclub through a stationary set.

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8 FRANKLIN D. TALL

The third has been annoying me for several years; I thought I had a proof,but there was a gap which I hope to finish filling in the near future. It is naturalto expect this to be true, since any other general theorem about when normalityimplies some form of collectionwise normality if true about first countable spacesseems to turn out to be true about locally compact spaces. [LTab], [Tal11a],[Tal11c] explore the consequences of an afirmative answer.

I had originally planned to put in a PFA(S)[S] proof into this survey, but infact there have been no very successful applications so far of Todorcevic’s methodsother than those he has done, none of which he has published. (The MAω1

(S)[S]technique is exhibited in [LTo02].) Such a proof would also add at least 10 pagesto this survey, which is already too long. However, I think it will be useful for thereader if we list a number of widely applicable set-theoretic or topological principleswhich hold under PFA(S)[S]:

(1) OCA [Far96].(2) Every Aronszajn tree is special [Far96].(3) b = ℵ2 = 2ℵ0 [Lar99].(4) P -ideal dichotomy [Tod].(5) Every compact, countably tight space is sequential [Tod].(6) Every compact hereditarily separable space is hereditarily Lindelof [Tod].(7) Every first countable hereditarily Lindelof space is hereditarily separable

[LTo02].(8)

∑∑∑[Tod] plus [Fis].

(9) Normal first countable spaces are ℵ1-collectionwise Hausdorff [LTaa].

If PFA(S) is forced in the usual fashion [Dev83, Lav78], we also obtain[LTab]:

(10) Fleissner’s Axiom R.

By starting with a particular ground model, we can obtain all the above con-sequences, and in addition:

(11) Every first countable normal space is collectionwise Hausdorff.

Proof [LTaa]. We first force to make a supercompact κ indestructible underκ-closed forcing [Lav78]. We then force GCH and ♦ for stationary systems at allregular uncountable λ by adjoining λ+ Cohen subsets of λ for all such λ via anEaston extension. Getting GCH is standard; obtaining ♦ for stationary systemsis due to W. Fleissner and can be found in [Tal88]. That the supercompactnessof κ is preserved by this forcing is because it is a product of κ-closed forcing witha forcing of size < κ. (“Mild” forcings preserve large cardinals — see e.g. 10.15of [Kan94]). Since we now have ♦, we can construct a coherent Souslin tree S[Lar99]. We then proceed to force PFA(S)[S] in the usual manner, establishing inparticular that normal first countable spaces are ℵ1-collectionwise Hausdorff. Theforcing to establish PFA(S)[S] is a κ-chain condition forcing of size κ, so preservesGCH as well as ♦ for stationary systems for all regular λ ≥ κ (Fleissner, see[Tal88]). By [Fle74] (or see [Tal84]), we then have that normal first countableℵ1-collectionwise Hausdorff spaces are collectionwise Hausdorff. But κ now equalsℵ2, so we are done. �

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3. Preservation under forcing

There is a very useful technique for proving preservation of covering properties(or non-covering properties) under Cohen and random reals. This technique wasintroduced in [DTW90a] as part of a reflection + forcing proof of the consistencyof the Normal Moore Space Conjecture, and has found several applications since:[GJT98], [Iwa07], [ST], [Kad]. It almost — but not quite — yields a groundmodel refinement for an open cover in a forcing extension.

Definition 3.1. An n-dowment is a family L of finite antichains of Fn(κ, 2)(the forcing adding κ Cohen reals) such that:

(1) For each maximal antichain A ⊆ Fn(κ, 2), there is an L ∈ L such thatL ⊆ A,

(2) For every p ∈ Fn(κ, 2) such that |domp| ≤ n and for every collectionL1, . . . , Ln of elements of L, there are q1 ∈ L1, . . . , qn ∈ Ln and there isan r ∈ Fn(κ, 2) such that r ≤ p and r ≤ qi, for every i ≤ n.

n-dowments are the Cohen real analogue of the fact that for random reals, givena condition p with measure > 1 \ (n−1

n )n, n ∈ ω, and finite antichains, L1, . . . , Ln,

each with measures summing up to at least n−1n , there are elements qi ∈ Li and

a condition r such that r ≤ p and r ≤ qi. They are named in honor of Alan Dowwho discovered their existence in connection with his work on remote points inStone-Cech remainders. A proof of their existence can be found in [DTW90a].Since the proof itself has not had any applications in our work, we omit it. Theway we use n-dowments is, given an open cover in a forcing extension composed ofground model open sets, we create an open cover in the ground model by takingintersections of the possible elements of the given cover given by the Ln’s, and usewhat we know about open covers in the ground model to force what we want tohold in the extension. Here is a typical proof; more difficult ones can be found inthe papers cited above.

Theorem 3.2. Lindelofness is preserved by Cohen reals.

Proof. Given an open cover U in the extension, without loss of generality wemay assume U is composed of ground model open sets. For each x ∈ X we canpick a maximal antichain Ax and open sets Up,x containing x such that for every

p ∈ Ax, p � Up,x ∈ U . For each n ∈ ω, let Ln be an n-dowment. Then for eachn ∈ ω, there is an An,x ⊆ Ax. An,x ∈ Ln. Define

Vn,x =⋂

{Up,x : p ∈ An,x}.

Then {Vn,x : x ∈ X} is an open cover of X in V and so has a countable subcoverVn.

Claim 3.3. � � (∀y ∈ X)(∃n ∈ ω)(∃V ∈ Vn)(∃U ∈ U)(y ∈ V ⊆ U).

Proof. We have to prove that:

(∀y ∈ X)(∀q)(∃r ≤ q)(∃n ∈ ω)(∃V ∈ Vn)(∃ an open U)(r � y ∈ V ⊆ U ∈ U).Fix y ∈ X and q ∈ Fn(κ, 2). Take n ≥ |dom q|. Since Vn covers X, there is aVnx ∈ Vn, such that y ∈ Vn,x. By the definition of n-dowment, there is a p ∈ An,x

such that p and q are compatible. Let r ≤ p, q. Then r � y ∈ Vn,x ⊆ Un,x ∈ U . �

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10 FRANKLIN D. TALL

By the claim, in V [G], for each y ∈ X, pick V yn,x such that y ∈ V y

n,x ⊆ U , forsome U ∈ U . Then {V y

n,x : y ∈ Y } is a countable refinement of U , so U has acountable subcover. �

The early work on n-dowments might lead one to believe that covering prop-erties behave the same under Cohen and random extensions. This is not the case;there are several counterexamples in [ST].

Normality can be thought of as a covering property. Let’s show that if Y is aclosed discrete subspace of X, and there exists a Z ⊆ Y such that there do not existdisjoint open sets about Z and Y − Z, then the same is true after adding Cohenreals.

In fact, for future use, let’s prove a more complicated version of this:

Definition 3.4. A discrete collection Y is normalized if for each Z ⊆ Y , thereexist disjoint open sets about

⋃{Y : Y ∈ Z} and

⋃{Y : Y ∈ Y−Z}. Y is separated

if there exist disjoint open sets UY ⊇ Y , for all Y ∈ Y .

Theorem 3.5. If Y is not normalized, it remains not normalized in any Cohenreal extension.

Proof. Let Z ⊆ Y witness that Y is not normalized in the ground model. Firstconsider the case of adding κ Cohen reals where κ ≥ |

⋃Y|. Let f be a supposed

normalization in the extension, i.e. f :⋃{Y : Y ∈ Y} → J , where J is the ground

model topology, and⋃{f(y) : y ∈ Y ∈ Z} ∩

⋃{f(y) : y ∈ Y ∈ Y − Z} = 0. We

can assume this, because J forms a basis for the topology in the extension. Usinga 2-dowment L2 and considering maximal antichains Ay of conditions deciding thevalue of f at y ∈ Y ∈ Y , we can define g :

⋃{Y : Y ∈ Y} → J in the ground model

as follows for each y ∈⋃{Y : Y ∈ Y}, Ly ∈ L2, Ly ⊆ Ay.

Let g(y) =⋂{U : for some p ∈ Ly, p � f(y) = U}. We claim

⋃{g(y) : y ∈

Y ∈ Y}∩⋃{g(y) : y ∈ Y ∈ Y −Z} = 0. We may assume without loss of generality

(by modifying the name for f , if necessary) that � � f is a normalization. Taker ≤ q0, q1, where q0 ∈ Ly0

and q1 ∈ Ly1, where y0 ∈ Y0 ∈ Z and y1 ∈ Y1 ∈ Z − Y .

Then r � g(y0) ⊆ f(y0) and g(y1) ⊆ f(y1), so r � g(y0) ∩ g(y1) = 0 so indeedg(y0) ∩ g(y1) = 0.

Now, consider the case when fewer than |⋃Y| reals are added. Add |

⋃Y| more

reals and get that Y is not normalized. But if there were disjoint open sets about⋃Z and

⋃{Y − Z} before adding the extra reals, these open sets would still be

disjoint after the extra addition. �A similar argument proves:

Theorem 3.6. If Y is not separated, it remains not separated in any Cohemreal extension.

Analogous results hold for random reals.One of the early preservation results was for property K forcing. A partial order

P satisfies property K if every uncountable subset of P includes an uncountablepairwise compatible uncountable subset. It is not difficult to show that both Cohenand random forcing satisfy property K, and it is by a now classic result that MAω1

implies all countable chain condition partial orders satisfy property K. Property Kforcing preserves Souslin trees, since an uncountable chain or antichain in the exten-sion yields one in the ground model [KT79]. Since this is an exercise in [Kun80],

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we shall leave it to the reader. As you will see, if some ground model object has anuncountable subset in the extension satisfying some “binary property”, it will alsohave such a subset in the ground model.

Here is the key idea in property K preservation, followed by an application.

Theorem 3.7. Suppose X ∈ V and we force with a property K partial orderP . Suppose Y ∈ [X]ℵ1 in the extension. Then:

(∃S ∈ [X]ω1)(∀T ∈ [S]ω1)(∃p ∈ P )(p � |T ∩ Y | = ℵ1)

Proof. Let � � “f : ω1 → X, f one-one, f“ω1 = Y ”. For each α ∈ ω1, takea maximal incompatible family of conditions {pαn}n<ω and xαn ∈ X, n < ω, such

that pαn � f(α) = xαn. Let S = {xαn : α < ω1, n < ω}. Then S is uncountable,for take an uncountable pairwise compatible subset of {pαn : α < ω1, n < ω} andobserve that the corresponding xαn’s are distinct. Now suppose, on the contrary,that for some T ∈ [S]ω1 , � � f“ω1 ∩ T is countable. Extend each pαn suchthat xαn ∈ T to a qαn for which there is a βαn such that for each γ ≥ βαn,qαn � f(γ) ∈ T . Since T is uncountable, there is an uncountable A ⊆ ω1 × ω suchthat {qαn : 〈α, n〉 ∈ A} is pairwise compatible. But now take some 〈α0, n0〉 ∈ A,and then take 〈α1, n1〉 ∈ A such that α1 ≥ βα0n0

. Then xα1n1∈ T and qα1n1

�f(α1) = xα1n1

, but qα0n0� f(α1) ∈ T , contradiction. �

Corollary 3.8 ([BT02]). Suppose X is a space in V and we force with aproperty K partial order. Then if X has a Lindelof subspace of size ℵ1 in theextension, it has one in V as well.

Proof. Suppose Y ∈ [X]ℵ1 is Lindelof in the extension. Form S as above.Claim S is Lindelof. We know that every uncountable T ⊆ S is forced to haveuncountable intersection with f“ω1 in some extension. Then, in that extension, Tis forced to have a complete accumulation point in f“ω1. Thus there are p, x, α inV such that p � f(α) = x and x is a complete accumulation point of T . But then xreally is a complete accumulation point of T , and since p is compatible with somepαn, x = xαn is in S. Thus S is a subspace of size ℵ1 in which every uncountableset has a complete accumulation point, so S is Lindelof. �

Another strengthening of the countable chain condition which is of interestfrom the point of view of preservation is that a partial order P be σ-centered. i.e.P =

⋃{Pn : n < ω}, where for each n and each finite F ⊆ Pn, F has a lower

bound in P (not necessarily in Pn). Again it is a classic result that MAω1implies

every subset of size ℵ1 of a countable chain condition partial order is σ-centered.This weakening of “σ-centered” is preserved by arbitrary finite-support products,but σ-centered itself is not. These can be seen by looking at the analogous resultsfor separable topological spaces.

It is quite clear how to try to show that a covering property in a σ-centeredextension pulls down to the ground model: it gives rise to countably many col-lections of open sets in the ground model, each having finitary properties like theoriginal object in the extension. Suppose for example that in the extension, everyopen cover of a ground model space X has a σ-disjoint open refinement. We claimthe same is true in the ground model. Let U be an open cover of X in V . Withoutloss of generality, we may assume its σ-disjoint refinement {Un : n < ω} in the ex-tension is composed of ground model open sets. Let P =

⋃n<ω Pn, where each Pn

is centered. Let Wnm = {W : W is open in V and for some p ∈ Pm, p � W ∈ Un}.

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Claim 3.9. W =⋃{Wnm : n,m < ω} is a σ-disjoint open refinement of U .

Proof. First of all, W is a refinement, since p � W ∈ Un implies p � (∃U ∈U)(W ⊆ U), so

(∀q ≤ p)(∃r ≤ q)(∃U ∈ U)r � W ⊆ U

But then W ⊆ U ∈ U .Second, W is an open cover, since � �

⋃{Un : n < ω} is a cover, so � � (∀x ∈

X)(∃n ∈ ω)(∃U ∈ Un)(x ∈ U). Then (∀x ∈ X)(∀p)(∃q ≤ p)(∃n)(∃U)(q � U ∈ Un

and x ∈ U). Then x ∈ U , and for some m, q ∈ Pm, so x ∈ U ∈ Wnm.

Finally, each Wnm is disjoint, for if p1 � W1 ∈ Un and p2 � W2 ∈ Un, wherep1, p2 ∈ Pm, take q ≤ p1, p2. Then q � W1 ∩ W2 = 0, so indeed, W1 ∩W2 = 0. �

4. Supercompact reflection

Quite often one would like to prove the consistency of some universal statement,e.g. “every hexacompact space with a purple diagonal (an HP space) is foreclos-able”. The first step would be to consider a particular hexacompact space with apurple diagonal and force it to be foreclosable. Another possibility would be to startwith a space which is not foreclosable, and force it to either be non-hexacompact orto not have a purple diagonal. Assume you could, for example, do the foreclosableforcing. If your forcing were sufficiently nice, you would be able to iterate it to forceother hexacompact spaces with purple diagonals to be foreclosable, but there aresome obvious difficulties in the way of establishing your desired consistency result.

(1) There could be too many HP spaces, so you could never deal with themall.

(2) The very act of making one HP space foreclosable could create other HPspaces, so that you are facing an unwinnable game of Whack-a-mole.

(3) Once you “kill” an HP space by making it foreclosable, perhaps it doesn’tstay “dead”, but rather is “revived”, i.e. made non-foreclosable again, bylater forcing.

Each of these problems needs to be dealt with; in some — but not all —circumstances, this can be accomplished. If the number of HP spaces is bounded,#1 is doable. For example, if every HP space has size ≤ ℵ1, and 2ℵ1 = ℵ2,once could deal with them all in an iteration of length ℵ2. One could also do abookkeeping argument as in the consistency proof for MAω1

to take care of #2.If the number of HP spaces is unbounded, there are two reasonable approaches.

One is to do class forcing: deal with all HP spaces of size ℵ1 in ℵ2 steps, all thoseof size ℵ2 in ℵ3 steps, etc. For an example of this sort of argument, see [Tal88]. Amore widely applicable technique is to use reflection: show that there is a cardinalκ such that if every HP space of size < κ is foreclosable, then every HP space is.One could then plausibly get that every HP space is foreclosable by an iteration oflength κ. One might be able to achieve such reflection by an elementary submodelargument, but, in general, one is likely to have to assume the existence of a largecardinal, in particular a supercompact one, and so we turn our attention to that.

Since the existence of large cardinals entails the existence of models of ZFC,by Godel’s Incompleteness Theorem, such existence cannot be proved consistent.Thus, proponents of large cardinal axioms must convince other mathematiciansthat they should adopt such axioms, or at least that their adoption is harmless.The arguments of the proponents resemble those of medieval philosophers for the

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existence of God; rather than explore that parallel, we shall follow the contempo-rary set-theoretic fashion and sleep soundly despite using supercompact cardinals.However, it seems interesting to note that the concept of reflection also appears inthe Kabbalah: as above, so below, i.e. what happens in Heaven is reflected here onEarth.

A typical use of supercompact cardinals in topology is to prove results of theform if every small subspace of X has property P , then X has property P . “Small”here means of size< κ, where κ is the supercompact cardinal. Such results are calledcompactness theorems. Taking the contrapositive, we obtain if X has property Q(= ∼P ), then some small subspace of X has the property Q. This is called areflection theorem. Such results are not so interesting, since κ is very large. Thegame, however, is to make κ relatively small, either by collapsing it to ℵ1 or ℵ2, or byblowing up the continuum to have size κ, and hope that the compactness/reflectionresults for that particular P survive the collapse or blow-up. Such results have apattern worth learning, which we shall now explain. The pattern involves generalmachinery concerning elementary embeddings.

Definition 4.1. An injection j from a model V into a model M of the samelanguage is an elementary embedding if for each n, for every formula φ with n freevariables, and any n elements x1, . . . , xn ∈ V ,

V � φ[x1, . . . , xn] if and only if M � φ[j(x1), . . . , j(xn)].

Definition 4.2. A cardinal κ is supercompact if there is for each cardinal λ ≥ κan elementary embedding jλ : V → M from the universe V into a transitive classM such that:

• for each ξ < κ, jλ(ξ) = ξ, but jλ(κ) > λ, and• λM ⊆ M .

Here, λM denotes the class of sequences of length λ, where the terms of thesequences are elements of M . It follows by transfinite induction that for all α ≤ λwe have Mα = Vα — that is, the cumulative hierarchy as computed in V and in Mcoincides at least up to λ.

The above definition of supercompactness is the most useful for applications,but prima facie is not expressible in ZFC because of the quantification over for-mulas. There is an equivalent definition which asserts the existence of certainultrafilters and thus avoids this difficulty. See [Kan94].

A typical proof using supercompactness exploits the interplay between j andj“, where:

j“ X = {j(x) : x ∈ X}.Clearly j(X) ⊇ j“(X), since if x ∈ X, by elementarity, j(x) ∈ j(X). Also,

j“κ = κ but j(κ) > κ if κ is supercompact and j is one of the jλ’s, λ ≥ κ. Thusj“X is a copy of X, while j(X) “looks just like X” (by elementarity) but can bemuch bigger, if |X| ≥ κ. We then have that j(X) has a smaller subset, namelyj“X, that looks like it, so by elementarity, X must have such a subset as well. If φis a first-order property of X, both j(X) and j“X will satisfy φ, so we will be ableto conclude that j(X) — and hence X — has a subset of smaller size satisfying φ.

The following are frequently used facts about elementary embeddings. We leavethe proofs to the reader as an exercise, but they can be found in [DTW90a].

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Lemma 4.3. Suppose j : V → M is an elementary embedding into a transitiveclass M . Then:

(i) For any A ∈ V, j“A ⊆ j(A),(ii) A ⊆ B ∈ V → j(A) ∩ j“B = j“A(iii) if j(α) = α for all α < κ and A ∈ V is such that |A| < κ, then j(A) = j“A.(iv) if j(α) = α for all α < κ, then for any a ∈ V of rank < κ, j(a) = a.

After we force with our partial order P to collapse κ, κ is no longer supercompactin the extension. We would like to make use of its former supercompactness. Todo this, we seek to extend the elementary embedding j : V → M to V [G], where Gis P-generic over V .

We cannot expect to extend j to an embedding j∗ : V [G] → M [G], since M [G]is a transitive subclass of V [G] and so such a j∗ would demonstrate that κ is stillsupercompact in V [G]. Instead we will (when certain conditions are met) extend jto j∗ : V [G] → M [G∗], where M [G∗] is a generic extension of M [G] (for instance, ifG adds κ Cohen reals, then G∗ adds j(κ) Cohen reals). The details and conditionsare developed below.

Definition 4.4. j∗ is a generic elementary embedding if there exists some Hwhich is generic over M [G] such that j∗ : V [G] → M [G][H].

The following technical lemmas give us sufficient conditions for extending anelementary embedding to a generic elementary embedding.

Lemma 4.5. Let j be an elementary embedding from V into a transitive class M .Suppose G is P-generic over V and G∗ is j(P)-generic over M . If p ∈ G → j(p) ∈G∗ for all p ∈ P, then j extends to an elementary embedding j : V [G] → M [G∗].

Proof. If x = val(τ,G), define j(x) = val(j(τ ), G∗). �

Lemma 4.6. Let j : V → M be an elementary embedding with critical point(the first ordinal moved) κ. Let G be P-generic over V . If j(P) is equivalent to(i.e., yields the same extension as) j“ P ∗ Q for some Q (call it j(P)/j“ P) andthere is a master condition m ∈ Q such that if H is a Q-generic filter over M [G]containing m, then p ∈ G → j(p) ∈ G ∗H, then G ∗H is j(P)-generic over M .

In particular, this is true if P is κ-c.c.

For the κ-c.c. case, the element � ∈ Q is a master condition.Often, the master condition will look like

⋃G, where G is P-generic over V ,

although sometimes extra work is necessary in order for this to make sense. Forexample, in the case that P is the usual poset to collapse κ to ω2,

⋃G is not a

condition in the second poset. Silver solved that problem by modifying the Levycollapse to the Silver collapse. For this and much more that was supposed to be inthe second volume of Kanamori [Kan94], see [KM78].

Let’s verify this in the simple case when P is Fn(κ, 2), the partial order foradjoining κ Cohen reals. Suppose H is any Q-generic filter over M [G] where G isP-generic over V . If p ∈ G, then j(p) = p, so j“ P = P and j(P) = Fn(j(κ), 2) =Fn(κ, 2)∗Fn(j(κ)\κ, 2). If H is any Fn(j(κ)\κ, 2)-generic over M [G] set (all suchsets contain �, the empty condition), then p ∈ G implies j(p)(= p) ∗ � ∈ G ∗H.

Having developed the general machinery, let’s start reaping the rewards.Let us first examine this interplay in the simpler context in which we don’t do

any forcing. Then, given a topological space 〈X,J 〉, j“J is a topology on j“X, and

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〈j“X, j“J 〉 is homeomorphic to 〈X,J 〉. 〈j(X), j(J )〉 is also a topological space; if〈j“X, j“J 〉 were a subspace of 〈j(X), j(J)〉, we could argue that since 〈j“X, j“J 〉is homeomorphic to 〈X,J 〉, if 〈X,J 〉 has property P , so does 〈j“X, j“J 〉. On theother hand, by elementarity, if 〈X,J 〉 has property P , so does 〈j(X), j(J )〉. Thus,taking j such that j(κ) > |X ∪ J |, we could get that 〈j(X), j(J )〉 would have asubspace of size < j(κ) with property P , whence by elementarity 〈X,J 〉 wouldhave a subspace of size < κ with property P .

The question, then, is when is j“X a subspace of j(X)? (We shall omit men-tioning the topology when it is clear from context.) There is a useful sufficientcondition:

Theorem 4.7. If each point of X has character less than κ, the critical pointof the elementary embedding j, then j“X is a subspace of j(X).

Proof. Let {N(p, α)}α<μ, μ < κ, be a neighbourhood base at p ∈ X. Thenj(N(p, α))α<j(μ) is a neighbourhood base at j(p) in j(X). {j“(N(p, α))}α<μ is aneighbourhood base at j(p) in j“X. j(μ) = μ and j(N(p, α)) ∩ j“X = j“(N(p, α)).

�

We shall illustrate the general procedure with a result from a recent paper[ST]. A classic problem of Hajnal and Juhasz [HJ76] asks whether every Lindelofspace has a Lindelof subspace of size ℵ1. There are some positive results for specialcases or with additional set-theoretic hypotheses in [BT02], but whether there isa consistent positive answer in general is unknown. A consistent counterexamplewas constructed in [KT02]. We shall strengthen “Lindelof” to “Rothberger” andshow that the general machinery establishes:

Theorem 4.8. If it’s consistent that there is a supercompact cardinal, it’s con-sistent that every Rothberger space of character ≤ ℵ1 has a Rothberger subspace ofsize ℵ1.

Definition 4.9. A space is Rothberger if whenever {Un : n < ω} is a se-quence of open covers, there are Un ∈ Un, n < ω such that {Un : n < ω} is acover.

The Rothberger property was introduced in the context of the real line by FritzRothberger; it is studied in detail in [ST], in which further references can be found.

Clearly, Rothberger spaces are Lindelof, but more is true:

Lemma 4.10. After countably closed forcing, a Rothberger space X remainsRothberger.

Given {Un}n<ω in the extension, we may assume each Un is composed of groundmodel open sets. If X is still Lindelof, we will be able to find countable subcoversfor each Un, and then go back and apply Rothberger in the ground model. Thus weneed only to show that the Lindelof property of Rothberger spaces is preserved bycountably closed forcing. A good way to see this is via certain topological games.We follow the treatment in [ST]. Consider an α-inning game, α an ordinal. PlayerI picks an open cover, Player II picks a member of that cover, etc., for each β < α.Player II wins if her choices form an open cover; Player I wins otherwise.

Lemma 4.11 ([Paw94]). A space is Rothberger if and only if Player I has nowinning strategy in the ω-length version of the covering game.

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16 FRANKLIN D. TALL

More surprising is:

Theorem 4.12 ([ST]). A space X is Lindelof in all countably closed forcingextensions if and only if Player I has no winning strategy in the ω1-length coveringgame for X.

If Player I has no winning strategy in the ω1-length game, she has no winningstrategy in the ω-length game, so by the above remarks, we see that:

Corollary 4.13. The Rothberger property is preserved by countably closedforcing.

Theorem 4.12 is proved from a result in [Tal95], which in turn is based on[She96]. We need the following definition:

Definition 4.14. A collection {Uf : f ∈⋃

α<ω1

αω} of open subsets of X isa covering tree if for every α ∈ ω1, and every f ∈ αω, {U

f {〈α,n〉} : n < ω} is a

cover of X.

Lemma 4.15 ([Tal95]). Let X be a Lindelof space. The following are equivalent:

(1) X is Lindelof in all countably closed forcing extensions,(2) X is Lindelof after adding one Cohen subset of ω1,(3) for each covering tree {Uf : f ∈

⋃α<ω1

αω} of X, there is an f ∈⋃

α<ω1

αωsuch that {Uf |β : β ∈ domf} covers X.

Proof. Trivially, a) implies b). Let’s show not a) implies not c). Supposethat in some countably closed forcing extension, there is an open cover U whichhas no countable subcover. We may assume without loss of generality that it iscomposed of ground model open sets. Form a tree indexed by

⋃β<ω1

βω, witheach node assigned a member pf of the offending poset as follows. At each node fpick countably many conditions extending pf and deciding pf and deciding someelement of U . By Lindelofness, we can pick those conditions such that the decidedelements cover. At limit levels, use countable closure to pick a condition below theconditions on a branch. The resulting covering tree witnesses the failure of c).

To see that not c) implies not b), note that a Cohen subset of ω1 determines abranch through the tree. By genericity, the open sets decided by that branch forma cover which has no countable subcover, else some pf would force that U had acountable subcover. We have thus shown a) ⇒ b) ⇒ c) ⇒ a). �

From Lemma 4.15, we can easily prove the important direction of Theorem4.12, for let {Uf : f ∈

⋃α<ω1

αω} be a covering tree of X. This implicitly definesa strategy for Player I in the ω1-length covering game; a winning play against thisstrategy yields a cover, but then Lindelofness cuts it down to some α-length branch.We omit the other direction. �

For λ < κ infinite regular cardinals let Lv(κ, λ) be the partially ordered setwhose elements are functions p such that |p| < λ, dom(p) ⊆ κ × λ and for all(α, ξ) ∈ dom(p), p(α, ξ) ∈ α, elements p and q are ordered by p ≤ q if and onlyif q ⊆ p. It is well-known that when κ is strongly inaccessible, each antichain ofLv(κ, λ) is of cardinality less than κ. Also, as λ is regular, Lv(κ, λ) is λ-closed. Itis also well-known that in the generic extension obtained by forcing with Lv(κ, λ),κ remains a cardinal, but is the successor of λ. The phrase “Levy-collapse . . . to ω2

with countable conditions” means “Force with Lv(κ, ω1) where κ is supercompact”.

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Theorem 4.16. Levy-collapse a supercompact cardinal to ω2 with countableconditions. Then every Rothberger space of character ≤ ℵ1 includes a Rothbergersubspace of size ≤ ℵ1.

Proof. Let κ be a supercompact cardinal in the ground model. Let G beLv(κ, ω1)-generic over V . In V [G], let 〈X,J 〉 be a Rothberger space of character≤ ℵ1 and cardinality larger than ℵ1. Let α be a cardinal such that X, all opencovers of X, and all sequences of open covers of X, as well as Lv(κ, ω1) and anyof its antichains are members of Vα[G]. Choose a regular cardinal λ > 2α. In V [G]let μ = |X|. Then μ is also a cardinal in V and in V , κ ≤ μ < α < λ.

By supercompactness of κ, fix an elementary embedding j : V → M withj(κ) > λ and λM ⊆ M . Since for each β ≤ λ we have Mβ = Vβ, it follows thatLv(κ, ω1) is Lv(κ, ω1)-generic over M if and only if it is over V . Moreover, for allβ ≤ λ, Vβ[G] = Mβ[G]. It follows that in fact X ∈ M [G]. Moreover, any familyof open covers for X in V [G] is also in M [G]. Thus, M [G] � “X is a Rothbergerspace of cardinality > ℵ1 and character ≤ ℵ1”.

Since Lv(κ, ω1) has no antichain of cardinality κ, there is a j(Lv(κ, ω1))-generic(over M) filter G∗ such that p ∈ G implies j(p) ∈ G∗. But then j extends to anelementary embedding we shall also call j : V [G] → M [G∗].

Until further notice we now work in M [G∗]. The equation j(Lv(κ, ω1)) =Lv(j(κ), ω1) = Lv(κ, ω1) × Lv(j(κ) \ κ, ω1) implies that M [G∗] is of the formM [G][H] where H is Lv(j(κ) \ κ, ω1)-generic over M [G]. By Corollary 4.13, Roth-berger is preserved by countably closed forcing, giving M [G∗] � “X is a Rothbergerspace”.

The bijection j|X from X to j“X induces a homeomorphic topology, say j“T ,on j“X: Then (j|X“X, T ) is a Rothberger space.

The subset j“X of j(X) inherits a topology from j(X), say S. Compare thetwo spaces (j“X, j“T ) and (j“X,S). First note that S contains sets of the formj(U) ∩ j“X, where U ⊆ X is open in X. Since j(U) ∩ j“X = j“U is an elementof the topology j“T on j“X, we have j“T ⊆ S. The character restriction ensuresthat S = j“T . Thus, j“X is a Rothberger subspace of j(X).

Since j is an elementary embedding, j(X) is a Rothberger space with character≤ ℵ1 and cardinality j(μ). Since j[X] is an uncountable subset of j(X), we concludethat in M [G∗] the statement

“j“X ⊆ j(X) is an uncountable Rothberger subspace of j(X)”

as well as the statement “j(κ) = ℵ2 and |μ| = ℵ1” are true. This implies:

• M [G∗] � “j(X) has a Rothberger subspace of cardinalty ℵ1”.

This concludes working in M [G∗]. Since j(ℵ1) = ℵ1 and j is an elementary embed-ding of V [G] into M [G∗], in V [G] it is true that X has a Rothberger subspace ofcardinality ℵ1. �

By different methods, Marion Scheepers [Sch] has recently been able to obtainthis result by Levy-collapsing only a measurable to ℵ2.

In the Cohen or random situation, our preservation results work no matterhow many reals we are adjoining. The situation is considerably more subtle inthe σ-centered case, because the iteration of σ-centered partial orders need not beσ-centered. A typical example of when one would want preservation by arbitraryiterations of σ-centered partial orders is when one wants to have reflection phenom-ena co-existing withMA (σ-centered), or equivalently, p = c [Bel81], [Wei84]. The

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18 FRANKLIN D. TALL

point is that j(P)/P — where P is the natural partial order that forces p = κ, whereκ is supercompact, will not be σ-centered, although it is an iteration of σ-centeredpartial orders, since it is an iteration of more that 2ℵ0 such orders. Nonetheless, inmany cases in which one has preservation by σ-centered forcing, one can actuallyprove preservation by finite support iterations of σ-centered forcing.

Definition 4.17. An α-stage finite support iteration P with typical successorstage Pβ+1 = Pβ ∗ Qβ is an absolute σ-centered iteration if in any forcing

extension, for any β < α, �β Qβ is σ-centered.

Lemma 4.18 ([Tal94a]). Suppose Φ(X) is preserved by σ-centered forcing. Letκ be a cardinal. Suppose there is a poset R such that R � κ ≤ 2ℵ0 and R preservesboth Φ(X) and ¬Φ(X). Then Φ(X) is preserved by finite support κ-stage absoluteσ-centered iterations.

Proof. Let P be a finite support κ-stage absolute σ-centered iteration, andsuppose for a contradiction that P does not preserve Φ(X). Then neither doesR × P; hence, the same goes for P × R. But R � κ ≤ 2ℵ0 , and P is still a finitesupport κ-stage σ-centered iteration after forcing with R. Therefore after forcingwith R, P is σ-centered. This is a contradiction, since Φ(X) is preserved by R andby σ-centered forcing. �

Thus, for example, if we can show that some forcing that adds a lot of realspreserves both Φ and ¬Φ(X), and σ-centered forcing preserves Φ(X), then arbi-trary finite support iterations of σ-centered forcing preserve Φ(X). There is anapplication of this in [Tal94a], involving submetrizability and random reals. Therandom real preservation argument is novel and ought to have additional applica-tions. Given a function d(x, y) bounded by 1 in a random real extension — e.g. ametric — we define ρ(x, y) in V which behaves like d:

ρ(x, y) :=∑

n∈N

μ(‖d(x, y) > 1n‖)

2n.

Preservation arguments akin to σ-centered ones arise in some applications ofhuge cardinal collapse. One can collapse κ to ℵ1 and j(κ) to ℵ2 such that j(P)/j“Pis ℵ1-centered ∗ ℵ2-closed. This enables the transfer of some properties — e.g.weak collectionwise Hausdorffness, from ℵ1 to ℵ2 [Tal94b].

5. Normality versus collectionwise normality

Once one has “reflection + preservation” in one’s mind, as well as “small char-acter implies j“X is a subspace of j(X)”, it is fairly easy to come up with a proofof the following result, the conclusion of which implies the Normal Moore SpaceConjecture:

Theorem 5.1 ([DTW90a]). Adjoin either supercompact many Cohen reals orsupercompact many random reals. Then every normal space in which each pointhas character < 2ℵ0 is collectionwise normal.

Proof. First check that these partial orders are candidates for the methodby checking that Pκ is completely embedded in j(Pκ) = Pj(κ). Suppose X werea counterexample. Then there would be a discrete unseparated subcollection Yin X. By Theorem 3.6, “unseparated” is preserved by Cohen or random forcing.

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Thus Z = {j“Y : Y ∈ Y} is unseparated in j“X and hence in j(X). It is alsoeasy to check it is (still) discrete because j“Y ⊆ j(Y ) and j(Y) is discrete. If wehave picked a j that sends κ past |

⋃Z|, we have Z is “a small discrete collection

of small sets”. Thus it suffices to show that if we adjoin either μ Cohen reals orμ random reals, where μ is regular, then in a normal space in which points havecharacter < 2ℵ0 = μ, discrete collections Y such that |

⋃Y| < μ are separated. This

is established by a slight variation of the proof of Theorem 3.5 — we need to showthat not only does normalized in the extension imply normalized in the groundmodel, but that it implies separated as well.

We will only do the Cohen argument. We may assume without loss of generalitythat Y ∈ V . We slightly vary the proof of Theorem 3.5 to show that if Y isnormalized in a Cohen extension, then it is separated in V . We may assume we haveadded more than |

⋃Y| Cohen reals. By some easy topology, if Y = {Yγ}γ<λ, λ < κ,

it suffices to separate {Yγ : γ ∈ κ \ domp}, where p � Y is normalized. Proceed asbefore, but also insist r is below {〈γ, 0〉, 〈γ′, 1〉}, in order to ensure that the opensets about Yγ and Yγ′ , γ = γ′, are disjoint. �

Many of the interesting questions involving reflection fit into the general frame-work we have outlined, but others do not, because the character requirement is toorestrictive. A case in point is the consistency from a supercompact cardinal ofevery locally compact normal space being collectionwise normal. The first proofby Balogh [Bal91] was unnecessarily difficult, and it was not clear what was go-ing on, or whether there was any generally applicable method. The proof of Dow[Dow92] was an improvement but still was not transparent, at least not to thosenot embedded in C∗(X). The proof in [GJT98] has the advantage that concep-tually it is a minor modification of the small character case, namely replacing “Xis homeomorphic to a subspace of j(X)” by “X is a perfect image of a subspaceof j(X)”. By definition, topological properties are preserved by homeomorphisms;many topological properties are also perfect invariants, i.e. preserved both directlyand inversely by perfect maps.

The exposition in [GJT98] is good, so we will confine ourself to just saying afew words about it. The key observation is that much of the reflection technologystill works in this perfect image situation. In particular, one can use the perfectmap to lift a discrete collection in X (which is homeomorphic to j“X) to j(X),use normality to separate it there, since the size of its union is < j(X), and thenuse the perfect map to bring down the separation to j“X. To obtain the perfectmap, send

⋂{j“V : V is open , x ∈ V } to j(x). The use of local compactness is

to prove that this intersection equals⋂{j“K : K is a compact Gδ, x ∈ K}. These

observations about perfect maps were generalized in [JT98] to the topology ofelementary submodels context (see Section 6). “Generalized” because j“Hθ is anelementary submodel of j(Hθ) = Hj(θ).

The proof that locally compact normal spaces are collectionwise normal in thismodel extends naturally to replace locally compact by pointwise countable type. Aspace has pointwise countable type if every point is contained in a compact set ofcountable character. The largest class of naturally defined spaces for which onecould hope to consistently prove that normality implies collectionwise normalityare the k-spaces, i.e. spaces in which a set is closed if and only if its intersectionwith every compact set is compact. I conjecture that this can be done, and theway to do it is to induct on the k-order of the spaces. k-order was defined by

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20 FRANKLIN D. TALL

Arhangel’skiı in [Arh65] in analogy to sequential order. The first step in theprogram was accomplished by Renata Prado [Pra99] in her Ph.D thesis, using aweakening of “perfect map” to adapt the pointwise countable type proof to theclass of k′-spaces, i.e. spaces such that whenever x ∈ F , F ⊆ X, then there is acompact K ⊆ X such that x ∈ F ∩K.

The consistency of normality implying collectionwise normality for k′-spaceshad been shown earlier by Daniels [Dan91], using the Product Measure ExtensionAxiom, but there was no obvious way to continue along that path.

Having seen the use of preservation arguments for Cohen and random forcing inreflection proofs, we are encouraged to examine other classes of partial orders with aview toward preservation. Where we would really like to be able to apply preserva-tion is in the case of countably closed forcing, for then we could get reflection afterLevy-collapsing a supercompact to ℵ2. Unfortunately, not much is known aboutthis. We would like to show that Lindelofness is preserved for T2 spaces with pointsGδ; instead, as we saw above, the stronger Rothberger property is preserved. Wewould like to show that non-collectionwise Hausdorff is preserved; instead Shelah[She77] showed this was for locally separable, first countable spaces. Solving eitherof these questions affirmatively would lead to the solution of long-outstanding openproblems: the first, a consistency proof for “every Lindelof T2 space with points Gδ

has cardinality ≤ 2ℵ0 = ℵ1” [Tal95]; the second, a consistency proof for “every firstcountable ℵ1-collectionwise Hausdorff space is collectionwise Hausdorff” [Tal07].

6. Topology of elementary submodels

Once one introduces a new mathematical technique for the purpose of attack-ing certain problems, the technique itself often becomes an object of study. Onesuch technique is that of elementary submodels. The best published exposition ofthe basic facts for set theorists is Chapter 24 of [JW97], which we shall assumethe reader is acquainted with. A large number of topological applications are in[Dow88]. The first paper to systematically study the technique itself is [JT98].There, given a topological space 〈X,J 〉 in an elementary submodel M of a suffi-ciently large Hθ, θ regular, we define XM to be X∩M with topology JM generatedby {U ∩M : U ∈ J ∩M}. For the student not so familiar with elementary submod-els, let us note that the incantation about Hθ is chanted in order to avoid dealingwith M ’s which are elementary submodels of V , because there are Godelian diffi-culties in doing do. For all practical purposes, we think of M as indeed being anelementary submodel of V . For elucidation of this point, see the chapter of [JW97]referred to above.

There are two natural questions associated with the study of XM :

(1) Given X and M , what does XM look like?(2) Given XM , what does X look like? In particular, if XM is homeomorphic

to some nice space, must XM = X?

XM is not the only way of forming a new space, given X and M . See e.g.Eisworth [Eis06] and Section 3.5 of [Pra99].

Question 1 is pretty well dealt with in [JT98], so we will not get into it here.However, one point I want to call attention to is that — in contrast to the abundanceof examples of different topological spaces with various combinations of properties— at present there seem to be only a few different kinds of elementary submodels.Perhaps this is a good subject for further study.

202202


Even powers of the two-point discrete space D reveal interesting questionsabout Question 2: if XM is homeomorphic to Dκ, does X = XM? In a seriesof papers, [Tal00b], [JT03], [Kun03], [Tal04], [JLT06], [Tal06], it transpiredthat the answer is positive for small cardinals, but can be negative for large ones.Moreover, the essential property of D turned out to be that it satisfies the countablechain condition. Some sample results are:

Proposition 6.1 ([Kun03]). Let κ be the first supercompact. Let X be acompact space of size ≥ κ. Then for some elementary submodel M containing X,XM is compact and not equal to X.

Proposition 6.2 ([Tal06]). If XM is compact and satisfies the countable chaincondition, and if X is not scattered and |X| < the first inaccessible, then X = XM .If the character of XM is in M then “inaccessible” can be replaced by “1-extendible”.

There remains one intriguing problem I wish to bring to the reader’s attention:

Problem 6.3. If XM is homeomorphic to the space of irrationals, does XM =X?

This is still open; partial results were achieved in [Tal02].This question seems natural; for such familiar spaces as R (the real line), Q

(the rationals), P (the irrationals), K (the Cantor set), we ask whether if XM ishomeomorphic to any of these, must X = XM?

For Q, there is a simple negative answer. Let X be any T3 space withoutisolated points. Suppose M is a countable elementary submodel of Hθ, for θ asufficiently large regular cardinal. Then XM is homeomorphic to Q. The reason isthat XM is a countable T3 space with a countable base and with no isolated points.All such spaces are homeomorphic to Q [Eng89, Exercise 6.2.A(d)]. �

Theorem 6.4 ([Tal00b], [Tal02]). If XM is a locally compact uncountableseparable metrizable space, then X = XM .

Proof. We first prove it suffices to show that [0, 1] ⊆ M — any definable spaceof power 2ℵ0 would do. It follows that ω1 ⊆ M . Now first observe that if XM is T2,so is X. Then recall that {xα}α<ω1

is left- (right-)separated if there exist open sets{Uα : α < ω1}, with α ∈ Uα such that Uα ∩ {xβ : β < α} ({xβ : β > α}) = 0. Ifω1 ⊆ M and X has an uncountable left- or right-separated subspace, so does XM

— simply relativize the assertion that there is such a subspace to M . Separablemetrizable spaces don’t have uncountable left- or right-separated subspaces sincethey are hereditarily separable and hereditarily Lindelof, so X doesn’t. XM isalso T3. It is a standard cardinal function exercise [Juh71], [Roi84] to showthat the non-existence of an uncountable left-separated subspace implies a space ishereditarily separable and therefore (by T3) has weight ≤ 2ℵ0 , and that the non-existence of an uncountable right-separated subspace implies a space is hereditarilyLindelof, and hence, if its weight is ≤ 2ℵ0 , has no more than 2ℵ0 open sets. But if[0, 1] ⊆ M and both X and its topology J have cardinality ≤ 2ℵ0 , then X and Jare included in M , so X = XM .

Now, to show [0, 1] ⊆ M , take a basic open subset V = U ∩M , U open in X,U ∈ M , of XM , with compact closure. Then U ∈ M . Observe that V = (U)M ; that(U)M = U ∩M ⊆ V is clear, if x ∈ V were not in U , there would be a W open inXM containing x such that W ∩U = 0. Take such a W ∈ M . Then W ∩M ∩U = 0,contradiction. �

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22 FRANKLIN D. TALL

We need a result of Lucia Junqueira [Jun00]:

Lemma 6.5. If YM is compact, so is Y , and, in fact, YM is a perfect image ofY .

Then U maps perfectly onto (U)M . U includes a copy of the Cantor set, so aclosed subspace of U — and hence, by the Tietze Extension Theorem — U itselfmaps onto [0, 1]M = [0, 1] ∩ M (by the first countability of [0, 1]). Since [0, 1] isseparable, [0, 1]∩M is dense in [0, 1]. But [0, 1]∩M is compact, so [0, 1]∩M = [0, 1],i.e. [0, 1] ⊆ M . �

For the Irrational Problem, the basic difficulty is that, unlike compactness,completeness — as far as I know — does not “lift” from XM to X. There are avariety of disparate sufficient conditions that ensure that XM = X. They all aimto either ensure that X includes a copy of the Cantor set, or else ensure that Mis sufficiently “fat”, so that if its cardinality is ≤ 2ℵ0 , it actually includes 2ℵ0 andhenceM . The question of whether elementary submodels of size ≥ 2ℵ0 (entailed e.g.by XM being uncountable, separable, and completely metrizable) must necessarilyinclude 2ℵ0 is an intriguing one. In [KT00], we proved this follows from the non-existence of 0#. Welch [Wel02] noted this also follows from the assumption that2ℵ0 is not a Jonsson cardinal. On the other hand, Chang’s Conjecture plus theContinuum Hypothesis implies there is an elementary submodel M of size 2ℵ0 with2ℵ0 ∩M countable [Tal00a].

Theorem 6.6. Suppose XM is an uncountable, separable, completely metrizablespace. Then X = XM if any of the following conditions hold:

a) X includes a copy of the Cantor set,b) X is of pointwise countable type,c) |R ∩M | is uncountable,d) |X| = 2ℵ0 .

Clause a) begs a question I do not know the answer to:

Problem 6.7. If XM includes a copy of a Cantor set, does X?

The proof of a) follows along the same lines as that of Theorem 6.4. We againwish to obtain a closed map from a closed — hence complete — subspace of XM

onto [0, 1]M = [0, 1]∩M . Closed metrizable images of completely metrizable spacesare completely metrizable [Eng89, 4.4.17] and hence absolute Gδ. If [0, 1] ∩M isa Gδ in [0, 1], it is all of [0, 1] by [Kec95, 0.11], and now we can finish as before.Unfortunately, just because there is a closed map from a closed subspace of Xonto [0, 1] does not mean there is a closed map from a closed subspace of XM onto[0, 1]∩M . The difficulty is that elementarity will tell us about basic closed subsetsof XM , but not necessarily about arbitrary closed sets. Nonetheless, one can pushthis idea for a proof through — see [Tal02]. As part of the proof, we use maps thatbehave nicely with respect to basic closed sets, but — unlike continuous maps —need not necessarily then behave nicely with respect to all closed sets. This topicdeserves further study.

Clause b) follows from:

Lemma 6.8 ([JT98]). If X is of pointwise countable type, then XM is a perfectimage of a subspace of X.

Clause c) follows from a theorem of Woodin [Tal02]:

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Lemma 6.9. If R ∩M includes a perfect set, then R ∩M = R.

We get a continuous map from a Gδ subspace of XM onto [0, 1] ∩M and thusget R∩M analytic. But an analytic uncountable set of reals includes a perfect set.

Clause d) follows directly from c), and is left for the reader. �

7. Lindelof problems

There are quite a few classical problems involving Lindelof spaces. Aspiring set-theoretic topologists could pick any one of these as a thesis problem. My philosophywith regards to thesis problems is that one should work on something significant.Even if one doesn’t solve it, one will likely amass sufficiently many related resultsso as to have a thesis. I consider this more worthwhile than just solving someinsignificant problem one’s supervisor already knows how to solve.

Here are some sample Lindelof problems I have been looking at:

Problem 7.1. Is there a Lindelof space which is not a D-space? A space is aD-space if for each funtion f assigning to each x ∈ X a neighbourhood N(x), thereis a closed discrete D such that

⋃{N(x) : x ∈ D} covers X.

For this problem, consult [Gru] in this volume.

Problem 7.2. Find a reasonable bound on the Lindelof number of a product

of two Lindelof spaces. Maybe 22ℵ0

will do, or perhaps 2ℵ0 consistently. It isconsistent that the product can have Lindelof number > 2ℵ0 — see [Juh84] and[Gor94]. The only absolute bound known is the first strongly compact cardinal,which is far too large. To obtain that bound, mindlessly generalize the TychonoffTheorem.

Problem 7.3. Call a space productively Lindelof if it is Lindelof and its prod-uct with every Lindelof space is Lindelof. Are productively Lindelof metrizablespaces always σ-compact? Indeed it is not hard to see that σ-compact spaces areproductively Lindelof. This problem is a variant of E. Michael’s classic problem asto whether there is a Lindelof space whose product with the space of irrationalsis not normal (such a space is called a Michael space). The latter problem hasnumerous connections with cardinal invariants of the continuum — see for example[Moo99].

Set theory is constantly rearing its playful head in the kind of topology I do;the latest example is:

Theorem 7.4 ([Tal09]). The Axiom of Projective Determinacy implies thatevery productively Lindelof projective set of reals is σ-compact if and only if thereis a Michael space.

There are connections between Problems 7.1 and 7.3. These are briefly surveyedin [Gru]. For details, see [Aur10], [AT], [Tal09], [Tal10a], [Tal10b]. The mostinteresting recent result on the Lindelof D problem is due to Leandro Aurichi[Aur10]. The classical Menger property is like the Rothberger property definedabove, except picking a finite subset rather than a singleton from each one of thecountable sequence of open covers.

Proposition 7.5 ([Aur10]). Every Menger space is D.

From this we obtain:

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24 FRANKLIN D. TALL

Proposition 7.6 ([Tal10b]). The Continuum Hypothesis implies that if theproduct of X with every Lindelof space is Lindelof, then X is a D-space.

We survey these and other Lindelof problems in [Tal11b]In conclusion, let me thank for their attention the graduate students and post-

docs who attended the course that led to this survey. I also thank the referee forhelpful comments.

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Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4,

Canada


209209


MENGER’S AND HUREWICZ’S PROBLEMS: SOLUTIONS FROM

“THE BOOK” AND REFINEMENTS

BOAZ TSABAN

Abstract. We provide simplified solutions of Menger’s and Hurewicz’s prob-lems and conjectures, concerning generalizations of σ-compactness. The readerwho is new to this field will find a self-contained treatment in Sections 1, 2,and 5.

Sections 3 and 4 contain new results, based on the mentioned simplifiedsolutions. The main new result is that there are concrete uncountable sets ofreals X (indeed, |X| = b), which have the following property:

Given point-cofinite covers U1,U2, . . . of X, there are for each nsets Un, Vn ∈ Un, such that each member of X is contained in allbut finitely many of the sets U1 ∪ V1, U2 ∪ V2, . . .

This property is strictly stronger than Hurewicz’s covering property. Millerand the present author showed that one cannot prove the same result if we areonly allowed to pick one set from each Un.

Dedicated to Professor Gideon Schechtman

Contents

1. Menger’s Conjecture2. Hurewicz’s Conjecture3. Strongly Hurewicz sets of reals, in ZFC4. A visit at the border of ZFC5. The Hurewicz ProblemAcknowledgmentsReferencesAppendix A. Sf (A ,B)

1. Menger’s Conjecture

In 1924, Menger [14] introduced the following basis property for a metric spaceX:

For each basis B for the topology of X, there are B1, B2, · · · ∈ Bsuch that limn→∞ diam(Bn) = 0, and X =

⋃n Bn.

Soon thereafter, Hurewicz [10] observed that Menger’s basis property can be refor-mulated as follows:

1991 Mathematics Subject Classification. Primary: 37F20; Secondary 26A03, 03E75 03E17 .Key words and phrases. Menger property, Hurewicz property, Rothberger property, Selection

principles, special sets of real numbers.

1



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211

121

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2 BOAZ TSABAN

For all given open covers U1,U2, . . . of X, there are finite F1 ⊆U1,F2 ⊆ U2, . . . such that

⋃n Fn is a cover of X.

We introduce some convenient notation, suggested by Scheepers in [20]. We saythat U is a cover of X if X =

⋃U ,1 but X /∈ U . Let X be a topological space, and

A ,B be families of covers of X. We consider the following statements.

S1(A ,B): For all U1,U2, · · · ∈ A , there are U1 ∈ U1, U2 ∈ U2, . . . such that{Un : n ∈ N} ∈ B.

Sfin(A ,B): For all U1,U2, · · · ∈ A , there are finite F1 ⊆ U1,F2 ⊆ U2, . . . suchthat

⋃n Fn ∈ B.

Ufin(A ,B): For all U1,U2, · · · ∈ A , none containing a finite subcover, thereare finite F1 ⊆ U1,F2 ⊆ U2, . . . such that {

⋃Fn : n ∈ N} ∈ B.

Let O(X) be the family of all open covers of X. We say that X satisfies S1(O,O)if the statement S1(O(X),O(X)) holds. This way, S1(O,O) is a property of topo-logical spaces. A similar convention applies to all properties of this type.

Hurewicz’s observation tells that for metric spaces, Menger’s basis property isequivalent to Sfin(O,O). This is a natural generalization of compactness. Notethat indeed, every σ-compact space (a countable union of compact spaces) satisfiesSfin(O,O). Menger made the following conjecture.

Conjecture 1.1 (Menger [14]). A metric space X satisfies Sfin(O,O) if, and onlyif, X is σ-compact.

Hurewicz proved that when restricted to analytic spaces, Menger’s Conjectureis true.

Recall that a set M ⊆ R is meager (or of Baire first category) if M is a union ofcountably many nowhere dense sets. A set L ⊆ R is a Luzin set if L is uncountable,and for each meager set M , L ∩M is countable.

Luzin sets can be constructed assuming the Continuum Hypothesis: Every mea-ger set is contained in a Borel (indeed, Fσ) meager set. Let Mα, α < ℵ1 be all Borelmeager sets. For each α < ℵ1, take xα ∈ R \

⋃β<α Mβ . Then L = {xα : α < ℵ1}

is a Luzin set.A subset of R is perfect if it is nonempty, closed, and has no isolated points. In

[11], Hurewicz quotes an argument of Sierpinski, proving the following.

Theorem 1.2 (Sierpinski). Every Luzin set satisfies Sfin(O,O), and is not σ-compact.

Proof. Let U1,U2, . . . be open covers of a Luzin set L ⊆ R. Let D = {dn : n ∈ N} bea dense subset of L. For each n, pick Un ∈ Un such that dn ∈ Un. Let U =

⋃n Un.

Then L\U is nowhere dense, and thus countable. Enumerate L\U = {xn : n ∈ N}.For each n, pick Vn ∈ Un such that xn ∈ Vn. Then L \ U ⊆

⋃n Vn, and thus

{Un, Vn : n ∈ N} is a cover of L, with at most two elements from each Un.2

Lemma 1.3 (Cantor-Bendixon). Every uncountable σ-compact set X ⊆ R containsa perfect set.

1We follow the set theoretic standard that, for a family of sets F ,⋃

F means the union of allelements of F .

2The interested reader may wish to show in a similar manner that actually, every Luzin setsatisfies S1(O,O). We will not use this fact.

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MENGER AND HUREWICZ PROBLEMS 3

Proof. By moving to a subset, we may assume that X is an uncountable compact,and thus closed, set. By the Cantor-Bendixon Theorem, X contains a perfectset. �

As perfect sets contain perfect nowhere dense subsets, a Luzin set cannot beσ-compact. �

Thus, Menger’s Conjecture is settled if one assumes the Continuum Hypothesis.In 1988, Fremlin and Miller [7] settled Menger’s Conjecture in ZFC. They used theconcept of a scale, which we now define. This concept is normally defined using NN,but for our purposes it is easier to work with P (N) (this will become clear later).

Let P (N) be the family of all subsets of N, and [N]<∞, [N]∞ ⊆ P (N) denotethe family of all finite subsets of N and the family of all infinite subsets of N,respectively. For a ∈ [N]∞ and n ∈ N, a(n) denotes the n-th element in theincreasing enumeration of a.

For a, b ∈ [N]∞, let a ≤∗ b mean: a(n) ≤ b(n) for all but finitely many n. Asubset Y of [N]∞ is dominating if for each a ∈ [N]∞ there is b ∈ Y such that a ≤∗ b.Let d denote the minimal cardinality of a dominating subset of [N]∞. A scale is adominating set S ⊆ [N]∞, which has a ≤∗-increasing enumeration S = {sα : α < d},that is, such that sα ≤∗ sβ for all α < β < d.

Scales require special hypotheses to be constructed. Indeed, say that a subsetY of [N]∞ is unbounded if it is unbounded with respect to ≤∗, that is, for eacha ∈ [N]∞ there is b ∈ Y such that b �≤∗ a. Let b denote the minimal cardinality ofan unbounded subset of [N]∞. b ≤ d, and strict inequality is consistent. (Indeed,b < d holds in the Cohen real model.)

Lemma 1.4 (folklore). There is a scale if, and only if, b = d.

Proof. (⇐) Let {dα : α < b} ⊆ [N]∞ be dominating. For each α < b, choose sα tobe a ≤∗-bound of {dβ , sβ : β < α}.

(⇒) Let S = {sα : α < d} be a scale, and assume that b < d. Let {bα : α <b} ⊆ [N]∞ be unbounded. For each α, take βα < d such that bα ≤∗ sβα

.Let c ∈ [N]∞ witness that {sβα

: α < b} is not dominating, and let γ < d besuch that c ≤∗ sγ . For each α < b, sγ �≤∗ sβα

, and thus bα ≤∗ sβα≤∗ sγ . Thus,

{bα : α < b} is bounded. A contradiction. �The canonical way to construct sets of reals from scales (more generally, from

subsets of P (N)) is as follows. P (N) is identified with Cantor’s space {0, 1}N, viacharacteristic functions. This defines the canonical topology on P (N). Cantor’sspace is homeomorphic to the canonical middle-third Cantor set C ⊆ [0, 1], and thehomeomorphism is (necessarily, uniformly) continuous in both directions. Thus,subsets of P (N) exhibiting properties preserved by taking (uniformly) continuousimages may be converted into subsets of [0, 1] with the same properties. We maythus work in P (N).

The critical cardinality of a (nontrivial) property P of set of reals, denotednon(P ), is the minimal cardinality of a set of reals X such that X does not havethe property P . The following is essentially due to Hurewicz [11].

Lemma 1.5 (folklore). non(Sfin(O,O)) = d.

Proof. (≥) Let X be a set of reals with |X| < d. Let U1,U2, . . . be open coversof X. Since X is Lindelof, we may assume that these covers are countable, andenumerate them Un = {Un

m : m ∈ N}.

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4 BOAZ TSABAN

Define for each x ∈ X a set ax ∈ [N]∞ by

ax(n) = min{m > ax(n− 1) : x ∈ Un1 ∪ Un

2 ∪ · · · ∪ Unm}.

As |{ax : x ∈ X}| < d, there is (in particular) c ∈ [N]∞ such that for each x ∈ X,ax(n) ≤ c(n) for some n. Take Fn = {Un

1 , . . . , Unc(n)} for all n. Then

⋃n Fn is a

cover of X.(≤) Let D be a dominating subset of [N]∞. Consider the open covers Un = {Un

m :m ∈ N}, n ∈ N, where

Unm = {a ∈ [N]∞ : a(n) = m}.

For all finite F1 ⊆ U1,F2 ⊆ U2, . . . , there is x ∈ D such that for all but finitelymany n, x(n) > max{m : Un

m ∈ Fn} (and thus x /∈⋃Fn).

But if X satisfies Sfin(O,O), then for all open covers U1,U2, . . . of X, there arefinite F1 ⊆ U1,F2 ⊆ U2, . . . , such that for each x ∈ X, x belongs to

⋃Fn for

infinitely many n: To see this, split the given sequence U1,U2, . . . into infinitelymany disjoint subsequences, and apply Sfin(O,O) to each of these subsequencesseparately.

Thus, dominating subsets of [N]∞ do not satisfy Sfin(O,O). �

Let κ be an infinite cardinal. A set of reals X is κ-concentrated on a set Q if,for each open set U containing Q, |X \ U | < κ.

Lemma 1.6 (folklore [23]). Assume that a set of reals X is c-concentrated on acountable set Q. Then X does not contain a perfect set.

Proof. Assume that X contains a perfect set P . Then P \ Q is Borel and un-countable. A classical result of Alexandroff tells that every uncountable Borel setcontains a perfect set. Let C ⊆ P \Q be a perfect set.3 Then U = R\C is open andcontains Q, and C = P \U ⊆ X \U has cardinality c. Thus, X is not c-concentratedon Q. �

Theorem 1.7 (Fremlin-Miller [7]). Menger’s Conjecture is false.

Proof. As perfect sets of reals have cardinality continuum, we have by Lemma 1.3that if b < d, then any set of reals of cardinality b is a counter-example.

Thus, assume that b = d (this is the interesting case), and let S = {sα : α <d} ⊆ [N]∞ be a scale (Lemma 1.4).

S ∪ [N]<∞ satisfies Sfin(O,O): This is similar to the argument about Luzin setssatisfying Sfin(O,O). Given open covers U1,U2, . . . of S∪ [N]<∞, take U1 ∈ U1, U2 ∈U2, . . . , such that [N]<∞ ⊆

⋃n Un. We can do that because [N]<∞ is countable.

Let U =⋃

n Un. P (N) \ U is closed and thus compact. For each n, the evaluationmap en : [N]∞ → N defined by en(a) = a(n) is continuous. Thus, en[P (N) \ U ] iscompact and thus finite, for all n. Therefore, there is a ≤∗-bound b for P (N) \ U .Take α < d such that b <∗ sα. Then

S \ U = S ∩ (P (N) \ U) ⊆ {sβ : β < d, sβ ≤∗ b} ⊆ {sβ : β < α}has cardinality < d, and thus satisfies Sfin(O,O). Let F1 ⊆ U1,F2 ⊆ U2, . . . besuch that S \ U ⊆

⋃n Fn. Then S ∪ [N]<∞ ⊆

⋃n Fn ∪ {Un}.

S ∪ [N]<∞ is not σ-compact: We have just seen that it is d-concentrated on thecountable set [N]<∞. Use Lemmata 1.3 and 1.6. �

3As Q is countable, one can alternatively prove directly that P \Q contains a perfect set.

214214


A reader not familiar with dichotomic proofs may be perplexed by the proof ofthe Fremlin-Miller Theorem 1.7. It gives a ZFC result by considering an undecidablestatement. Indeed, it shows that there is a certain set of reals, but does not tell uswhat this set is (unless we know in advance whether b < d or b = d). Another wayto view this is as follows.

Sets of reals X satisfying P because |X| < non(P ) are in a sense trivial examplesfor this property. From this point of view, the real question is, given a property P ,whether there are sets of reals of cardinality at least non(P ), which satisfy P . Theproof of Theorem 1.7 answers this in the positive only when b = d. However, witha small modification we get a complete answer.

Definition 1.8. A d-scale is a dominating set S = {sα : α < d} ⊆ [N]∞, such thatfor all α < β < d, sβ �≤∗ sα.

Lemma 1.9. There are d-scales.

Proof. Let {dα : α < d} ⊆ [N]∞ be dominating. For each α < d, choose sα to be awitness that {sβ : β < α} is not dominating, such that in addition, dα ≤∗ sα. �

An argument similar to that in the proof of Theorem 1.7 gives the following.

Lemma 1.10. Every d-scale is d-concentrated on [N]<∞. �

We therefore have the following.

Theorem 1.11 (Bartoszynski-Tsaban [3]). For each d-scale S, S ∪ [N]<∞ satisfiesSfin(O,O), and is not σ-compact. In other words, S ∪ [N]<∞ is a counter-exampleto Menger’s Conjecture. �

Theorem 1.11 is generalized in Tsaban-Zdomskyy [23].We conclude the section with some easy improvements of statements made above.Define the following subfamily of O(X): U ∈ Γ(X) if U is infinite, and each

element of X is contained in all but finitely many members of U . If U ∈ Γ(X), thenevery infinite subset of U belongs to Γ(X). Thus, we may assume for our purposesthat elements of Γ(X) are countable.

Corollary 1.12 (Just, et al. [12]). S1(Γ,O) implies Sfin(O,O).

Proof. Let X be a set of reals satisfying S1(Γ,O), and let U1,U2, · · · ∈ O(X). Theclaim is trivial if some Un contains a finite subcover. Thus, assume that this is notthe case.

As sets of reals are Lindelof, we may assume that each Un is countable, sayUn = {Un

m : m ∈ N}. Let

Vn =

⎧⎨

⎩

⋃

k≤m

Unk : m ∈ N

⎫⎬

⎭.

Then Vn ∈ Γ(X). Applying S1(Γ,O) there are mn, n ∈ N, such that {⋃

k≤mnUnk :

n ∈ N} is a cover of X. For each n, the finite sets Fn = {Unk : k ≤ mn} ⊆ Un are

as required in the definition of Sfin(O,O). �

A modification of the proof of Lemma 1.5 yields the following.

Lemma 1.13 (Just, et al. [12]). non(S1(Γ,O)) = d. �

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6 BOAZ TSABAN

Proof. By Corollary 1.12 and Lemma 1.5,

non(S1(Γ,O)) ≤ non(Sfin(O,O)) = d.

To prove the remaining inequality, assume that |X| < d, and U1,U2, · · · ∈ Γ(X). Wemay assume that for each n, Un is countable, and enumerate it Un = {Un

m : m ∈ N}.For each x ∈ X, let

ax(n) = min{k > ax(n− 1) : (∀m ≥ k) x ∈ Unm}

for all n. (In the case n = 1, omit the restriction k > ax(n − 1).) |{ax : x ∈X}| < d. Let d ∈ [N]∞ exemplify that {ax : x ∈ X} is not dominating, and takeFn = {Un

1 , . . . , Und(n)}. Then each x ∈ X belongs to

⋃Fn for infinitely many n. �

Corollary 1.14. Each set which is d-concentrated on a countable subset, satisfiesS1(Γ,O). �Corollary 1.15 (Bartoszynski-Tsaban [3]). For each d-scale S, S∪ [N]<∞ satisfiesS1(Γ,O). �

S1(Γ,O) is strictly stronger that Sfin(O,O). While every σ-compact set satisfiesthe latter, we have the following.

Lemma 1.16 (Just, et al. [12]). If X satisfies S1(Γ,O), then X has no perfectsubsets.

Proof. We give Sakai’s proof [18, Lemma 2.1]. Assume that X has a perfect subsetand satisfies S1(Γ,O). Then X has a subset C homeomorphic to Cantor’s space{0, 1}N. C is compact, and thus closed in X, and therefore satisfies S1(Γ,O) aswell.4 Thus, it suffices to show that {0, 1}N does not satisfy S1(Γ,O). We showinstead that its homeomorphic copy ({0, 1}N)N does not satisfy S1(Γ,O).

Let C1, C2, . . . be pairwise disjoint nonempty clopen subsets of {0, 1}N. LetU1, U2, . . . be the complements of C1, C2, . . . , respectively. For each n, let πn :({0, 1}N)N → {0, 1}N be the projection on the n-th coordinate. Then Un = {π−1

n [Um] :m ∈ N} ∈ Γ(X) for all n. But for all π−1

1 [Um1] ∈ U1, π

−12 [Um2

] ∈ U2, . . . , we havethat ΠnCn is disjoint of

⋃n π

−1n [Umn

]. �

2. Hurewicz’s Conjecture

Hurewicz suspected that Menger’s Conjecture was false. For this reason, he in-troduced in [10] a formally stronger property, which in our notation is Ufin(O,Γ). Itis easy to see that every σ-compact set satisfies, in fact, Ufin(O,Γ), and analogouslyto Menger, Hurewicz made the following.

Conjecture 2.1 (Hurewicz [10]). A metric space X satisfies Ufin(O,Γ) if, and onlyif, X is σ-compact.

The following easy fact is instructive.

Lemma 2.2. X satisfies Ufin(O,Γ) if, and only if, for all U1,U2, . . . , none havinga finite subcover of X, there is a decomposition X =

⋃k Xk, such that for each k,

there are finite subsets Fk1 ⊆ U1,Fk

2 ⊆ U2, . . . , such that for each x ∈ Xk, x ∈⋃Fk

n

for all but finitely many n.

4It is easy to see that all properties involving open covers, considered in this paper, are hered-itary for closed subsets [12].

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Proof. For each n, take Fn =⋃

k≤n Fkn . Then {

⋃Fn : n ∈ N} ∈ Γ(X). �

S ⊆ R is a Sierpinski set if S is uncountable, and for each Lebesgue measure zeroset N , S∩N is countable. Since every perfect set contains a perfect set of Lebesguemeasure zero, a Sierpinski set cannot contain a perfect subset, and therefore is notσ-compact (Lemma 1.3). A construction similar to that of a Luzin set describedabove, shows that the Continuum Hypothesis implies the existence of Sierpinskisets. We do not know when the following observation was made first.

Theorem 2.3 (folklore). Every Sierpinski set satisfies Ufin(O,Γ).

Proof. The following proof is a slightly simplified version of the one given in [12].Let S be a Sierpinski set. S =

⋃n S ∩ [−n, n], and thus by Lemma 2.2, we may

assume that the outer measure p of S is finite. Since S is Sierpinski, p > 0.5 LetB ⊇ S be a Borel set of measure p.

Let U1,U2, . . . be open covers of S. We may assume that each Un is countable,and enumerate Un = {Un

m : m ∈ N}. We may assume that all Unm are Borel subsets

of B. For each n,⋃

m Unm ⊇ S, and thus has measure p for each n. Thus, for

each N there is fN ∈ NN such that⋃fN (n)

k=1 Unk has measure ≥ (1− 1/2n+N )p, and

consequently, AN =⋂

n

⋃fN (n)k=1 Un

k has measure ≥ (1− 1/2N )p.Then A =

⋃N AN has measure p, and thus S \ A is countable. The countable

decomposition S = (S\A)∪⋃

N AN is as required in Lemma 2.2, by the countabilityof S \A and the definition of AN . �

A stronger statement can be proved in a similar manner.

Theorem 2.4 (Just, et al. [12]). Every Sierpinski set satisfies S1(Γ,Γ) (even whenwe consider Borel covers instead of open ones).

Proof. Replace, in the proof of Theorem 2.3, Unm by

⋂k≥m Un

k . Let f ∈ NN be such

that for each x ∈ S \ A, x ∈⋂

k≥f(n) Unk for all but finitely many n. Let g be a

≤∗-bound of {fN : N ∈ N} ∪ {f}. Then the choice U1g(1) ∈ U1, U

2g(2) ∈ U2, . . . is as

required. �

Thus, the Continuum Hypothesis implies the failure of Hurewicz’s Conjecture. Acomplete refutation, however, was only discovered in 1996, by Just, Miller, Scheep-ers, and Szeptycki, in their seminal paper [12].

Theorem 2.5 (Just, et al. [12]). Hurewicz’s Conjecture is false.

We will not provide the full solution from [12] here (since we provide a simplerone), but just discuss its main ingredients. The argument in [12] is dichotomic.Recall that b is the minimal cardinality of a set B ⊆ [N]∞ which is unbounded withrespect to ≤∗. A proof similar to that of Lemma 1.5 gives the following two results,which are also essentially due to Hurewicz [11].

Lemma 2.6 (folklore). An unbounded subset of [N]∞ cannot satisfy Ufin(O,Γ). �Lemma 2.7 (folklore). non(S1(Γ,Γ)) = non(Ufin(O,Γ)) = b. �

Thus, if b > ℵ1 then any set of cardinality ℵ1 is a counter-example to Hurewicz’sConjecture.

5Otherwise, S would have measure zero, and thus be countable.

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8 BOAZ TSABAN

Definition 2.8. A b-scale is an unbounded set {bα : α < b} ⊆ [N]∞, such that theenumeration is increasing with respect to ≤∗ (i.e., bα ≤∗ bβ whenever α < β < b).

Like d-scales, b-scales can be constructed without special hypotheses.

Lemma 2.9 (folklore). There are b-scales.

Proof. Let {xα : α < b} ⊆ [N]∞ be unbounded. For each α < b, choose bα to be a≤∗-bound of {bβ : β < α}, such that xα ≤∗ bα. �

The argument in [12] proceeds as follows. We have just seen that the case b > ℵ1

is trivial. Thus, assume that b = ℵ1. Then there is a b-scale B = {bα : α < b} ⊆[N]∞ such that in addition, for all α < β < b, bβ \ bα is finite.6 It is proved in [12]that for such B, B ∪ [N]<∞ satisfies Ufin(O,Γ). An argument similar to the onegiven in Theorem 1.7 for scales shows the following.

Lemma 2.10. Every b-scale B is b-concentrated on [N]<∞. In particular, B ∪[N]<∞ is not σ-compact. �

Unfortunately, the existence of b-scales as in the proof of [12] is undecidable.This is so because Scheepers proved that for this type of b-scales, B∪ [N]<∞ in factsatisfies S1(Γ,Γ) [21] (see also [16]), and we have the following.

Theorem 2.11 (Miller-Tsaban [16]). It is consistent that for each set of realssatisfying S1(Γ,Γ), |X| < b. Indeed, this is the case in Laver’s model.

Bartoszynski and Shelah have discovered an ingenious direct solution to Hurewicz’sConjecture, which can be reformulated as follows.

Theorem 2.12 (Bartoszynski-Shelah [2]). For each b-scale B, B ∪ [N]<∞ satisfiesUfin(O,Γ).

We provide a simplified proof of this theorem, using a method of Galvin andMiller from [8]. For natural numbers n,m, let [n,m) = {n, n+ 1, . . . ,m− 1}.Lemma 2.13 (folklore). Let Y ⊆ [N]∞. The following are equivalent:

(1) Y is bounded;(2) There is s ∈ [N]∞ such that for each a ∈ Y , a ∩ [s(n), s(n+1)) �= ∅ for all

but finitely many n.

Proof. (1 ⇒ 2) Let b ∈ [N]∞ be a ≤∗-bound for Y . Define inductively s ∈ [N]∞ by

s(1) = b(1)

s(n+ 1) = b(s(n)) + 1

For each a ∈ Y and all but finitely many n, s(n) ≤ a(s(n)) ≤ b(s(n)) < s(n + 1),that is, a(s(n)) ∈ [s(n), s(n+1)).

(2 ⇒ 1) Let s be as in (2). s has countably many cofinite subsets. Let b ∈ [N]∞

be a ≤∗-bound of all cofinite subsets of s. Let a ∈ Y and choose n0 such that foreach n ≥ n0, a∩[s(n), s(n+1)) �= ∅. Choosem0 such that a(m0) ∈ [s(n0), s(n0+1)).By induction on n, we have that (a(n) ≤)a(m0 + n) ≤ s(n0 + 1 + n) for all n. Forlarge enough n, we have that s(n0 + 1 + n) ≤ b(n), thus a ≤∗ b. �

6We will not use this fact here, but here is a proof: Fix an unbounded family {xα : α < b} ⊆[N]∞. At step α, we have a countable set Bα = {bβ : β < α} such that for all γ < β < b,bβ \ bγ is finite. In particular, each finite subset of Bα has an infinite intersection. EnumerateBα = {sn : n ∈ N}, and for each n pick mn ∈ s1 ∩ · · · ∩ sn such that mn > mn−1. Let c be a≤∗-bound of Bα, and let bα be a subset of {mn : n ∈ N}, such that max{c, xα} ≤∗ bα.

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Lemma 2.14 (Galvin-Miller [8]). Assume that [N]<∞ ⊆ X ⊆ P (N). For eachU ∈ Γ(X),7 there are a ∈ [N]∞ and distinct U1, U2, · · · ∈ U , such that for eachx ⊆ N, x ∈ Un whenever x ∩ [a(n), a(n+1)) = ∅.Proof. Let a(1) = 1. For each n ≥ 1: As U ∈ Γ(X), each finite subset of X iscontained in infinitely many elements of U . Take Un ∈ U\{U1, . . . , Un−1}, such thatP ([1, a(n))) ⊆ Un. As Un is open, for each s ⊆ [1, a(n)) there is ks such that for eachx ∈ P (N) with x ∩ [1, ks) = s, x ∈ Un. Let a(n+ 1) = max{ks : s ⊆ [1, a(n))}. �

Given the methods presented thus far, the following proof boils down to the factthat, if we throw fewer than n balls into n bins, at least one bin remains empty.

Proof of Theorem 2.12. Let B = {bα : α < b} be a b-scale. Let U1,U2, . . . ∈Γ(B ∪ [N]<∞).

For each n, take an and distinct Un1 , U

n2 , . . . for Un as in Lemma 2.14. We may

assume that an(1) = 1. Let α be such that I = {n : an(n+ 1) < bα(n)} is infinite.As |{xβ : β < α}| < b, {xβ : β < α} satisfies S1(Γ,Γ) (Lemma 2.7). Thus, thereare mn, n ∈ I, such that {Un

mn: n ∈ I} ∈ Γ({xβ : β < α}). Take Fn = ∅ for n /∈ I,

and Fn = {Un1 , . . . , U

nn } ∪ {Un

mn} for n ∈ I.

As {⋃Fn : n ∈ N} = {

⋃Fn : n ∈ I} ∪ {∅}, it suffices to show that for each

x ∈ X, x ∈⋃Fn for all but finitely many n ∈ I. If x ∈ [N]<∞, then for each

large enough n ∈ I, x ∩ [an(n), an(n+1)) = ∅ (because an(n) ≥ n), and thusx ∈ Un

n ∈ Fn. For β < α, bβ ∈ Unmn

⊆⋃Fn for all large enough n.

For β ≥ α (that’s the interesting case!) and all but finitely many n ∈ I,bβ(n) ≥ bα(n) > an(n + 1). Thus, |bβ ∩ [1, an(n + 1))| < n. As [1, an(n +1)) =

⋃ni=1 [an(i), an(i+1)) is a union of n intervals, there must be i ≤ n such

bβ ∩ [an(i), an(i+1)) = ∅, and thus bβ ∈ Uni ⊆

⋃Fn. �

A multidimensional version of the last proof gives the following.

Theorem 2.15 (Bartoszynski-Tsaban [3]). For each b-scale B, all finite powers ofthe set B ∪ [N]<∞ satisfy Ufin(O,Γ). �

Indeed, Zdomskyy and the present author proved in [23] that any finite product(B1 ∪ [N]<∞)× . . .× (B1 ∪ [N]<∞), with B1, . . . , Bk b-scales, satisfies Ufin(O,Γ).

In a work in progress, the method introduced here is used to prove the following,substantially stronger, result.

Theorem 2.16 (Miller-Tsaban-Zdomskyy). For each b-scale B and each set ofreals H satisfying Ufin(O,Γ), (B ∪ [N]<∞)×H satisfies Ufin(O,Γ).

3. Strongly Hurewicz sets of reals, in ZFC

Consider, for each f ∈ NN, the following selection hypothesis.

Uf (A ,B): For all U1,U2, · · · ∈ A , none containing a finite subcover, thereare finite F1 ⊆ U1,F2 ⊆ U2, . . . such that such that |Fn| ≤ f(n) for all n,and {

⋃Fn : n ∈ N} ∈ B.

Remark 3.1. One may require in the definition of Uf (A ,B) that each Fn isnonempty. This will not change the property when A ,B ∈ {O,Γ}, since we mayassume that the given covers get finer and finer. This can be generalized to mosttypes of covers considered in the field.

7Less than that is required of the given covers. See the proof.

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10 BOAZ TSABAN

Uf (A ,B) depends only on lim supn f(n).

Lemma 3.2. Assume that for each V ∈ B, {∅} ∪ V ∈ B. For all f, g ∈ NN withlim supn f(n) = lim supn g(n), Uf (A ,B) = Ug(A ,B).

Proof. The argument is as in the proofs of [9, 3.2–3.5] and [24, Lemma 3], concerningsimilar concepts in other contexts.

Let U1,U2, · · · ∈ A (X). Let m1 < m2 < . . . be such that f(n) ≤ g(mn) for alln. Apply Uf (A ,B) to the sequence Um1

,Um2, . . . , to obtain Fm1

⊆ Um1,Fm2

⊆Um2

, . . . , such that |Fmn| ≤ f(n) for all n, and {

⋃Fmn

: n ∈ N} ∈ B(X). Fork /∈ {mn : n ∈ N} we can take Fk = ∅. Then {

⋃Fn : n ∈ N} = {∅} ∪ {

⋃Fmn

: n ∈N} ∈ B(X), and |Fn| ≤ g(n) for all n. �

Thus, for each f ∈ NN with lim supn f(n) = ∞, Uf (A ,B) = Uid(A ,B), whereid is the identity function, id(n) = n for all n. We henceforth use the notation

Un(A ,B)

for Uid(A ,B).Our proof of Theorem 2.12 shows the following.

Theorem 3.3. For each b-scale B, B ∪ [N]<∞ satisfies Un(Γ,Γ).

Proof. In the proof of Theorem 2.12 we show that B ∪ [N]<∞ satisfies Un+1(Γ,Γ).By Lemma 3.2, this is the same as Un(Γ,Γ). �

We will soon show that Un(Γ,Γ) is strictly stronger than Ufin(O,Γ).A cover U of X is multifinite [22] if there exists a partition of U into infinitely

many finite covers of X. Let A be a family of covers of X. A)ג ) is the family ofall covers U of X such that: Either U is multifinite, or there exists a partition P ofU into finite sets such that {

⋃F : F ∈ P} \ {X} ∈ A [19].

The special case (Γ)ג was first studied by Kocinac and Scheepers [13], whereit was proved that Ufin(O,Γ) = Sfin(Ω, .((Γ)ג Additional results of this type areavailable in Babinkostova-Kocinac-Scheepers [1], and in general form in Samet-Scheepers-Tsaban [19].

Theorem 3.4 (Samet, et al. [19]). Ufin(Γ, ((Γ)ג = Sfin(Γ, .((Γ)ג

Theorem 3.5. Un(Γ,Γ) implies S1(Γ, .((Γ)ג

Proof. We prove the following, stronger statement: Assume thatX satisfies Un(Γ,Γ),and let s(n) = 1 + · · · + n = (n + 1)n/2. For all U1,U2, · · · ∈ Γ(X), there are

U1 ∈ U1, U2 ∈ U2, . . . , such that for each x ∈ X, x ∈⋃s(n+1)

k=s(n) Uk for all but finitelymany n.

Let U1,U2, · · · ∈ Γ(X). We may assume that for each n, Un+1 refines Un. ApplyUn(Γ,Γ) to Us(1),Us(2), . . . to obtain U1 ∈ Us(1), U2, U3 ∈ Us(2), . . . , such that for

each x ∈ X, x ∈⋃s(n+1)

k=s(n)+1 Uk for all but finitely many n. For each n and each

k = s(n) + 1, . . . , s(n+ 1), replace Uk by an equal or larger set from Uk. �Remark 3.6. The statement at the beginning of the last proof is in fact a charac-terization of Un(Γ,Γ).

Remark 3.7. In general, if every pair of elements of A has a joint refinement in A ,and B is finitely thick in the sense of [22], then Un(A ,B) implies S1(A , .((B)ג

In particular, when B = O, (B)ג = O, and thus Un(A ,O) = S1(A ,O). Forexample, Un(Γ,O) = S1(Γ,O).

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Thus, the Bartoszynski-Shelah Theorem tells that for each b-scale B, B∪ [N]<∞

satisfies Sfin(Γ, ,((Γ)ג whereas Theorem 3.3 tells that it indeed satisfies S1(Γ, .((Γ)גAs Ufin(O,Γ) does not even imply S1(Γ,O) (Lemma 1.16), we have that Un(Γ,Γ)is strictly stronger than Ufin(O,Γ).

Theorem 3.8 (Tsaban-Zdomskyy [17]). Assume the Continuum Hypothesis (orjust b = c). There is a b-scale B such that no set of reals containing B ∪ [N]<∞

satisfies S1(Γ,Γ).

By Theorems 3.3 and 3.8, Un(Γ,Γ) �= S1(Γ,Γ). Thus, Un(Γ,Γ) is strictly inbetween S1(Γ,Γ) and Ufin(O,Γ).

A natural refinement of the Problem 9, solved in Theorem 3.8, is the following.

Problem 3.9 (Zdomskyy). Is there a set of reals X without perfect subsets, suchthat X satisfies Ufin(O,Γ) but not Un(Γ,Γ)?

4. A visit at the border of ZFC

By Lemma 3.2, there are only the following kinds of (strongly) Hurewicz proper-ties: Ufin(Γ,Γ), Un(Γ,Γ), and Uc(Γ,Γ), for constants c ∈ N. For c = 1, Uc(Γ,Γ) =S1(Γ,Γ), and thus by the results of the previous section, at least three of theseproperties are distinct. (We consider properties distinct if they are not provablyequivalent.)

By Theorem 2.11, U1(Γ,Γ) may be trivial. The next strongest property isU2(Γ,Γ). We prove that it is not trivial.

Definition 4.1. Let s, a ∈ [N]∞. s slaloms8 a if a ∩ [s(n), s(n+1)) �= ∅ for all butfinitely many n. s slaloms a set Y ⊆ [N]∞ if it slaloms each a ∈ Y .

By Lemma 2.13, a set Y ⊆ [N]∞ is bounded if, and only if, there is s whichslaloms Y .

Definition 4.2. A slalom b-scale is an unbounded set {bα : α < b} ⊆ [N]∞, suchthat bβ slaloms bα for all α < β < b.

By Lemma 2.13, we have the following.

Lemma 4.3. There are slalom b-scales. �We are now ready to prove the main result of this paper.

Theorem 4.4. For each slalom b-scale B, B ∪ [N]<∞ satisfies U2(Γ,Γ).

Proof. Let B = {bα : α < b} be a slalom b-scale. Let U1,U2, · · · ∈ Γ(B ∪ [N]<∞).For each n, take an ∈ [N]∞ and distinct Un

1 , Un2 , . . . for Un as in Lemma 2.14. We

may assume that an(1) = 1. Let a ∈ [N]∞ slalom {an : n ∈ N}. As B is unbounded,there is by Lemma 2.13 α < b, such that I = {m : [a(m), a(m+3)) ∩ bα = ∅} isinfinite. (Otherwise, {a(3n) : n ∈ N} would slalom B.) For each n, let

In = {m ≥ n : [an(m), an(m+2)) ∩ bα = ∅}.As a slaloms an, In is infinite, and therefore {Un

m : m ∈ In} ∈ Γ(B ∪ [N]<∞).As |{xβ : β < α}| < b, {xβ : β < α} satisfies S1(Γ,Γ) (Lemma 2.7), and thus,

there are mn ∈ In, n ∈ N, such that {Unmn

: n ∈ N} ∈ Γ({xβ : β < α}). We claimthat

{Unmn

∪ Unmn+1 : n ∈ N} ∈ Γ(B ∪ [N]<∞).

8Short for “is a slalom for”.

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12 BOAZ TSABAN

If x ∈ [N]<∞, then for each large enough n, x ∩ [an(mn), an(mn+1)) = ∅ (becausemn ≥ n), and thus x ∈ Un

mn. For β < α, bβ ∈ Un

mnfor all large enough n, by the

choice of mn.For β ≥ α (that’s the interesting case), we have the following: Let mn ∈ In, and

let k be such that

bα(k) < an(mn) < an(mn + 2) ≤ bα(k + 1).

If n is large, then k is large, and as bβ slaloms bα, there is i such that

bβ(i) ≤ bα(k) < an(mn) < an(mn + 2) ≤ bα(k + 1) < bβ(i+ 2).

There are two possibilities for an(mn+1): If an(mn+1) ≤ bβ(i+1), then [an(mn), an(mn+1))∩bβ = ∅, and thus bβ ∈ Un

mn. Otherwise, an(mn +1) > bβ(i + 1), and thus

[an(mn+1), an(mn+2)) ∩ bβ = ∅. Therefore, bβ ∈ Unmn+1 in this case. �

Theorem 4.5. Assume the Continuum Hypothesis (or just b = c). There is aslalom b-scale B such that B∪[N]<∞ satisfies U2(Γ,Γ), but no set of reals containingB ∪ [N]<∞ satisfies S1(Γ,Γ).

Proof. Consider the proof of Theorem 3.8, given in [17]. We need only make surethat in Proposition 2.5 of [17], B can be constructed in a way that it is a slalomb-scale. This should be taken care of in the second paragraph of page 2518.

At step α < b of this construction, we are given a set Y with |Y | = |α| < b, anda set aα ∈ [N]∞. Take an infinite bα ⊆ aα such that bα slaloms Y . (E.g., take aslalom b for Y , and then define bα ⊆ aα by induction on n, such that for each n,|b∩ [bα(n), bα(n+1))| ≥ 2.) By induction on n, thin out bα such that it satisfies thedisplayed inequality there for all n. bα remains a slalom for Y .

Theorem 4.4 guarantees that B ∪ [N]<∞ satisfies U2(Γ,Γ). �By Theorem 2.11, it is consistent that S1(Γ,Γ) is trivial, whereas by Theorem

4.4, U2(Γ,Γ) is never trivial. The following remains open.

Conjecture 4.6. U2(Γ,Γ) is strictly stronger than Un(Γ,Γ).

5. The Hurewicz Problem

In the same 1927 paper Hurewicz asked the following.

Problem 5.1 (Hurewicz [11]). Is there a metric space satisfying Sfin(O,O), butnot Ufin(O,Γ)?

In a footnote added at the proof stage (the same one mentioned before Theorem1.2), Hurewicz quotes the following, which solves his problem if the ContinuumHypothesis is assumed.

Theorem 5.2 (Sierpinski). Every Luzin set satisfies Sfin(O,O), but not Ufin(O,Γ).

Proof. Let L be a Luzin set. We have already proved that L satisfies Sfin(O,O)(Theorem 1.2). It remains to show that L does not satisfy Ufin(O,Γ).

As L contains no perfect sets, R\L is dense in R. Fix a countable denseD ⊆ R\L.R \ D is homeomorphic to R \ Q,9 which in turn is homeomorphic to [N]∞ (e.g.,using continued fractions).

9D is order-isomorphic to Q. An order isomorphism f : D → Q extends uniquely to and orderisomorphism f : R → R by setting f(r) = sup{f(d) : d < r}. The restriction of f to R \ D is ahomeomorphism.

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As L ⊆ R \D, we may assume that L ⊆ [N]∞.10 By Lemma 2.6, it suffices toshow that L is unbounded. For each b ∈ [N]∞, the set

{a ∈ [N]∞ : a ≤∗ b} =⋃

n∈N

{a ∈ [N]∞ : (∀m ≥ n) a(m) ≤ b(m)},

with each {a ∈ [N]∞ : (∀m ≥ n) a(m) ≤ b(m)} nowhere dense. Thus, {a ∈ [N]∞ :a ≤∗ b} is meager, and therefore does not contain L. �

Hurewicz’s problem remained, however, open until the end of 2002.

Theorem 5.3 (Chaber-Pol [6]). There is a set of reals satisfying Sfin(O,O) but notUfin(O,Γ).

Chaber and Pol’s proof is topological and uses a technique due to Michael. Thefollowing combinatorial proof contains the essence of their proof.

Proof of Theorem 5.3. The proof is dichotomic. If b < d, then any unbounded B ⊆[N]∞ of cardinality b satisfies Sfin(O,O) (Lemma 1.5) but not Ufin(O,Γ) (Lemma2.6).

Lemma 5.4. For each s ∈ [N]∞, there is a ∈ [N]∞ such that: ac = N \ a ∈ [N]∞,a �≤∗ s, and ac �≤∗ s.

Proof. Letm1 > s(1). For each n > 1, letmn > s(mn−1). Let a =⋃

n[m2n−1,m2n).For each n:

a(m2n) ≥ m2n+1 > s(m2n);

ac(m2n−1) ≥ m2n > s(m2n−1). �

So, assume that b = d. Fix a scale {sα : α < d} ⊆ [N]∞. For each α < d, useLemma 5.4 to pick aα ∈ [N]∞ such that:

(1) acα = N \ aα is infinite;(2) aα �≤∗ sα; and(3) acα �≤∗ sα.

Let A = {aα : α < d}. For b ∈ [N]∞, let α < d be such that b <∗ sα. Then{β : aβ ≤∗ b} ⊆ α. As in the proof of Theorem 1.7, this implies that A is d-concentrated on [N]<∞, and thus A∪ [N]<∞ satisfies Sfin(O,O) (indeed, S1(Γ,O) –Corollary 1.14).

On the other hand, A ∪ [N]<∞ is homeomorphic to Y = {xc : x ∈ A ∪ [N]<∞},which is an unbounded subset of [N]∞ (by item (3) of the construction). By Lemma2.6, Y (and therefore A ∪ [N]<∞) does not satisfy Ufin(O,Γ). �

The advantage of the last proof is its simplicity. However, it does not providean explicit example, and in the case b < d gives a trivial example, i.e., one ofcardinality smaller than non(Sfin(O,O)). We conclude with an explicit solution.

Theorem 5.5 (Tsaban-Zdomskyy [23]). There is a set of reals of cardinality d,satisfying Sfin(O,O) (indeed, S1(Γ,O)), but not Ufin(O,Γ).

10If L is a Luzin set in a topological space X and f : X → Y is a homeomorphism, then f [L]is a Luzin set in Y , since “being meager” is preserved by homeomorphisms.

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14 BOAZ TSABAN

Our original proof uses in its crucial step a topological argument. Here, we givea more combinatorial argument, based on a (slightly amended) lemma of Milden-berger.

A set Y ⊆ [N]∞ is groupwise dense if:

(1) a ⊆∗ y ∈ Y implies a ∈ Y ; and(2) For each a ∈ [N]∞, there is an infinite I ⊆ N such that

⋃n∈I [a(n), a(n+1)) ∈

Y .

For Y satisfying (1), Y is groupwise dense if, and only if, Y is nonmeager [4].

Proof of Theorem 5.5. Fix a dominating set {dα : α < d}. Define aα ∈ [N]∞ byinduction on α < d. Step α: Let Y = {dβ , aβ : β < α}. |Y | < d.

The following is proved by Mildenberger as part of the proof of [15, Theorem2.2], except that we eliminate the “next” function from her argument.

Lemma 5.6 (Mildenberger [15]). For each Y ⊆ [N]∞ with |Y | < d, G = {a ∈[N]∞ : (∀y ∈ Y ) a �≤∗ y} is groupwise dense.

Proof. Clearly, G satisfies (1) of the definition of groupwise density. We verify (2).We may assume that Y is closed under maxima of finite subsets. Let g ∈ [N]∞ be

a witness that Y is not dominating. Then the family of all sets {n : y(n) < g(n)},y ∈ Y , can be extended to a nonprincipal ultrafilter U .

Let a ∈ [N]∞. By thinning out a, we may assume that g(a(n)) < a(n+1) for alln. For i = 0, 1, 2, let

ai =⋃

n∈N

[a(3n+ i), a(3n+ i+ 1)).

Then there is i such that ai ∈ U . We claim that ai+2 mod 3 ∈ G. Let y ∈ Y . For eachk in the infinite set {n : y(n) < g(n)}∩ai, let n be such that k ∈ [a(3n+i), a(3n+i+1)).Then

y(k) < g(k) < g(a(3n+ i+ 1)) < a(3n+ i+ 2) ≤ ai+2 mod 3(k),

because a(3n+ i+2) is the first element of ai+2 mod 3 greater or equal to k, andai+2 mod 3(k) ≥ k. �

Let G = {a ∈ [N]∞ : (∀y ∈ Y ) a �≤∗ y}. As G is groupwise dense, thereis aα ∈ G such that acα is infinite and acα �≤∗ dα. To see this, take an intervalpartition as in the proof of Lemma 5.4. Then there is an infinite subfamily of theeven intervals, whose union aα is in G. For each n such that [m2n−1,m2n) ⊆ aα,ac(m2n−1) ≥ m2n > s(m2n−1).

11

Thus, there is

aα ∈ {a ∈ [N]∞ : (∀y ∈ Y ) a �≤∗ y} \ {a ∈ [N]∞ : ac ≤∗ dα}.Continue exactly as in the above proof of Theorem 5.3. �

Chaber and Pol’s Theorem in [6] is actually stronger than Theorem 5.3 above,and establishes the existence of a set of reals X such that X does not satisfyUfin(O,Γ),12 but all finite powers of X satisfy Sfin(O,O).

11Alternatively, note that {a : ac ≤∗ dα} is homeomorphic to the meager set {a : a ≤∗ dα},and thus cannot contain a groupwise dense (i.e., nonmeager) set.

12And thus neither any finite power of X, since X is a continuous image of Xk for each k.

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Their proof shows that if b = d, then there is such an example of cardinality d.The assumption “b = d” was weakened to “d is regular” by Tsaban and Zdomskyy[23], but the following remains open.

Problem 5.7. Is there, provably in ZFC, a nontrivial (i.e., one of cardinality atleast d) example of a set of reals such that X does not satisfy Ufin(O,Γ), but allfinite powers of X satisfy Sfin(O,O)?

In other words, the question whether there is a nondichotomic proof of Chaberand Pol’s full theorem remains open.

Acknowledgments. We thank Gabor Lukacs, Lyubomyr Zdomskyy and the ref-eree for their useful comments, which lead to improvements in the presentation ofthis paper.

References

[1] L. Babinkostova, L. Kocinac, and M. Scheepers, Combinatorics of open covers (VIII), Topol-ogy and its Applications 140 (2004), 15–32.

[2] T. Bartoszynski and S. Shelah, Continuous images of sets of reals, Topology and its Appli-cations 116 (2001), 243–253.

[3] T. Bartoszynski and B. Tsaban, Hereditary topological diagonalizations and the Menger-Hurewicz Conjectures, Proceedings of the American Mathematical Society 134 (2006), 605–615.

[4] A. Blass, Combinatorial cardinal characteristics of the continuum, in: Handbook of SetTheory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Kluwer Academic Publishers,Dordrecht, to appear. http://www.math.lsa.umich.edu/~ablass/hbk.pdf

[5] L. Bukovsky and K. Ciesielski, Spaces on which every pointwise convergent series of con-tinuous functions converges pseudo-normally, Proceedings of the American MathematicalSociety 133, 605–611.

[6] J. Chaber and R. Pol, A remark on Fremlin-Miller theorem concerning the Menger propertyand Michael concentrated sets, unpublished note (October 2002).

[7] D. Fremlin and A. Miller, On some properties of Hurewicz, Menger and Rothberger, Funda-menta Mathematica 129 (1988), 17–33.

[8] F. Galvin and A. Miller, γ-sets and other singular sets of real numbers, Topology and itsApplications 17 (1984), 145–155.

[9] S. Garcia-Ferreira and A. Tamariz-Mascarua, Some generalizations of rapid ultrafilters andId-fan tightness, Tsukuba Journal of Mathematics 19 (1995), 173–185.

[10] W. Hurewicz, Uber eine Verallgemeinerung des Borelschen Theorems, MathematischeZeitschrift 24 (1925), 401–421.

[11] W. Hurewicz, Uber Folgen stetiger Funktionen, Fundamenta Mathematicae 9 (1927), 193–204.

[12] W. Just, A. Miller, M. Scheepers, and P. Szeptycki, The combinatorics of open covers II,Topology and its Applications 73 (1996), 241–266.

[13] L. Kocinac and M. Scheepers, Combinatorics of open covers (VII): Groupability, FundamentaMathematicae 179 (2003), 131–155.

[14] K. Menger, Einige Uberdeckungssatze der Punktmengenlehre, Sitzungsberichte der WienerAkademie 133 (1924), 421–444.

[15] H. Mildenberger, Groupwise dense families, Archive for Mathematical Logic 40 (2001), 93–112.

[16] A. Miller and B. Tsaban, Point-cofinite covers in Laver’s model, Proceedings of the AmericanMathematical Society 138 (2010), 3313–3321.

[17] D. Repovs, B. Tsaban, and L. Zdomskyy, Hurewicz sets of reals without perfect subsets,Proceedings of the American Mathematical Society 136 (2008), 2515–2520.

[18] M. Sakai, The sequence selection properties of Cp(X), Topology and its Applications 154(2007), 552–560.

[19] N. Samet, M. Scheepers, and B. Tsaban, Partition relations for Hurewicz-type selectionhypotheses, Topology and its Applications 156 (2009), 616–623.

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[20] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology and its Applications69 (1996), 31–62.

[21] M. Scheepers, Cp(X) and Arhangel’skiı’s αi spaces, Topology and its Applications 89 (1998),265–275.

[22] B. Tsaban, Strong γ-sets and other singular spaces, Topology and its Applications 153 (2005),620–639.

[23] B. Tsaban and L. Zdomskyy, Scales, fields, and a problem of Hurewicz, Journal of the Euro-

pean Mathematical Society 10 (2008), 837–866.[24] J. Valueva, On some fan-tightness type properties, Commentationes Mathematicae Universi-

tatis Carolinae 39 (1998), 415–421.[25] J. Valueva, A remark on combinatorics of open covers and Cp-spaces, Questions and Ansers

in General Topology 16 (1998), 183–187.

Appendix A. Sf (A ,B)

Properties closely related to our Uf (A ,B) were considered in the literature.Consider, for each f ∈ NN, the following selection hypothesis.

Sf (A ,B): For all U1,U2, · · · ∈ A , there are finite F1 ⊆ U1,F2 ⊆ U2, . . . suchthat such that |Fn| ≤ f(n) for all n, and

⋃n Fn ∈ B.

In [9, 5] it is proved that for each f ∈ NN, Sf (O,O) = S1(O,O). Indeed, byRemark 3.7 we have that for all A ,

Sf (A ,O) = Un(A ,O) = S1(A ,O).

A family B of open covers of X is finitely thick [22] if:

(1) If U ∈ B and for each U ∈ U :FU is a finite nonempty family of open sets such that for eachV ∈ FU , U ⊆ V �= X,

then⋃

U∈U FU ∈ B.

(2) If U ∈ B and V = U ∪ F where F is finite and X /∈ F , then V ∈ B.13

Many families of “rich” covers considered in the literature, including O,Ω,Γ [20, 12],are finitely thick. Also, for each of these families, each pair of elements has a jointrefinement in the same family.

The case A = B = Ω of the following theorem was proved in [9, 25].

Theorem A.1. Assume that each pair of elements of A has a joint refinement inA , and B is finitely thick. For each f ∈ NN, Sf (A ,B) = S1(A ,B).

Proof. As 1 ≤ f(n) for all n, S1(A ,B) implies Sf (A ,B). To prove the remainingimplication, assume that X satisfies Sf (A ,B).

Let U1,U2, · · · ∈ A (X). Let s(n) = f(1) + f(2) + · · · + f(n) for all n. For eachn, take Vn ∈ A (X) refining U1, . . . ,Us(n).

Apply Sf (A ,B) to the sequence V1,V2, . . . , to obtain F1 ⊆ V1,F2 ⊆ V2, . . . ,such that |Fn| ≤ f(n) for all n, and

⋃n Fn ∈ B(X).

Fix n. For each k ∈ {s(n−1)+1, . . . , s(n)}, pick Uk ∈ Uk such that each memberof Fn is contained in some Uk. As B is finitely thick, {Uk : k ∈ N} ∈ B(X). �

Thus, in our context, the scheme Sf (A ,B) does not introduce new properties.As we have seen in the present paper, this is not the case for Uf (A ,B).

Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

E-mail address: [email protected]: http://www.cs.biu.ac.il/~tsaban

13We will not use Item (2) of the definition of finitely thick here.

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A trichotomy theorem in natural models of AD+

Andres Eduardo Caicedo and Richard Ketchersid

Abstract. Assume AD+ and that either V = L(P(R)), or V = L(T,R) forsome set T ⊂ ORD. Let (X,≤) be a pre-partially ordered set. Then exactlyone of the following cases holds: (1) X can be written as a well-ordered union ofpre-chains, or (2) X admits a perfect set of pairwise ≤-incomparable elements,and the quotient partial order induced by (X,≤) embeds into (2α,≤lex) forsome ordinal α, or (3) there is an embedding of 2ω/E0 into (X,≤) whose rangeconsists of pairwise ≤-incomparable elements.

By considering the case where ≤ is the diagonal on X, it follows that forany set X exactly one of the following cases holds: (1) X is well-orderable, or(2) X embeds the reals and is linearly orderable, or (3) 2ω/E0 embeds into X.In particular, a set is linearly orderable if and only if it embeds into P(α) forsome α. Also, ω is the smallest infinite cardinal, and {ω1,R} is a basis for theuncountable cardinals.

Assuming the model has the form L(T,R) for some T ⊂ ORD, the result

is a consequence of ZF+DCR together with the existence of a fine σ-completemeasure on Pω1(R) via an analysis of Vopenka-like forcing. It is known that inthe models not covered by this case, ADR holds. The result then requires moreof the theory of determinacy; in particular, that V = OD((< Θ)ω), and theexistence and uniqueness of supercompactness measures on Pω1 (γ) for γ < Θ.

As an application, we show that (under the same basic assumptions)Scheepers’s countable-finite game over a set S is undetermined whenever S isuncountable.

Contents

1. Introduction2. Preliminaries3. AD+

4. The dichotomy theorem5. The countable-finite game in natural models of AD+

6. QuestionsReferences

2010 Mathematics Subject Classification. 03E60, 03E25, 03C20.Key words and phrases. Determinacy, AD+, ADR, ∞-Borel sets, ordinal determinacy,

Vopenka forcing, Glimm-Effros dichotomy, countable-finite game.The first author wants to thank the National Science Foundation for partial support through

grant DMS-0801189.


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2 ANDRES EDUARDO CAICEDO AND RICHARD KETCHERSID

1. Introduction

This paper deals with consequences of the strengthening AD+ of the axiom ofdeterminacy AD for the general theory of sets, not just for sets of reals or sets ofsets of reals.

Particular versions of our results were known either in L(R) or under the addi-tional assumption of ADR. They can be seen as generalizations of well-known factsin the theory of Borel equivalence relations.

We consider “natural” models of AD+, namely, those that satisfy V = L(P(R)),although our results apply to a slightly larger class of models. The special form ofV is used in the argument, not just consequences of determinacy.

Although an acquaintance with determinacy is certainly desirable, we strive tobe reasonably self-contained and expect the paper to be accessible to readers with aworking understanding of forcing, and combinatorial and descriptive set theory. Westate explicitly all additional results we require, and provide enough backgroundto motivate our assumptions. Jech [15] and Moschovakis [25] are standard sourcesfor notation and definitions. For basic consequences of determinacy, some of whichwe will use without comment, see Kanamori [16].

1.1. Results.Our main result can be seen as a simultaneous generalization of the Harrington-Marker-Shelah [10] theorem on Borel orderings, the Dilworth decomposition the-orem of Foreman [7], the Glimm-Effros dichotomy of Harrington-Kechris-Louveau[9], and the dichotomy theorem of Hjorth [13].

Recall that a pre-partial ordering ≤ on a set X (also called a quasi-orderingon X) is a binary relation that is reflexive and transitive, though not necessarilyanti-symmetric. Recall that E0 is the equivalence relation on 2ω defined by

xE0y ⇐⇒ ∃n ∀m ≥ n(x(m) = y(m)

).

Theorem 1.1. Assume AD+ holds and either V = L(T,R) for some T ⊂ ORD,or else V = L(P(R)). Let (X,≤) be a pre-partially ordered set. Then exactly oneof the following holds:

(1) X is a well-ordered union of ≤-pre-chains.(2) There are perfectly many ≤-incomparable elements of X, and there is an

order preserving injection of the quotient partial order induced by X into(2α,≤lex) for some ordinal α.

(3) There are 2ω/E0 many ≤-incomparable elements of X.

The argument can be seen in a natural way as proving two dichotomy theorems,Theorems 1.2 and 1.3.

Theorem 1.2. Assume AD+ holds and either V = L(T,R) for some T ⊂ ORD,or else V = L(P(R)). Let (X,≤) be a pre-partially ordered set. Then either:

(1) There are perfectly many ≤-incomparable elements of X, or else(2) X is a well-ordered union of ≤-pre-chains.

Theorem 1.3. Assume AD+ holds and either V = L(T,R) for some T ⊂ ORD,or else V = L(P(R)). Let (X,≤) be a partially ordered set. Then either:

(1) There are 2ω/E0 many ≤-incomparable elements of X, or else(2) There is an order preserving injection of (X,≤) into (2α,≤lex) for some

ordinal α.

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A TRICHOTOMY THEOREM IN NATURAL MODELS OF AD+ 3

It is easy to see that R injects into 2ω/E0, and it is well-known that, underdeterminacy, ω1 does not inject into R, and 2ω/E0 is not linearly orderable andtherefore cannot embed into any linearly orderable set. This shows that the casesdisplayed above are mutually exclusive.

Theorem 1.2 generalizes a theorem of Foreman [7] where, among other results,it is shown (in ZF+AD+DCR) that if ≤ is a Suslin/co-Suslin pre-partial orderingof R without perfectly many incomparable elements, then R is a union of λ-manySuslin sets, each pre-linearly-ordered by ≤, where λ is least such that both ≤ andits complement are λ-Suslin.

By considering the case ≤= {(x, x) : x ∈ X}, the following corollary, a gener-alization of the theorem of Silver [29] on co-analytic equivalence relations, followsimmediately:

Theorem 1.4. Assume AD+ holds and either V = L(T,R) for some T ⊂ ORD,or else V = L(P(R)). Let X be a set. Then either:

(1) R embeds into X, or else(2) X is well-orderable.

The corollary gives us the following basis result for infinite cardinalities:

Corollary 1.5. Assume AD+ holds and either V = L(T,R) for some T ⊂ORD, or else V = L(P(R)). Let S be an infinite set. Then:

(1) ω embeds into S.(2) If κ is a well-ordered cardinal, and S is strictly larger than κ, then either

κ+ or κ ∪ R embeds into S. In particular, ω1 and R form a basis for theuncountable cardinals. �

Note that there are no assumptions in Theorems 1.2–1.4 on the set X. If, inTheorem 1.4, the set X is a quotient of R by, say, a projective equivalence relation,one can give additional information on the length of the well-ordering. This hasbeen investigated by several authors including Harrington-Sami [11], Ditzen [5],Hjorth [12], and Schlicht [28].

Theorems 1.2 and 1.4 were our original results, and we consider Theorem 1.2the main theorem of this paper. After writing a first version of the paper, we foundHjorth [13], where the version of Theorem 1.4 for L(R) is attributed to Woodin.Hjorth [13] investigates in L(R) what happens when alternative 1 in Theorem 1.4holds but the quotient R/E0 does not embed into X; much remains to be exploredin this area. We remark that the argument of Hjorth [13] easily combines withour techniques, so we in fact have Theorem 1.3, a simultaneous generalization offurther results in Foreman [7], and the main result in Hjorth [13]. The followingcorollary is immediate:

Corollary 1.6. Assume AD+ holds and either V = L(T,R) for some T ⊂ORD, or else V = L(P(R)). Let X be a set. Then either:

(1) 2ω/E0 embeds into X, or else(2) X embeds into P(α) for some ordinal α. �

In particular:

Corollary 1.7. Assume AD+ holds and either V = L(T,R) for some T ⊂ORD, or else V = L(P(R)). Then a set is linearly orderable if and only if it embedsinto P(α) for some ordinal α. �

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Since it is slightly easier to follow, we arrange the exposition around the proofof Theorem 1.4, and then explain the easy adjustments to the argument that allowus to obtain Theorem 1.2, and the modifications required to the argument in Hjorth[13] to obtain Theorem 1.3.

Weak versions of some of these results were known previously in the context ofADR. It is thanks to the use of ∞-Borel codes in our arguments that we can extendthem in the way presented here.

As an application of our results, we show:

Theorem 1.8. Assume AD+ holds and either V = L(T,R) for some T ⊂ ORD,or else V = L(P(R)). Then the countable-finite game CF (S) is undetermined forall uncountable sets S.

This is a slightly amusing situation in that we have a family of games that areobviously determined under choice, but are undetermined in the natural models ofdeterminacy.

Theorem 1.8 seems of independent interest, since it is still open whether, underchoice, player II has a winning 2-tactic in CF (R). Theorem 1.8 seems to indicatethat the answer to this question only depends on the cardinal c rather than on anyparticular structural properties of the set of reals.

We also present detailed proofs of two additional results, not due to us. First,directly related to our approach is Woodin’s theorem characterizing the ∞-Borelsets:

Theorem 1.9 (Woodin). Assume ZF+ DCR + μ is a fine σ-complete measureon Pω1

(R). Then a set of reals A is ∞-Borel iff A ∈ L(S,R), for some S ⊂ ORD.

For models of AD+ of the form L(T,R) for some T ⊂ ORD, Theorems 1.2 and1.3 are in fact consequences of the assumptions of Theorem 1.9, this we establishvia an analysis of ∞-Borel codes by means of Vopenka-like forcing.

In the models not covered by this case, ADR holds, and the results require twoadditional consequences of determinacy due to Woodin, namely, that

V = OD((< Θ)ω),

and the uniqueness of supercompactness measures on Pω1(γ) for γ < Θ. We omit

the proofs of these two facts.Second, we also present a proof of the following result of Jackson:

Theorem 1.10 (Jackson). Assume ACω(R). Then there is a countable pairingfunction, i.e, a map

F : [P(R)]≤ω → P(R)

satisfying:

(1) F (A) is independent of any particular way A is enumerated, and(2) Each A ∈ A is Wadge-reducible to F (A).

It is because of Theorem 1.10 that our approach to Theorem 1.2 in the ADR

case is different from the approach when V = L(T,R) for some T ⊂ ORD.

1.2. Organization of the paper.Section 2 provides the required general background to understand our results, andincludes a brief (and perhaps overdue) motivation for AD+, a quick discussion of

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the known methods for obtaining natural models of determinacy, and a descriptionof Scheepers’s countable-finite game.

In Section 3 we state without proofs some specific consequences of AD+ thatour argument needs. We also prove Jackson’s Theorem 1.10.

In Section 4 we prove Woodin’s Theorem 1.9, and the dichotomy Theorem1.4. The argument divides in a natural way into two cases, according to whetherV = L(T,R) for some T ⊂ ORD, or V = L(P(R)). In the latter case, we may alsoassume ADR, that we use to derive the result from the former case. The argumentin the ADR case was suggested by Hugh Woodin. We also explain how to modifythe argument to derive our main result, Theorem 1.2, and sketch how to extendthe argument in Hjorth [13] to prove Theorem 1.3. The deduction of Corollary 1.7from the argument of Theorem 1.3 is standard.

In Section 5 we analyze the countable-finite game CF (S) in ZF, and use thedichotomy Theorem 1.4 to show that in models of AD+ of the forms stated above,the game is undetermined for all uncountable sets S. Since trivially player II hasa winning strategy if S is countable, this provides us with a complete analysis ofthe game in natural models of AD+. We have written this section in a way thatreaders mainly interested in this result, can follow the argument without needingto understand the proofs of our main results.

Finally, in Section 6 we close with some open problems.

1.3. Acknowledgments.We want to thank Marion Scheepers, for introducing us to the countable-finitegame, which led us to the results in this paper; Steve Jackson, for allowing us toinclude in Subsection 3.3 his construction of a pairing function; Matthew Foreman,for making us aware of Foreman [7], which led us to improve Theorem 1.4 intoTheorem 1.2; and Hugh Woodin, for developing the beautiful theory of AD+, forhis key insight regarding the dichotomy Theorem 1.4 in the ADR case, and forallowing us to include a proof of Theorem 1.9.

2. Preliminaries

The purpose of this section is to provide preliminary definitions and back-ground. In particular, we present a brief discussion of AD+ in Subsection 2.2, oftwo methods for obtaining models of determinacy in Subsection 2.3, and of thecountable-finite game in Subsection 2.5.

2.1. Basic notation.ORD denotes the class of ordinals. Whenever we write S ⊂ ORD, it is understoodthat S is a set. Given a set X, we endow Xω with the (Tychonoff’s) producttopology of ω copies of the discrete space X, so basic open sets have the form

[s] = {f ∈ Xω : s � f},where s ∈ X<ω. This will always be the case, even if X is an ordinal or carriessome other natural topology.

R will always mean Baire space, ωω, that is homeomorphic to the set of irra-tional numbers.

Definition 2.1. A tree T on a finite product∏

i<n Xi (typically for us, n = 1or 2) is a subset of (

∏i<n Xi)

<ω that is closed under restrictions. It is customary

to identify T with a subset of∏

i<n(X<ωi ) such that:

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(1) Whenever (pi : i < n) ∈ T , then all pi, i < n, have the same length.(2) T is closed under restrictions, in the following sense: If

p = (pi : i < n) ∈ T

and m is smaller than the common length of the pi, i < n, then

p � m := (pi � m : i < n) ∈ T.

If T is a tree on X × Y and x ∈ X<ω, then

Tx = {y ∈ Y <ω : (x, y) ∈ T}

and if x ∈ Xω, then

Tx =⋃

n

Tx�n,

so Tx is a tree on Y .We denote by [T ] the set of infinite branches through T and, if T is a tree on

X × Y , then

p[T ] = {f ∈ Xω : ∃g ∈ Y ω((f, g) ∈ [T ]

)} = {f :Tf is ill-founded}.

As usual, an infinite branch through T is a function f : ω → T such that forall n, f � n ∈ T .

2.1.1. Games.We deal with infinite games, all following a similar format: For some (fixed) setX, two players I and II alternate making moves for ω many innings, with I movingfirst. In each move, the corresponding player plays an element of X:

I x0 x2 . . .II x1 x3

(Specific games may impose restrictions on what elements are allowed as theplay progresses.) This way both players collaborate to produce an element x =〈x0, x1, x2, . . . 〉 of Xω.

Given A ⊆ Xω, we define the game �X(A) by following the format just de-scribed, and declaring that player I wins iff x ∈ A.

A strategy is a function σ : X<ω → X. Player I follows the strategy σ iff eachmove of I is dictated by σ and the previous moves of player II:

I σ(〈〉) σ(〈x0〉) σ(〈x0, x1〉)II x0 x1 . . .

Similarly one defines when II follows σ. A strategy σ is winning for I in a game� on X iff, for all x = 〈x0, x1, . . . 〉 ∈ Xω, player I wins the run

σ ∗ x

of the game, produced by I following σ against player II, who plays x bit by bit.Similarly we define when σ is winning for II.

We say that a game is determined when there is a winning strategy for one ofthe players. When the game is �X(A) for some A ⊆ Xω, it is customary to saythat A is determined.

Definition 2.2 (AD). In ZF, the axiom of determinacy, AD, is the state-ment that all A ⊆ R are determined.

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A standard consequence of AD is the perfect set property for sets of reals: AnyA ⊆ R is either countable or contains a perfect subset. It follows that AD isincompatible with the existence of a well-ordering of the reals and, in fact, with theweaker statement ω1 � R, that ω1 injects (or embeds) into R.

Since determinacy contradicts the axiom of choice, it should be understood asholding not in the universe V of all sets but rather in particular inner models, suchas L(R). When our results below assume, for example, that V = L(P(R)) and thatAD holds, this could then be understood as a result about all inner models M ofZF that satisfy AD+ V = L(P(R)).

2.2. AD+.At first the study of models of determinacy might appear to be a strange enterprise.However, as the theory develops, it becomes clear that one is really studying theproperties of “definable” sets of reals. The notion of definability is inherently vague;however, under appropriate large cardinal assumptions, any reasonable notion of “Ais a definable set of reals” is equivalent to “A is in an inner model of determinacycontaining all the reals.” Thus the study of properties of definable sets of realsbecomes the focus.

2.2.1. The theory AD+.AD+ is a strengthening of AD. The theory of models of AD+ is due to Woodin, seefor example Woodin [34, Section 9.1]. All unattributed results and definitions inthis section are either folklore, or can be safely attributed to Woodin.

The starting point for this study is the collection of Suslin sets.

Definition 2.3. A set A ⊆ Xω is κ-Suslin iff A = p[S] for some tree S onX × κ.

A set A is co-κ-Suslin if Xω \A is κ-Suslin and we say that A is Suslin/co-Suslin if A is both κ-Suslin and co-κ-Suslin for some κ. That A is κ-Suslin is alsoexpressed by saying that A has a κ-(semi)-scale. In this paper, we have no use forscales other than the incumbent Suslin representation, so we say no more aboutthem.

Let

Sλ = {A ⊆ R :A is λ-Suslin}.Being Suslin is obviously one notion of being definable, and the classically studieddefinable sets of reals are all Suslin assuming enough determinacy or large cardinals.Actually, choice implies that all sets of reals are Suslin, so under choice one actuallystudies which sets of reals are in Sλ for specific cardinals λ. Without choice, it isnot necessarily the case that all sets of reals are Suslin.

Definition 2.4. κ is a Suslin cardinal iff Sκ \⋃

λ<κ Sλ �= ∅.

For example, one can prove in ZF that the first two Suslin cardinals are ω andω1. Also, Sω = Σ

˜

11, the class of projections of closed subsets of R2; note that the

notion of Σ˜11 sets also makes sense for subsets of Rn for n > 1. Assuming some

determinacy, then Sω1= Σ

˜

12, the class of projections of complements of Σ

˜

11 sets.

It is a classical theorem of Suslin that “A is Borel” is equivalent to “A isω-Suslin/co-Suslin.” Being Borel is a notion of definability which is obviously ex-tendible by taking longer well-ordered unions. This leads to the notion of ∞-Borelsets, that we describe carefully below, in § 2.2.3.

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For now, define “A is ∞-Borel with code (φ, S)” to mean that S ⊂ ORD, φ isa formula in the language of set theory and, for any x ∈ R,

x ∈ A ⇐⇒ L[S, x] |= φ(S, x).

Clearly, if T witnesses that A is Suslin, then T also witnesses that A is ∞-Borel,since

x ∈ A ⇐⇒ L[T, x] |= Tx is ill-founded.

There are multiple senses in which a code for A is easy to calculate from A, assumingthat A is ∞-Borel. One of these will be discussed later, see Theorem 3.4 and § 4.1.1,and another is given by Theorem 2.5 below.

First, we need a couple of basic notions. Define

Θ = sup{| · |≤ : ≤ is a pre-well-ordering of a subset of R},where | · |≤ is the rank of the pre-well-ordering ≤. Equivalently,

Θ = sup{α : ∃f : R −−→onto

α}.

Suppose that A ⊆ Rn for some n ∈ ω, and define Σ˜

11(A) as the smallest col-

lection of subsets of Rm with m varying in ω, that contains A and is closed underinteger quantification, finite unions and intersections, continuous reduction, and ex-istential real quantification. As usual, define Π

˜11(A) to be the class of complements

of Σ˜

11(A) sets, Σ

˜

12(A) = ∃RΠ

˜

11(A), etc. Each of these classes has a canonical uni-

versal set U1n(A). See Moschovakis [25] for notation, the definition of universality,

and this fact.If ≤ is a pre-well-order of length γ, then we say that S ⊆ γ is Σ

˜

1n(≤) in the

codes iff there is a real x such that for ξ ∈ γ,

ξ ∈ S ⇐⇒ ∃y[|y|≤ = ξ and U1

n(≤)(x, y)].

The Moschovakis Coding Lemma, see Moschovakis [25], states that, under deter-minacy, given any pre-well-order ≤ of R of length γ, any S ⊆ γ is Σ

˜

11(≤) in the

codes.This yields that ifM andN are transitive models of AD with the same reals, and

γ < min{ΘM ,ΘN}, then P(γ)M = P(γ)N . We then have the following regarding∞-Borel codes.

Theorem 2.5 (Woodin). Assume AD and that A is ∞-Borel. Then there is aγ < Θ, a pre-well-order ≤ in Π

˜

12(A) of length γ, and a code S ⊆ γ for A. By the

coding Lemma, S is Σ˜

11(≤) in the codes. So S is Σ

˜

13(A) in the codes. �

In particular, if M and N are transitive models of AD with RM = RN , andA ∈ M is Suslin (or just ∞-Borel) in N , then A is ∞-Borel in M , although it neednot be the case that A is also Suslin in M .

The following is essentially contained in results of Kechris-Kleinberg-Moschova-kis-Woodin [17], see also Jackson [14].

Theorem 2.6. Assume AD, and suppose that λ < Θ and that A ⊆ λω isSuslin/co-Suslin. Then the game �λ(A) is determined. �

Suppose that M is a transitive model of AD, λ < ΘM , and f : λω → R is in Mand continuous. Let A be a set of reals in M , and consider the (A, f)-induced gameon λ, �λ(f

−1[A]). Suppose moreover that there is a transitive model N of AD withthe same reals as M , and such that A is Suslin/co-Suslin in N . Then, by Theorem

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2.6, in N , �λ(f−1[A]) is determined and hence, by the Coding Lemma, this game

is determined in M , since the winning strategy can be viewed as a subset of λ.Finally, recall that Suslin subsets R ⊆ R2 can be uniformized, see Moschovakis

[25], so that there is a partial function f : R → R such that whenever x ∈ R andthere is a y ∈ R with xRy then, in fact, x ∈ dom(f) and xRf(x).

Suppose that M is a transitive model of AD, and that R ⊆ R2 is a relation inM such that for any x ∈ R there is a y ∈ R such that xRy. If there is a transitivemodel N of AD, with the same reals as M , and such that R is Suslin in N , thenR is uniformizable in N . If f is a uniformizing function for R in N , then for anyreal x0 ∈ N there is then a real x ∈ N coding the sequence 〈xn :n < ω〉 wherexn+1 = f(xn) for all n ∈ ω. Since M and N have the same reals, then x andtherefore 〈xn :n < ω〉 are in M . This shows that DCR holds in M (see § 2.2.2below for the definition of DCR).

In summary, we have that if M is a transitive model of AD such that for eachA ∈ P(R)M , there is a transitive N such that:

(1) N models AD,(2) N has the same reals as M and,(3) in N , A is Suslin,

then the following hold in M :

• DCR.• All sets of reals are ∞-Borel.• For all ordinals λ < ΘM , all continuous functions f : λω → R, and all

A ⊆ R, the (A, f)-induced game on λ is determined.

This situation is axiomatized by AD+.

Definition 2.7 (Woodin). Over the base theory ZF, AD+ is the conjunctionof

• DCR.• All sets of reals are ∞-Borel.• < Θ-ordinal determinacy, i.e., all (A, f)-induced games on ordinals λ < Θare determined, for any A ⊆ R and any continuous f : λω → R.

The following is a consequence of the preceding discussion.

Theorem 2.8. If M is a transitive model of ZF + AD such that every set ofreals in M is Suslin in some transitive model N of ZF + AD with the same reals,then M |= AD+. �

In fact, in Theorem 2.8, it suffices that M and N satisfy the restriction of ZFto Σn sentences, for an appropriate sufficiently large value of n.

Remark 2.9. Suppose that M and N are transitive models of AD with thesame reals. Let θ = min{ΘM ,ΘN}. Then, by the Coding Lemma,

(⋃

γ<θ

P(γ))M

=(⋃

γ<θ

P(γ))N

.

In particular, if A ∈ M ∩ N is a set of reals, and A is κ-Suslin in N , for someκ < ΘM , then A is κ-Suslin in M as well.

Recall that Wadge-reducibility of sets of reals is given by

A ≤W B

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iff there is a continuous function f : R → R such that A = f−1[B]. It is a basicconsequence of determinacy that ≤W is well-founded. We can then assign a rankto each set of reals. The rank of ≤W itself is exactly Θ. Obviously, a continuousreduction can be coded by a real. With M and N as above, we then have that ifA ∈ M ∩N is a set of reals, then |A|M≤W

= |A|N≤W. It follows that if A is not Suslin

in M but it is Suslin in N , then P(R)M � P(R)N and ΘM < ΘN .

A benefit of considering AD+ rather than AD is that much of the fine analysisof L(R) under the assumption of determinacy actually lifts to models of the formL(P(R)) under the assumption of AD+. Whether AD+ actually goes beyond AD isa delicate question, still open. We will briefly touch on this below.

2.2.2. DCR.Recall that DCR, or DCω(R), is the statement that whenever R ⊆ R2 is such thatfor any real x there is a y with xRy, then there is a function f : ω → R such thatfor all n, f(n)Rf(n+ 1). It is easy to see that this is equivalent to the claim thatany tree T on R with no end nodes has an infinite branch.

Two straightforward (and well-known) observations are worth making: First, inZF, assume that DCR holds and that T ⊂ ORD. Then DCR holds in L(T,R). Second,if DCR holds in L(T,R) then, in fact, L(T,R) satisfies the axiom of dependentchoices, DC.

It is shown in Solovay [30] that for models satisfying V = L(P(R)) and in fact,more generally, for models of V = OD(P(R)), if AD+ DCR holds, then

cf(Θ) > ω =⇒ DC.

Under AD, there are interesting relationships and variations of DCR, due to theexistence of certain measures. Let D denote the set of Turing degrees. A set A ⊆ D

is a cone iff there is an a ∈ D such that

A = {b ∈ D : a ≤T b},where ≤T denotes the relation of Turing reducibility. Define the Martin measureμM on D, by

A ∈ μM ⇐⇒ A contains a Turing cone.

Martin proved that μM is a σ-complete measure on D. We have:

DC =⇒∏

ORD/μM is well-founded =⇒∏

ω1/μM is well-founded =⇒ DCR.

The first and second implications are trivial. Here is a quick sketch of the third:

Lemma 2.10 (Woodin). Over ZF, assume that μM is a measure, and that∏ω1/μM is well-founded. Then DCR holds.

Proof. Let T be a tree on R. For d ∈ D, let Td be the tree T restricted tonodes recursive in d. Td is in essence a tree on ω and, since DCω(ω) certainly holds,Td is ill-founded iff Td has an infinite branch. If Td is ill-founded for any d, thenthere is an infinite branch through T , so assume that all trees Td are well-founded.For each �x ∈ R<ω, we can define a partial function

h�x : D → ω1

byh�x(d) = rkTd

(�x),

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leaving h�x(d) undefined if �x /∈ Td. Note that h�x(d) is defined for μM -a.e. degree d.By assumption, [h�x]μM

is an ordinal α�x, and the map

�x �→ α�x

ranks the original tree T and hence T is not a counterexample to DCR. �

Clearly, in this argument, the Turing degree measure could be replaced by anyσ-complete, fine measure μ on Pω1

(R) satisfying that∏

ω1/μ is well-founded.Under AD+ − DCR we actually have the equivalence

∏ORD/μM is well-founded ⇐⇒ DCR.

The left-hand side of this equivalence was part of Woodin’s original formalizationof AD+.

There are models of AD+ + cf(Θ) = ω. In these models, DC fails, so just thewell-foundedness of ultrapowers by fine measures on Pω1

(R) does not give DC.2.2.3. ∞-Borel sets.

Essentially the ∞-Borel sets are the result of extending the usual Borel hierarchyby allowing arbitrary well-ordered unions.

Work in ZF. Without choice it is better to work with “codes” for sets (descrip-tions of their transfinite Borel construction) rather than with the sets themselves(the output of such a construction), hence an ∞-Borel set is any set with an ∞-Borel code. For example, it might be the case that for all α < γ, Aα is ∞-Borel,but there is no sequence of codes cα and hence

⋃α<γ Aα might not be ∞-Borel.

There are several equivalent definitions of ∞-Borel codes. For definiteness, wepresent an official version, and then some variants, and leave it up to the reader tocheck that the notions are equivalent, and even locally equivalent when required.

Definition 2.11. Fix a countable set of objects

N ={¬,

∨}∪ {n :n ∈ ω}

with N disjoint from ORD; e.g., ¬ = (0, 0),∨

= (0, 1), and n = (1, n) wouldsuffice. The ∞-Borel codes (BC) are defined recursively by: T ∈ BC iff one of thefollowing holds:

• T = 〈n〉.• T =

∨α<κ Tα = {〈

∨, α〉s :α < κ and s ∈ Tα} where each Tα ∈ BC.

• T = ¬S = {〈¬〉s : s ∈ S} where S ∈ BC.

Hence a code is essentially a well-founded tree on ORD∪N , and we will identify∞-Borel codes with these trees without comment. Set

BCκ = BC ∩ {T :T is a well-founded tree of rank < κ}.For κ a limit ordinal, BCκ is closed under finite joins. If cf(κ) > ω, then BCκ

is σ-closed and, if κ is regular, then BCκ is < κ-closed. Clearly for regular κ,BCκ = BC ∩H(κ).

Definition 2.12. A set of reals is ∞-Borel iff it is the interpretation of someT ∈ BC. We denote this interpretation by AT , and define it by recursion as follows:

• An = {x ∈ R :x(n0) = n1}, where n ↔ (n0, n1) is a recursive bijectionbetween R and R2.

• A∨α<κ

Tα=

⋃α<κ ATα

.

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• A¬T = R \AT .

The predicates “T ∈ BC” and “x ∈ AT ” are Σ1 and absolute for any model ofKP+Σ1-separation. (Just KP is not enough, since the code must be well-founded.)

Let B∞ denote the collection of ∞-Borel sets, and let Bκ be the subset of B∞consisting of those sets with codes in BCκ. In particular, if ω1 is regular, then Bω1

is just the algebra of Borel sets.The following gives a few alternate definitions for the ∞-Borel sets. The equiv-

alence of the first three is local in the sense that it is absolute to models of KP+Σ1-separation. The equivalence with the fourth one is still reasonably local, certainlyabsolute to models of ZF, and the definition itself can be formalized in any theorystrong enough to allow the definability of the satisfiability relation for the classesL[S, x].

• A is ∞-Borel.• There is a tree T on κ×ω such that A(x) iff player I has a winning strategyin the game �T,x given by: Players I and II take turns playing ordinalsαi < κ so in the end they play out f ∈ κω. Player I wins iff (f, x) ∈ [T ].Note that the game �T,x is closed for I and hence determined.

(In this case T is taken as the code and AT = {x : I has a winningstrategy in �T,x}.)

• There is a Σ1 formula φ (in the language of set theory, with two freevariables) and S ⊆ γ for some γ, such that

A(x) ⇐⇒ L[S, x] |= φ(S, x).

(Here (φ, S) is taken to be the code and A = Aφ,S is the set coded.)• There is a formula φ and S ⊆ γ for some γ, such that

A(x) ⇐⇒ L[S, x] |= φ(S, x).

(Once again, (φ, S) is taken to be the code and A = Aφ,S is the set coded.)

It is thus natural to identify codes with sets of ordinals, and we will often doso.

For example, as mentioned above, Suslin sets are ∞-Borel. On the other hand,Suslin subsets of R × R can be uniformized, while in general there can be non-uniformizable sets in a model of AD+, so it is not true that all ∞-Borel sets areSuslin.

Under fairly mild assumptions, being ∞-Borel already entails many of the niceregularity properties shared by the Borel sets. In particular, suppose that S is acode witnessing that AS is ∞-Borel, and suppose that

|P(Pc) ∩ L[S]|V = ω,

where Pc = Add(ω, 1) is the Cohen poset (essentially ω<ω). Then AS has the

property of Baire. Similarly, if |P(PL) ∩ L[S]|V = ω, where PL = Ranω is randomforcing, then AS is Lebesgue measurable. In general, if ωV

1 is inaccessible in L[S],then AS has all the usual regularity properties.

Note that Theorem 1.9 provides us, over the base theory ZF+DCR+“there is afine measure on Pω1

(R),” with yet another equivalence for the notion of ∞-Borel;however, we know of no reasonable sense in which this version would be local asthe previous ones.

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2.2.4. Ordinal determinacy.AD states that all games on ω are determined. One may wonder whether it isconsistent with ZF that, more generally, all games on ordinals are determined.This is not the case; in fact, it is well-known that there is an undetermined gameon ω1.

To see this, consider two cases. If AD fails, we are done, and there is in factan undetermined game on ω. If AD holds, then ω1 �� R. Consider the game whereplayer I begins by playing some α < ω1, and player II plays bit by bit a real codingω + α. Since any countable ordinal can be coded by a real, it is clear that player Icannot have a winning strategy. Were this game determined, player II would havea winning strategy σ. But it is straightforward to define from σ an uncountableinjective sequence of reals, and we reach a contradiction.

It follows that some care is needed in the way the payoff of ordinal games ischosen if we want them to be determined, and this is why < Θ-determinacy isstated as above.

Note that ordinal determinacy indeed implies determinacy, so AD+ strengthensAD. One consequence of ordinal determinacy that we will use is the following:

Theorem 2.13 (Woodin). Assume AD+. Then, for every Suslin cardinal κ,there is a unique normal fine measure μκ on Pω1

(κ). In particular, μκ ∈ OD. �If κ is below the supremum of the Suslin cardinals, this follows from Woo-

din [33], where games on ordinals are simulated by real games, in particular,giving the result under ADR (which is the case that interests us). For the AD+

result, Woodin’s argument must be integrated with the generic coding techniquesin Kechris-Woodin [19] to produce ordinal games that are determined under AD+.The result is that the supercompactness measure coincides with the weak club filter,where S ⊆ Pω1

(κ) a is weak club iff⋃S = κ and, whenever σ0 ⊆ σ1 ⊆ · · · are in

S, then⋃

i∈ω σi ∈ S.Let κ be a Suslin cardinal. For any γ < κ, define μγ = πκ,γ(μκ) where

πκ,γ : Pω1(κ) → Pω1

(γ)

is defined by σ �→ σ ∩ γ. This gives a canonical sequence of ω1-supercompactnessmeasures on all γ less than the supremum of the Suslin cardinals.

2.2.5. ADR.Over ZF, ADR is the assertion that for all A ⊆ Rω, the game �R(A) is determined.

DCR is an obvious consequence of ADR, and Woodin has shown that ADR

yields that all sets of reals are ∞-Borel. However, as far as we know, the onlyproof of ADR =⇒ AD+ uses an argument of Becker [2] for getting scales fromuniformization, and Becker’s proof uses DC. The minimal model of ADR doesnot satisfy DC, but does satisfy AD+; this requires a different argument basicallyanalysing the strength of the least place where AD + ¬AD+ could hold. Woodinhas shown from ADR + AD+ that all sets are Suslin, without appeal to Becker’sargument. At the moment, the lack of a proof (not assuming DC) that ADR =⇒AD+, and hence that ADR =⇒ all sets are Suslin , seems to be a weakness in thetheory. To make results easy to state, from here on ADR will mean ADR + AD+.

Letκ∞ = sup{κ :κ is a Suslin cardinal}.

Assuming AD,κ∞ = Θ ⇐⇒ all sets of reals are Suslin.

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Theorem 2.14 (Steel, Woodin). The following hold in ZF:

(1) AD+ DCR implies that the Suslin cardinals are closed below κ∞.(2) ADR is equivalent to AD+ κ∞ = Θ.(3) AD+ is equivalent to AD + DCR together with “the Suslin cardinals are

closed below Θ.” �

(For a sketch of the proof, see Ketchersid [20].)Thus, at least in the presence of DCR, if there is a model of AD+ ¬AD+, then

in this model κ∞ < Θ and κ∞ is not a Suslin cardinal. The main open problem inthe theory of AD is whether AD does in fact (over ZF) imply AD+.

2.2.6. L(R).It is not immediate even that L(R) |= AD → AD+. This is the content of thefollowing results:

Theorem 2.15 (Kechris [18]). Assume V = L(R) |= ZF+AD. Then DCR (andtherefore DC) holds. �

As mentioned previously, in the context of choice, it is automatic that DC holdsin L(R), regardless of whether AD does. Woodin has found a new proof of Kechris’sresult using his celebrated derived model theorem, stated in Subsection 2.3.

The basic fine structure for L(R) yields that, working in L(R), if Γ(x) is thelightface pointclass consisting of all sets of reals Σ1-definable from x, then Γ(x) =Σ2

1(x), the collection of all sets A of reals such that

y ∈ A ⇐⇒ ∃B ⊆ Rφ(B, x, y)

for some Π12 formula φ. As usual, Π2

1(x) is the collection of complements of Σ21(x)

sets, and Δ21(x) is the collection of sets that are both Σ2

1(x) and Π21(x).

Solovay’s basis theorem, see Moschovakis [25], goes further to assert that thewitnessing set can in fact be chosen to be Δ2

1(x), that is,

x ∈ A ⇐⇒ ∃B ∈ Δ21(x)φ(B, x).

In Martin-Steel [23], it is shown that, under AD, ΣL(R)1 has the scale property.

For us, this means that every set in ΣL(R)1 is Suslin. Combining these two results

gives that any ΣL(R)1 fact about a real x has a Suslin/co-Suslin witness.

Let n be as in the paragraph following Theorem 2.8. The theory ZFn resultingfrom only considering those axioms of ZF that are at most Σn sentences, is finitelyaxiomatizable.

Suppose L(R) failed to satisfy AD+. Then the following ΣL(R)1 statement holds:

∃M[R ⊆ M and M |= ZFn + ¬AD+

].

By the basis theorem together with the Martin-Steel result, the witness M canbe coded by a Suslin/co-Suslin set. Thus M ⊂ L(R) are two transitive models ofZFn+AD with the same reals, and one can check that each set of reals inM is Suslinin L(R). It follows from Theorem 2.8 that M |= AD+ and this is a contradiction.This proves:

Corollary 2.16. L(R) |= AD → AD+. �

Two results that hold for L(R) whose appropriate generalizations are relevantto our results are the fact that in L(R) every set is ordinal definable from a real,and the following:

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Theorem 2.17 (Woodin). L(R) |= ∃S ⊆ Θ(HOD = L[S]). �The set S as in Theorem 2.17 is obtained by a version of Vopenka forcing due

to Woodin that can add R to HODL(R). Variants of this forcing are very usefulat different points during the development of the AD+ theory, the general versionbeing:

Theorem 2.18 (Woodin). Suppose that AD+ holds and that V = L(P(R)).Then there is S ⊆ Θ such that HOD = L[S]. �

S can be taken to code the Σ1-theory of Θ in L(P(R)). If V = L(T,R) forsome set T ⊂ ORD, then S can be obtained by a generalization of the version ofVopenka forcing hinted at above. The stronger statement that P(R) ⊂ L(S,R) isfalse in general. For example, it implies that ADR fails, as claimed in Woodin [34,Theorem 9.22].

2.3. Obtaining models of AD+.Here we briefly discuss two methods by which (transitive, proper class) models ofAD+ (that contain al the reals) can be obtained; this illustrates that there is a wideclass of natural models to which our results apply:

2.3.1. The derived model theorem.The best understood models of AD+ come from a construction due to Woodin, thederived model theorem. In a precise sense, this is our only source of natural modelsof AD+.

The derived model theorem carries two parts, first obtaining models of determi-nacy from Woodin cardinals, and second recovering models of choice with Woodincardinals from models of determinacy. Although the full result remains unpub-lished, proofs of a weaker version can be found in Steel [31, 32] and Koellner-Woodin [21].

Theorem 2.19 (Woodin). (ZFC) Suppose δ is a limit of Woodin cardinals. LetV (R∗) be a symmetric extension of V for Coll(ω,< δ), so

R∗ =⋃

α<δ

RV [G�α]

for some Coll(ω,< δ)-generic G over V . Then:

(1) R∗ = RV (R∗), V (R∗) �|= AC, and V (R∗) |= DC iff δ is regular.(2) Define

Γ = {A ⊆ R∗ :A ∈ V (R∗) and L(A,R∗) |= AD+ }.Then L(Γ,R∗) |= AD+. �

Notice thatL(Γ,R∗) |= V = L(P(R))

and that, in particular, the theorem implies Γ �= ∅.Remark 2.20. If δ as above is singular, then R∗ � RV [G].

It is the fact that the theorem admits a converse that makes it the optimalresult of its kind, in the sense that it captures all the L(P(R))-models of AD+:

Theorem 2.21 (Woodin). Suppose V = L(P(R)) + AD+. There exists P suchthat, if G is P-generic over V then, in V [G], one can define an inner model N |=ZFC such that:

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(1) ωV1 is limit of Woodin cardinals in N .

(2) N(RV ) is a symmetric extension of N for Coll(ω,< ωV1 ).

(3) V = N(RV ). �

Remark 2.22. N is not an inner model of V . If it were, every real of V wouldbe in a set-generic extension of a (fixed) inner model of V by a forcing of size < ωV

1 .AD prevents this from happening, as it is a standard consequence of determinacythat any subset of ω1 is constructible from a real.

The point here is that to be a symmetric extension is first order, as the follow-ing well-known result of Woodin indicates (see Bagaria-Woodin [1] or Di Prisco-Todorcevic [4] for a proof):

Lemma 2.23 (Woodin). Suppose N |= ZFC, let δ be a strong limit cardinal ofN , and let σ ⊆ R. Then N(σ) is a symmetric extension of N for Coll(ω,< δ) iff

(1) Whenever x, y ∈ σ, then R ∩N [x, y] ⊆ σ,(2) Whenever x ∈ σ, then x is P-generic over N for some P ∈ N such that

|P|N < δ, and

(3) supx∈σ ωN [x]1 = δ. �

Let us again emphasize that all the models obtained using the constructiondescribed in the derived model theorem satisfy V = L(P(R)), and they also satisfyAD+.

2.3.2. Homogeneous trees.The second method we want to mention is via homogeneously Suslin representationsin the presence of large cardinals. We briefly recall the required definitions. Thekey notion of homogeneous tree was isolated independently by Kechris and Martinfrom careful examination of Martin’s proof of Π

˜11-determinacy from a measurable

cardinal.

Definition 2.24. Let 1 ≤ n ≤ m < ω. For X a set and A ⊆ Xm, let

A � n := {u � n :u ∈ A }.

Let κ be a cardinal, and let μ and ν be measures on κn and κm, respectively. Wesay that μ and ν are compatible iff

∀A ⊆ κm (A ∈ ν ⇒ A � n ∈ μ)

or, equivalently, iff B ∈ μ ⇒ {u ∈ κm :u � n ∈ B } ∈ ν.

Definition 2.25. Let T be a tree on ω × κ. We say that 〈μu :u ∈ ω<ω 〉 is ahomogeneity system for T iff

(1) For each u ∈ ω<ω, μu is an ω1-complete ultrafilter on Tu (i.e., Tu ∈ μu),(2) For each u � v ∈ ω<ω, μu and μv are compatible, and(3) For any x ∈ R, if x ∈ p[T ] and Ai ∈ μx�i for all i < ω, then there is

f : ω → κ such that ∀i (f � i ∈ Ai).

We say that T is a homogeneous tree just in case it admits a homogeneitysystem, and we say it is κ-homogeneous iff it admits a homogeneity system

⟨μu :u ∈ ω<ω

⟩

where each μu is κ-complete.

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Note that if μ is a homogeneity system for T and x /∈ p[T ] then, setting Ai =Tx�i, there is no f such that ∀i (f � i ∈ Ai). Thus, item 3 of Definition 2.25 gives acharacterization of membership in p[T ].

The key fact relating determinacy and the notion of homogeneous trees is thefollowing:

Theorem 2.26 (Martin). If A = p[T ] for some homogeneous tree T , then�ω(A) is determined. �

Definition 2.27. A set A ⊆ R is homogeneously Suslin iff there is a ho-mogeneous tree T such that A = p[T ].

A is κ-homogeneously Suslin (or κ-homogeneous) iff it is the projectionof a κ-homogeneous tree.

A is ∞-homogeneous iff it is κ-homogeneous for all κ.

For example, Π˜

11-sets are homogeneously Suslin: For any measurable κ and any

Π˜

11-set A, there is a κ-homogeneous tree T on ω × κ with A = p[T ].All the proofs of determinacy from large cardinals have actually shown that

the pointclasses in question are not just determined, but consist of homogeneouslySuslin sets. Under large cardinal hypotheses, the ∞-homogeneous sets are closedunder nice operations. For example:

Theorem 2.28 (Martin-Steel [24]). Let δ be a Woodin cardinal. Suppose thatA ⊆ R2 is δ+-homogeneous and

B = ∃R¬A := {x : ∃y ((x, y) /∈ A)}.Then B is κ-homogeneous for all κ < δ. �

This allows us to identify, from enough large cardinals, nice pointclasses Γ ⊆P(R) such that

L(Γ,R) |= AD.

In fact, although this is not a straightforward adaptation of the sketch presentedfor L(R), the arguments establishing that sets in Γ are (sufficiently) homogeneousalso allow one to show that L(Γ,R) |= AD+.

Notice that, once again,

L(Γ,R) |= V = L(P(R)).

A posteriori, it follows that these models arise by applying the derived model the-orem to a suitable forcing extension of an inner model of V .

2.4. Canonical models of AD+.AD+ is essentially about sets of reals; in particular, if AD+ holds, then it holds inL(P(R)). We informally say that models of this form are natural and note that,for investigating global consequences of AD+, these are indeed the natural innermodels to concentrate on.

There are however, other canonical inner models of AD+, typically of the formL(P(R))[X] for various nice X. Niceness here means that the models satisfy anappropriate version of condensation. For example, L(R)[μ] where μ a fine measureon Pω1

(R) which is moreover normal in the sense of Solovay [30]; or L(R)[E ] for E acoherent sequence of extenders. We will not consider these more general structuresin this paper.

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As explained in the previous subsection, the best known methods of producingmodels of determinacy actually give us models of AD+ + V = L(P(R)). Of course,not all known models of AD+ have a nice canonical form, but they are typicallyobtained from these models, for example, by going to a forcing extension, as inWoodin’s example in Kechris [18] of a model of AD+ + ¬ACω obtained by forcingover L(R).

Woodin has shown that any model of AD+ of the form L(P(R)) either satisfiesV = L(T,R) for some set T ⊂ ORD, or else it is a model of ADR; a precise statementwill be given in Theorem 3.1 below. This may help explain the hypothesis in thestatement of our results in Subsection 1.1.

2.5. The game CF (S).Scheepers [26] introduced the countable-finite game around 1991. It is a perfectinformation, ω-length, two-player game relative to a set S. We denote it by CF (S).

I O0 O1 . . .II T0 T1

At move n, player I plays On, a countable subset of S, and player II respondswith Tn, a finite subset of S.

Player II wins iff⋃

n On ⊆⋃

n Tn.Obviously, under choice, player II has a winning strategy. Scheepers [26, 27]

investigates what happens when the notion of strategy is replaced with the morerestrictive notion of k-tactic for some k < ω: As opposed to strategies, that receiveas input the whole sequence of moves made by the opponent, in a k-tactic, only theprevious k moves of the opponent are considered.

Tactics being much more restrictive, additional conditions are then imposed onthe players:

• Player I must play increasing sets: O0 � O1 � . . . .• Player II wins iff

⋃n On =

⋃n Tn.

This setting is not completely understood yet. In ZFC:

• Player I does not have a winning strategy, and therefore no winning k-tactic for any k.

• Player II does not have a winning 1-tactic for any infinite S. (Scheepers[27])

• Player II has a winning 2-tactic for S if |S| < ℵω. (Koszmider [22])• Under reasonably mild assumptions (namely, that all singular cardinalsκ of cofinality ω are strong limit cardinals and carry a very weak squaresequence in the sense of Foreman-Magidor [8]), player II has a winning2-tactic for any S. (Koszmider [22])

• It is still open whether (in ZFC) player II has a winning 2-tactic forCF (ℵω) or for CF (R).

In view of the open problems just mentioned, it is natural to consider thecountable-finite game in the absence of choice, to help clarify whether AC reallyplays a role in these problems.

This was our original motivation for showing the dichotomy Theorem 1.4, sothat we could deduce Theorem 1.8 explaining that in natural models of determinacy,the game CF (S) is undetermined for all uncountable sets S.

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3. AD+

We work in ZF for the remainder of the paper. In this section we state withoutproof some consequences of AD+ that we require.

3.1. Natural models of AD+.To help explain the hypothesis of Theorems 1.1–1.3, we recall the following result.

Given a set T ⊂ ORD, the T -degree measure μT is defined as follows: First,say that a ≤μT

b for a, b ∈ R iff a ∈ L[T, b], and let the μT -degree of a be the set ofall b such that a ≤μT

b ≤μTa.

Letting DμTdenote the set of μT degrees, we define cones and the measure

μT just as they where defined for the set D of Turing degrees in § 2.2.2. The sameproof showing that, under determinacy, μM is a measure on D gives us that μT isa measure on DμT

for all T ⊂ ORD.

Theorem 3.1 (Woodin). Assume AD++V = L(P(R)) and suppose that κ∞ <Θ. Let T ⊆ (ω×κ∞)<ω be a tree witnessing that κ∞ is Suslin. Then V = L(T ∗,R)where T ∗ =

∏x T/μT . �

This immediately gives us, via Theorem 2.14:

Corollary 3.2 (Woodin). Assume AD+ + V = L(P(R)). Then either V is amodel of ADR, or else V = L(T,R) for some T ⊂ ORD. �

On the other hand, no model of the form L(T,R) for T ⊂ ORD can be a modelof ADR.

Ultrapowers by large degree notions, as in the theorem above, will be essentialtowards establishing our result in the L(T,R) case. For models of ADR, a differentargument is required, and the following result is essential to our approach:

Theorem 3.3 (Woodin). Assume ADR + V = L(P(R)). Then

V = OD((< Θ)ω),

where (< Θ)ω =⋃

γ<Θ γω. �3.2. Closeness of codes to sets.

There are a couple of ways in which ∞-Borel codes are “close” to the sets they code.One way is expressed by Theorem 2.5 above. More relevant to us is the following:

Theorem 3.4 (Woodin). Assume AD+ + V = L(P(R)). Let T ⊂ ORD and letA ⊆ R be ODT . Then A has an ODT ∞-Borel code. �

Just as an example of how determinacy can be separated from its structuralconsequences, the preceding theorem essentially is proved by showing:

Theorem 3.5 (Woodin). Suppose that V = L(P(γ)) |= ZF + DC and that μis a fine measure on Pω1

(P(γ)) in V . Then, for all T ⊂ ORD and A ⊆ R, ifA ∈ ODT,μ, then A is ∞-Borel and has an ∞-Borel code in HODT,μ. �

Fact 3.6. Under AD, there is an OD measure on Pω1(P(γ)) for all γ < Θ. �

As a corollary, if AD+ holds, V = L(P(γ)) for γ < Θ, and A ∈ ODT ∩ P(R),then A has an ODT ∞-Borel code.

This almost gives Theorem 3.4 since, assuming AD+ + V = L(P(R)), we haveV = L(

⋃γ<Θ P(γ)).

On the other hand, note that Theorem 3.4 is not immediate from Theorem 1.9,even if V = L(S,R).

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3.3. A countable pairing function on the Wadge degrees.Our original approach to the dichotomy Theorem 1.2 required the additional as-sumption that cf(Θ) > ω. Both when trying to generalize this approach to thecase cf(Θ) = ω, and while establishing Theorem 1.8 on the countable-finite gamein general, an issue we had to face was whether countable choice for finite sets ofreals could fail in a model of ADR.

That this is not the case follows from the existence of a pairing function. SteveJackson found (in ZF) an example of such a function. Although this is no longerrelevant to our argument, we believe the result is interesting in its own right. Belowis Jackson’s construction.

Theorem 3.7 (Jackson). In ZF, there is a function

F : P(R)× P(R) → P(R)

satisfying:

(1) F (A,B) = F (B,A) for all pairs (A,B), and(2) Both A and B Wadge-reduce to F (A,B).

Proof. If A = B, simply set F (A,B) = A. If A ⊆ B or B ⊆ A, set F (A,B) =(0 ∗ S) ∪ (1 ∗ T ) where S is the smaller of A,B, and T is the larger. Here, 0 ∗ S ={0a : a ∈ S} and similarly for 1 ∗ T .

If A \B and B \A are both non-empty, we proceed as follows:Let X(A,B) ⊆ RZ be defined by saying that, if f : Z → R, then f ∈ X(A,B)

iff there is an i such that f(i) ∈ A \ B (or B \ A) and, for each j, f(j) ∈ A if|j − i| is even, and f(j) ∈ B if |j − i| is odd (and reverse the roles of A,B here iff(i) ∈ B \A).

The set X(A,B) is an invariant set (with respect to the shift action of Z onRZ), and X(A,B) = X(B,A). (Thus the points of A \ B and B \A have to occurat places of different parity; while points of A ∩B can occur anywhere.)

Given X(A,B), we can compute A (and also B) as follows: Fix z ∈ A \ B.Then x ∈ A iff

∃f ∈ X(A,B) ∃i ∃j (f(i) = z and f(j) = x and |j − i| is even).This shows that A is Σ

˜

11(X(A,B)). If we replace X(A,B) with X ′(A,B), the Σ1

1-jump of X(A,B), then A is Wadge reducible to X(A,B). Finally, we use that thereis a Borel bijection between RZ and R, and define F (A,B) as the image of X ′(A,B)under this map. �

As pointed out by Jackson, essentially the same argument shows the following;recall that ACω(R) is a straightforward consequence of determinacy, so Theorem3.8 applies in models of AD:

Theorem 3.8 (Jackson). Assuming ZF+ACω(R), there is a countable pairingfunction.

Proof. Let 〈Ai : i ∈ ω〉 be a sequence of distinct sets of reals. Call f ∈ (R×2)ω

n-honest iff, whenever f(i) = (x, k), then

k = 1 ⇐⇒ x ∈ An,

so f is n-honest iff it is a countable approximation to the characteristic function ofAn. Let

B = {f : ∃n (f is n-honest)}.

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Clearly, B does not depend on the ordering of the sets Ai.Let 〈xi : i ∈ ω〉 be a sequence of elements of

⋃n An such that for each i �= j,

there is k with xk ∈ Ai�Aj . That there is such a sequence of reals follows fromACω(R).

Let gn(2k) = (xk, 1) if xk ∈ An and gn(2k) = (xk, 0) otherwise. Then:

• gn is the even part of an n-honest function,• gn cannot be the even part of a j-honest function for j �= n,• x ∈ An ⇔ ∃f ∈ B

(f ⊃ gn and ∃k (gn(k) = (x, 1))

).

This shows that An is Σ˜

11 in B. �

As a consequence, it follows that for no λ < Θ there is a sequence

〈Aγ : γ < λ〉such that each Aγ is a countable subset of P(R) and

⋃γ<λ Aγ is cofinal in the

Wadge degrees. This is trivial when Θ is regular, but does not seem to be when Θis singular. Essentially because of this obstacle, the argument for Theorem 1.2 inthe ADR case is different from the argument in the V = L(T,R) case.

4. The dichotomy theorem

Our goal is to establish the dichotomy Theorem 1.4. Our argument utilizesideas originally due to H. Woodin.

Before we begin, a few words are in order about the way the result came to be.We first proved the dichotomy for models where V = L(T,R) for T ⊂ ORD, andfor models of ADR of the form L(P(R)) where cf(Θ) > ω. For the general case,we only succeeded in showing the undeterminacy of the games CF (S), the mainadditional tool in the ADR case being Theorem 3.3. A key suggestion of Woodinallowed the argument for the dichotomy to be extended to this case as well. Thenew idea was the weaving together of different well-orderings using the uniquenessof the supercompactness measures for Pω1

(γ) as γ varies below Θ.

4.1. The L(T,R) case.We work throughout under the base theory

(BT) ZF+ DCR + μ is a fine σ-complete measure on Pω1(R).

It follows from DCR that L(T,R) |= DC for all T ⊂ ORD. So, when working insidemodels of the form L(T,R), we may freely use DC. In particular, ultrapowers ofwell-founded models are well-founded. Below, whenever we refer to L(T,R), HODS ,etc., we will tacitly assume that T, S ⊂ ORD.

For any X ∈ L(T,R), there is an r ∈ R such that X ∈ ODL(T,R)T,r . For α ∈ ORD,

and ϕ a formula, let Xϕ,α consist of those elements x of X such that, in L(T,R),for some real t, x is the unique v such that ϕ(v, T, r, α, t). If |R| ≤ |Xϕ,α|, then weare done, so suppose |R| � |Xϕ,α| for all ϕ and α.

Define a map from R onto Xϕ,α ∪ {∅} by setting xt to be the ODL(T,R)T,r -least

element of X definable from T, r, α, and t, via ϕ, if such an element exists, andotherwise xt = ∅. Let

t Eϕ,α t′ ⇐⇒ xt = xt′ ,

so Eϕ,α is an ODL(T,R)T,r equivalence relation on R. Clearly, the map

φ : R/Eϕ,α1-1−−→onto

Xϕ,α ∪ {∅}

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sending the class of t ∈ R to xt, is ODL(T,R)T,r . Thus if we show that R/Eϕ,α ⊂

ODT,r,μ, then it follows thatXϕ,α ⊂ ODT,r,μ.

Consequently, X ⊂ ODT,r,μ and so, clearly, X is well-orderable.

Definition 4.1. An equivalence relation E on R in thin iff R �� R/E. Other-wise, E is thick.

The theorem we prove is:

Theorem 4.2. Assume BT, and suppose that E is an ODL(T,R)T,r thin equivalence

relation. Then R/E ⊂ ODT,r,μ.

4.1.1. The extent of ∞-Borel sets.The proof goes through an analysis of ∞-Borel sets.

Here we show that, assuming BT, every A ⊆ R in L(S,R) is ∞-Borel. To showthis, it suffices to show that the ∞-Borel sets are closed under ∃R. Once this hasbeen established, the result follows by induction over the levels Lα(S,R) and, foreach such level, by induction in the complexity of the definitions of new sets ofreals.

Remark 4.3. It is clear that, in L(S,R), every set comes with a description ofhow to build that set using well-ordered unions, negations, and the quantifier “∃R”.

That every A ⊆ R in L(S,R) actually admits an ODS,μ ∞-Borel code requiresan additional argument, since it is not clear that ∞-Borel sets are closed underwell-ordered unions, due to an inability to uniformly pick codes. We omit thisadditional argument since it would take us too far from our intended goal.

There are in general many descriptions attached to a single set, but the pointis that to each description for a set of reals we can attach an ∞-Borel code so longas we have a way to pass from an ∞-Borel code for AS to one for ∃RAS .

Notice that we are not claiming that L(S,R) thinks that every set is ∞-Borel;in particular, ∃RS (see below) might not be in L(S,R). One would need μ to be inL(S,R) to get that all sets in L(S,R) admit ∞-Borel codes in L(S,R). This is thecase under AD where μ = μM is Martin’s measure.

We now explain how to pass from an ∞-Borel code S for A to an ∞-Borel codefor ∃RAS which we call ∃RS. The map S �→ ∃RS is ODS,μ.

If μS = πS(μ) where πS : Pω1(R) → Pω1

(R) is defined by

πS(σ) = R ∩ L(S, σ),

then μS is a fine measure. That R ∩ L(S, σ) is countable is a consequence of thefollowing discussion, since σ ∈ L[S, x] for some real x.

Let κ = ωV1 , and note that κ is measurable in V since, defining π : Pω1

(R) → ω1

byσ �→ sup

x∈σωck1 (x),

then ν = π(μ) is a σ-complete (hence κ-complete) measure on κ. It is clear that νis non-principal, so κ is indeed measurable.

The fact that κ is measurable in V yields that κ is (strongly) Mahlo in everyinner model of choice. To see this, let N be any class of ordinals coding themembership relation of a well-ordered transitive model of choice. Clearly,

HODN,ν ⊆ HODN,μ,

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and κ is measurable in HODN,ν . Since this is a model of choice, and N ⊂ HODN,ν ,

the model N coded by N satisfies N |= κ is Mahlo.Now let

S = {γ < κ : γ is N -inaccessible},and note that if S is N -non-stationary, then S /∈ ν. In particular, each πS(σ) iscountable, as claimed.

For any σ ∈ Pω1(R), let

HσS = HOD

L(S,σ)S ,

and let κσS be the least inaccessible δ of Hσ

S such that δ ≥ ΘL(S,σ). Define ∼σS on

the set BCσS of ∞-Borel codes of Hσ

S , as follows: For T, T′ ∈ BCσ

S , set

T ∼σS T ′ ⇐⇒ (AT = AT ′)L(S,σ).

Let

QσS = BCσ

S/ ∼σS .

QσS is like the Vopenka algebra of L(S, σ), except that OD ∞-Borel sets are used

in place of OD subsets of R. This is made clear by the following lemma whose easyproof we leave to the reader:

Lemma 4.4. For x ∈ L(S, σ), let

GσS(x) = {b ∈ Qσ

S :x ∈ (Ab)L(S,σ)}.

Then GσS(x) is Hσ

S -generic, and

HσS [x] = Hσ

S [GσS(x)].

Moreover, for any b ∈ QσS with b �= 0Qσ

S, there is x ∈ L(S, σ) with b ∈ Gσ

S(x). �

For κ a cardinal of HσS , let BC

σκ,S denote the set BCκ in the sense of Hσ

S . Nowset

QσS = BCσ

κσS,S/ ∼σ

S .

In HσS , Q

σS is κσ

S-cc (in fact, ΘL(S,σ)-cc) since, otherwise, there would be a sequence〈bα :α < κσ

S〉 of non-zero and incompatible elements in QσS . But then, in L(S, σ),

〈Abα :α < κσS〉 would give a pre-well-order of RL(S,σ) of length ≥ ΘL(S,σ).

Since κσS is regular, Qσ

S is κσS-cc and κσ

S-complete, and therefore QσS is complete.

So QσS = Qσ

S and we may identify QσS with a subset of κσ

S in HσS .

Since κσS is inaccessible and Qσ

S is κσS-cc, we have a canonical enumeration

DσS = 〈Dσ

S,α :α < κσS〉

of maximal antichains of QσS in Hσ

S . In fact, we enumerate every sequence

〈Tγ : γ < α〉from BCσ

κσS,S that becomes such an antichain upon moding out by ∼σ

S .

Again, DσS can be coded in a canonical way by a subset of κσ

S in HσS . Let bσS

be the “minimal” element of BCσκσS,S such that bσS ∼σ

S S, and define Sσ as∧

α<κσS

∧

T,T ′∈DσS,α

¬(T ∧ T ′) ∧ bσS ∧∧

α<κσS

∨Dσ

S,α.

Modulo ∼σS , S

σ is just bσS , but before passing to the quotient, we have:

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Lemma 4.5. For any real x (anywhere)

x ∈ ASσ ⇐⇒ x is HσS -generic over Qσ

S and HσS [x] |= x ∈ AS .

Proof. Suppose x ∈ ASσ , and define

GσS(x) = {b ∈ Qσ

S :∃α ∃T ∈ DσS,α(x ∈ T and b ∼σ

S T )}.Clearly, Gσ

S(x) meets every antichain of QσS in Hσ

S . If T, T ′ ∈ GσS(x), then T, T ′

are compatible in QσS , since otherwise there is some α with T, T ′ in Dσ

S,α, but

Sσ explicitly precludes x from being in two distinct elements of DσS . So Gσ

S(x) isHσ

S -generic.

Now, (HσS )

QσS |= “x ∈ Abσ

S⇐⇒ x ∈ AS” since this holds for all x ∈ L(S, σ). It

follows that

HσS [x] |= “x ∈ Abσ

S⇐⇒ x ∈ AS”

and, by choice of Sσ, HσS [x] |= x ∈ Abσ

Sand thus Hσ

S [x] |= x ∈ AS . This finishesthe left-to-right direction. The converse is easier. �

So, whereas AbσSonly needs to agree with AS on reals of L(S, σ), Sσ has a very

strong agreement with AS, extending even to reals in outer models of V .We are now in a position to establish Woodin’s Theorem 1.9 that, arguing in

BT, A is ∞-Borel iff A ∈ L(S,R), for some S ⊂ ORD. This follows immediatelyfrom the following:

Lemma 4.6. Assume BT and let S ⊂ ORD be an ∞-Borel code for a subset ofR2. Then there is a canonical ∞-Borel code ∃RS such that

∃y((x, y) ∈ AS

)⇐⇒ x ∈ A∃RS .

Proof. The point is that

∃y((x, y) ∈ AS

)⇐⇒ for μ-a.e. σ,

(Hσ

S [x])Coll(ω,κσ

S) |= ∃y ASσ(x, y).

In the right-to-left direction, fix in V a Coll(ω, κσS)-generic g over Hσ

S [x] such that

HσS [x][g] |= ∃y

((x, y) ∈ ASσ

).

Since (x, y) ∈ ASσ , then HσS [x, y] |= (x, y) ∈ AS , by the previous lemma. So

(x, y) ∈ AS and hence ∃y((x, y) ∈ AS

).

For the left-to-right direction, just fix y so that AS(x, y), and take any σ withx, y ∈ σ. Then (x, y) is Qσ

S-generic over HσS , and hence satisfies Sσ. It is a Σ1

1(x, b)statement about any real coding Sσ that there is a real y such that (x, y) ∈ ASσ .Thus there is such a real in Hσ

S [x][g] for any g enumerating Sσ.It should be noted that we do not need to use all of Hσ

S above. Instead, wecould work with L[Sσ], that is (letting ∀∗μ abbreviate “for μ-a.e.”)

∃y((x, y) ∈ AS

)⇐⇒ ∀∗μσ, L[Sσ, x]Coll(ω,κσ

S) |= ∃y((x, y) ∈ ASσ

).

Set

L[S∞, x] =∏

σ

L[Sσ, x]/μ.

Then

∃y((x, y) ∈ AS

)⇐⇒ L[S∞, x] |= ∃y

((x, y) ∈ AS∞

)⇐⇒ L[S∞, x] |= ϕ(S∞, x),

so (ϕ, S∞) “is” the ∞-Borel code ∃RS. �

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Notice that we actually showed that from a description dA of how to build aset of reals in L(S,R) we can canonically pass to an ∞-Borel code SdA

associated

to that description. A and dA are in L(S,R), and in fact ODL(S,R)S,t , while SdA

is inV , and in fact ODS,t,μ. This clearly generalizes so that, given a sequence of sets of

reals �A = 〈Aα :α < γ〉 ∈ L(S,R) and an associated description d �A ∈ ODL(S,R)S,t , we

produce a corresponding sequence �S of ∞-Borel codes, with �S ∈ ODS,t,μ.

Remark 4.7. This argument should illustrate the general technique behindour approach and, really, behind many applications of determinacy that rely on∞-Borel sets. Namely, the “localization” of ∞-Borel sets we established allows oneto argue about them as if they were actually Borel sets, and then lift the results viaabsoluteness. The proofs of Theorems 1.2–1.4 are further illustrations of this idea.

4.1.2. The first dichotomy.

Theorem 4.8. Assume BT. Then, for every X ∈ L(T,R), if |R| � |X|, thenX ⊂ ODT,t,μ for some t ∈ R.

For X ∈ L(T,R), X is ODL(T,R)T,t for some t ∈ R. The conclusion of Theorem

4.8 could be strengthened to X ∈ ODT,t,μ for any t ∈ R such that X ∈ ODL(T,R)T,t .

First, we make a useful reduction to equivalence relations on reals. For X ∈OD

L(T,R)T,t and α ∈ ORD, let Xα be the collection of elements of X definable in

L(T,R) from α and a real. Take γ so thatX =⋃

α<γ Xα. To eachXα we can canon-

ically associate an equivalence relation Eα on R and a bijection φα : R/Eα1-1−−→onto

Xα

with φα, Eα ∈ ODL(T,R)T,t . We have that 〈Eα :α < γ〉 is an OD

L(T,R)T,t -sequence of

sets of reals and so, by the comment at the end of § 4.1.1, we get a sequence�S = 〈Sα :α < γ〉 of ∞-Borel codes with �S ∈ ODT,t,μ.

Theorem 4.9. Assume BT. If E is a thin ∞-Borel equivalence relation withcode S, then R/E ⊂ ODS,μ.

This will complete the argument: If |R| � Xα for all α, then R/Eα ⊂ ODSα,μ ⊆ODT,t,μ. So Xα ⊂ ODT,t,μ for all α < γ and hence X ⊂ ODT,t,μ, as claimed.

Proof. Fix an ∞-Borel code S for a thin equivalence relation E. We will

use the previously established notation: HσS = HOD

L(S,σ)S , Qσ

S , etc. Let H∞S be

the ultrapower of HσS and, similarly, define Q∞

S , B∞, etc. It is clear, using Los’stheorem, that the following hold:

• Every real in V is Q∞S -generic over H∞

S , since

∀∗μσ (x is QσS-generic over Hσ

S ).

• Similarly, for T, T ′ ∈ Q∞S ,

T ∼∞S T ′ ⇐⇒ (AT = AT ′)V ,

so Q∞S is a subalgebra of B∞.

Write b∞S for the ultrapower of the codes bσS , Eb∞S

for Ab∞S, and define

W∞S = {p ∈ Q∞

S : (p, p) �H∞S

Q∞S

×Q∞S

r0 Eb∞Sr1}.

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IfW∞S is dense, then every x ∈ R is in Ap for some p ∈ W∞

S , and clearly |Ap/E| = 1.We say that p captures the E-class x/E if |Ap/E| = 1 and Ap ∩ x/E �= ∅. By ourassumption on W∞

S , all E-classes are captured, and we can define

φS : R/E → Q∞S ⊆ κ∞

S

by letting φS(x/E) be the least member of Q∞s that captures x/E. This is clearly

ODS,μ.If W∞

S is not dense then, by Los’s theorem, we can find a μ-measure one set ofσ on which this fact is true of W σ

S . Fix σ and p ∈ QσS such that (writing ESσ for

ASσ )

∀p′ ≤QσSp ∃p0, p1 ≤Q

σSp′ Hσ

S |= “(p0, p1) � r0 �ESσ r1.”

We can enumerate (in V ) the dense subsets of QσS in Hσ

S , and use the above tobuild a tree of conditions ps, s ∈ 2<ω, so that for each f ∈ 2ω, Gf = {pf�i : i ∈ ω}generates a generic filter for Hσ

S with corresponding real rf (in V ) such that

HσS [rf , rf ′ ] |= rf �ESσ r′f .

Recall that E = AS , and Sσ has the property that ESσ = E on reals QσS-generic

over HσS . Thus we have that rf �E rf ′ for f, f ′ ∈ 2ω with f �= f ′. This shows that

E is not thin. �

4.2. The main theorem for L(T,R).Now we indicate how to generalize Theorem 4.8 to obtain Theorem 1.2 when V =L(T,R) for T ⊂ ORD. As in the proof of Theorem 4.8, we reduce to the case of an∞-Borel code S whose interpretation ≤S= AS is a pre-partial ordering on R, andone needs only modify the definition of W∞

S . The relevant set becomes

W∞S = {p ∈ Q∞

S : (p, p) �H∞S

Q∞S

×Q∞S

r0 ≤b∞S

r1 or r1 ≤b∞S

r0},

where ≤b∞S= Ab∞

S. If W∞

S is not dense, just as before, we can find a copy of 2ω

consisting of ≤S-pairwise incomparable elements. If the set is dense, then Ap is apre-chain for p ∈ W∞

S , and every x ∈ R is in Ap for some such p.

4.3. The ADR case.Assume AD+ and V = L(P(R)) yet V �= L(T,R) for any T ⊂ ORD. We begin byexplaining how to obtain Theorem 1.4. As mentioned previously, the argument inthis case was suggested by Woodin.

Given X, find some γ < Θ and s0 ∈ γω such that X ∈ ODs0 . This is possible,by Theorem 3.3.

The key idea is to define, for σ ∈ [< Θ]ω,

Xσ,α = {a ∈ X :∃t ∈ R (a is definable from σ, s0, α, t)}.The reason for relativizing to σ will become apparent soon. Notice that if σ ⊆ τand a ∈ ODσ,s0,t for some t, then there is t′ ∈ R so that a ∈ ODτ,s0,t′ .

Let Eσ,α be the equivalence relation on R induced by Xσ,α. If any Eσ,α is thick,then we are done. Otherwise, uniformly in α, there is an ODσ,s0 ∞-Borel code Sσ,α

for Eσ,α and a corresponding φσ,α : R/Eσ,α → γα inducing Eσ,α.In particular (by the argument for the previous case) Xσ,α ⊂ ODσ,s0 and thus

Xσ ⊂ ODσ,s0 , where Xσ =⋃

α Xσ,α. Let <σ be the ODσ,s0 well-order of Xσ.For each ξ < Θ let Xξ =

⋃σ∈Pω1

(ξ) Xσ, and notice that Xξ ⊆ Xξ′ whenever

ξ < ξ′.

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Woodin’s main observation here is that the supercompactness measures can beused to uniformly well-order the sets Xξ and hence to obtain a well-order of X.Namely, set

a <ξ a′ ⇐⇒ ∀∗μξσ [a <σ a′].

This shows that Xξ ⊂ ODs0 and hence X ⊂ ODs0 . So X is well-orderable.This argument can be easily modified so we also obtain Theorem 1.2. Namely,

from the previous subsection, we can assume each ≤� Xξ is a well-ordered unionof pre-chains; this is uniform in ξ, and just as before we use the supercompactnessmeasures to obtain that ≤ itself is a well-ordered union of pre-chains.

4.4. The E0-dichotomy.Finally, we very briefly sketch how to prove Theorem 1.3. The argument in Hjorth[13] greatly resembles the construction in Harrington-Kechris-Louveau [9] and theproof above, and we only indicate the required additions, and leave the details tothe interested reader. For a more general result, see Caicedo-Ketchersid [3].

Assume AD+ and that V = L(T,R) for some T ⊂ ORD, or else V = L(P(R)).Let (X,≤) be a partially ordered set. First, the techniques above and Theorem2.5 of Hjorth [13] generalize straightforwardly to give us that, if X is a quotient of2ω by an equivalence relation E, then either there is an injection of 2ω/E0 into Xwhose image consists of pairwise ≤-incomparable elements, or else for some α thereis a sequence

(Aβ :β < α)

such that for any x, y ∈ R,

[x]E ≤ [y]E ⇐⇒ ∀β < α (x ∈ Aβ → y ∈ Aβ).

For this, just vary slightly the definition of A(�f�μ) in page 1202 of Hjorth [13].For example, using notation as in Hjorth [13], in L(R), we would now set A(�f�μ)as the set of those y for which there is an x0 such that (letting ≤T denote Turing

reducibility) for all x ≥T x0, letting A be the f(x)-th ODL[S,x]S subset of (2ω)L[S,x],

then for all ρ, if [ρ]E ≥ [y]E , if [ρ]E ∩ L[S, x] �= ∅, then [ρ]E ∩ A �= ∅. Thisstraightforwardly generalizes to the L(T,R) setting, under BT.

A similar adjustment is then needed in the definition of the embedding of E0

into E to ensure that points in the range are ≤-incomparable.(See also Foreman [7] for a proof from ADR under a slightly stronger assump-

tion; this approach can be transformed into a proof from AD+ of Foreman’s result,by using the AD+-version of Solovay’s basis theorem mentioned in page 14. Otherproofs are also possible.)

Using this, Theorem 1.3 follows immediately, first for models of the formL(T,R), just as in Theorem 2.6 of Hjorth [13], and then for models of ADR us-ing the ‘weaving together’ technique from the previous subsection.

5. The countable-finite game in natural models of AD+

In this section we work in ZF and prove Theorem 1.8. We are interested in thecountable-finite game in the absence of choice; here are some obvious observations:

Fact 5.1 (ZF). Player II has a winning strategy in CF (S) whenever S iscountable or Dedekind-finite.

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Proof. This is obvious if S is countable. Recall that S is Dedekind-finite iffω �� S. It follows that if S is Dedekind-finite, then each move of player I must bea finite set. �

Fact 5.2 (ZF). Assume every uncountable set admits an uncountable linearlyorderable subset.

Given a set S, player I has a winning strategy in CF (S) iff some uncountablesubset of S is the countable union of countable sets.

Proof. Suppose first that S admits an uncountable subset that can be writtenas a countable union of countable sets. We may as well assume that S itself admitssuch a representation, and that S is linearly orderable. It suffices to show that anycountable union of finite subsets of S is countable. For this, let < linearly order S,and let (Sn :n ∈ ω) be a sequence of finite subsets of S. We may as well assumethey are pairwise disjoint. We can then enumerate their union S∗ =

⋃n Sn by

listing the elements of each Sn in the order given by <, and listing the elements ofSn before those of Sm whenever n < m. This gives an ordering of S∗ in order typeat most ω.

Conversely, suppose any countable union of countable subsets of S is countable,and let F be a strategy for player I. Define a sequence (Cn)n∈ω of subsets of S asfollows:

• C0 = F (〈〉),• For n > 0, Cn =

⋃{xi : i<n}⊆

⋃i<n

CiF (〈{xi} : i < n〉).

By induction, each Cn is countable, and therefore so is⋃

n Cn. Using an enu-meration of this last set, it is straightforward for player II to win a run of CF (S)(by playing singletons) against player I following F . It follows that F is not winningfor player I. �

From the argument above, we see that it is consistent that player I has a winningstrategy in CF (S) for some S. For example, player I has a winning strategy inCF (ω1) whenever cf(ω1) = ω, and in CF (R) in the model of Feferman-Levy [6],where R is a countable union of countable sets.

From now on, assume that AD+ holds and that V = L(T,R) for some T ⊂ ORD,or V = L(P(R)). The dichotomy Theorem 1.4 immediately gives the basis theoremfor cardinalities, Corollary 1.5.

It follows that there are no infinite Dedekind-finite sets, and that (since ω1 isregular) any countable union of countable sets is countable.

By Fact 5.2, we now have:

Corollary 5.3. Assume that AD+ holds and that V = L(T,R) for someT ⊂ ORD, or V = L(P(R)). Then, for no set S, player I has a winning strategy inCF (S). �

It remains to study when player II has a winning strategy in CF (S). We mayassume that S is uncountable, and analyze the two possibilities ω1 � S and R � Sseparately.

Lemma 5.4 (ZF). Assume ω1 �� R. If ω1 � S, then player II has no winningstrategy in CF (S).

Recall that AD implies that ω1 �� R.

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Proof. From a winning strategy F for player II, we can find enumerationsof all countable ordinals: Without loss, ω1 ⊆ S. Consider the run of the gamewhere player I plays α, α+ 1, α+ 2, . . . . Then α is covered by the finite subsets ofα that player II plays by turns following F , and these finite sets provide us withan enumeration of α in order type ω. But it is trivial to turn such a sequence ofenumerations into an injective ω1-sequence of reals. �

Lemma 5.5 (ZF). Assume ACω(R) and that there is a fine measure on Pω1(R).

If R � S then player II has no winning strategy in CF (S).

AD implies both that ACω(R) holds, and that there is such a measure; thelatter can be obtained, for example, by lifting either Solovay’s club measure on ω1,or Martin’s cone measure on the Turing degrees.

Proof. We may assume S = R. Assume player II has a winning strategy F .Fix a fine measure μ on Pω1

(R). We find a μ-measure one set C such that playerII always plays the same (following F ) for any valid play of player I using membersof C. Since C is uncountable, this shows that player I can defeat F , contradiction.

Notice that we can identify Pω(R) with R. Using the σ-completeness of μ, thereis a measure 1 set A0 and a fixed finite set T0 such that for all σ ∈ A0, F (〈σ〉) = T0.To see this, notice that (identifying T0 with a real) for each i ∈ ω there is a uniqueji ∈ ω and a measure 1 set Ai

0 such that if σ ∈ Ai0 then F (σ)(i) = ji, and we can

set A0 =⋂

i Ai0.

Similarly, there is a measure 1 set A1 ⊆ A0 and a fixed finite set T1 such thatfor all σ, σ′ ∈ A1 with σ′ ⊇ σ, F (〈σ, σ′〉) = T1.

Continue this way to define sets A0, A1, . . . and finite sets T0, T1, . . . . LetA =

⋂i Ai. Then A has measure 1. In particular,

⋃A is uncountable. However,

for any σ0 ⊆ σ1 ⊆ . . . with all the σi in A, F (〈σ0, . . . , σi〉) = Ti. Since⋃

i Ti

is countable, we can find r, σ with r ∈ σ, σ ∈ A, r /∈⋃

i Ti, and from this itis straightforward to construct a run of CF (R) where player I defeats player IIfollowing F , and so F was not winning after all. �

From the basis theorem, Corollary 1.5, we now have:

Corollary 5.6. Assume that AD+ holds and that V = L(T,R) for someT ⊂ ORD, or V = L(P(R)). Then, for no uncountable set S, player II has awinning strategy in CF (S). �

Combining this with Corollary 5.3, Theorem 1.8 follows immediately.

6. Questions

Recall that the main step of the proof of the dichotomy Theorem 1.2 consists ofpassing from an ∞-Borel code S to a local version Sσ that correctly computes AS

on suitable inner models Nσ that satisfy choice and, moreover, this computation ispreserved by passing to forcing extensions of Nσ.

Question 6.1. Does our analysis extend to models of the form L(P(R))[X]for sets X that satisfy some appropriate form of condensation, so that Theorem 1.2holds for these models as well?

Vaguely, the point is that condensation might provide enough absoluteness ofthe structure so that the process of passing to countable structures and then takingan ultrapower produces the appropriate ∞-Borel codes.

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In a different direction, one can ask:

Question 6.2. To what extent can we recover the local bounds on the wit-nessing ordinals known previously in particular cases of Theorems 1.2–1.4?

For example, it is not too difficult to combine our analysis with known tech-niques, to see that, as in Harrington-Marker-Shelah [10], a thin Borel partial orderis a countable union of chains, or that quotients of R by projective equivalencerelations can be well-ordered in type less than δ

˜

1n for an appropriate n, as shown

in Harrington-Sami [11]. But it seems that, in general, the passing to ultrapowersblows up the bounds beyond their expected values. What we are asking, then, isfor a quantitative difference between κ-Borel sets and κ-Suslin sets, expressed interms of some cardinal associated to κ.

Let c = |R|. Under determinacy, ω1 + c is an immediate successor of c. It isa known consequence of ADR (probably going back to Ditzen [5]) that |2ω/E0| isalso an immediate successor of c; in fact, any cardinal strictly below |2ω/E0| injectsinto c. We have proved this result under AD+, see Caicedo-Ketchersid [3].

Let

S1 ={a ∈ Pω1

(ω1) : sup(a) = ωL[a]1

}.

In Woodin [35] it is shown, under ZF+DC+ADR, that |S1| is yet another immediatesuccessor of c.

On the other hand, in ZF + AD+ + ¬ADR, Woodin [35] shows that there isat least one cardinal intermediate between c and |S1|, and there is also at leastone cardinal intermediate between c and c · ω1 incomparable with ω1. We haveshown that this cardinal turns out to be an immediate successor of c, but we do notknow of a complete classification of immediate successors of c under our workingassumptions, or whether this is even possible.

Question 6.3. Is it possible to classify, under AD+ + V = L(P(R)), the im-mediate successors of |R|?

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Department of Mathematics, Boise State University, 1910 University Drive, Boise,

ID 83725-1555

URL: http://math.boisestate.edu/∼caicedo/E-mail address: [email protected]

Miami University, Department of Mathematics, Oxford, OH 45056

URL: http://unixgen.muohio.edu/∼ketchero/E-mail address: [email protected]

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The coinitialities of Efimov spaces

Stefan Geschke

Abstract. We use ♦ to construct Efimov spaces of countable and uncountablecoinitiality, showing that at least consistently there are compact spaces ofuncountable cofinality without uncountable dyadic families.

1. Introduction

In [9] Koppelberg defined the cofinality cof(B) of an infinite Boolean algebraB to be the least infinite cardinal κ such that there is a strictly increasing sequence(Bα)α<κ of subalgebras of B with B =

⋃α<κ Bα. (See also [5].) Observe that we

can equivalently define the cofinality of an infinite Boolean algebra B to be theleast limit ordinal δ such that B is the union of an increasing chain of length δ ofproper subalgebras.

Clearly, the cofinality of an infinite Boolean algebra B is an regular cardinalbounded by the size of B. If C is an infinite quotient of B, then cof(B) ≤ cof(C).Koppelberg showed that P(ω), and in fact every infinite complete Boolean algebra,has cofinality ℵ1. Moreover, since every infinite Boolean algebra has an infinitequotient of size ≤ 2ℵ0 , there is no Boolean algebra whose cofinality exceeds 2ℵ0 .Koppelberg asked whether there can be any Boolean algebra with a cofinality > ℵ1.

Let us call a Boolean algebra of cofinality > ℵ1 a Koppelberg algebra. If Bis Koppelberg, then it cannot have an infinite countable quotient or an infinitecomplete quotient. Now by Stone duality, the Stone space Ult(B) cannot containa nontrivial converging sequence or a copy of βω. In other words, Ult(B) has to bean Efimov space. The consistency of the existence of an Efimov space was shownby Fedorchuk [4]. Nowadays, various constructions of Efimov spaces are available.However, none of these constructions seems to have the potential of producing theStone space of a Koppelberg algebra.

On the other hand, CH implies that there are no Koppelberg algebras. It is alsoknown that there are models of set theory in which 2ℵ0 is large but every Booleanalgebra has cofinality ≤ ℵ1. (See for example [10], [8] and [7]).

2000 Mathematics Subject Classification. Primary: 03E35, 54A25; Secondary: 06E05, 06E15,46L05.

The author was supported by NSF grant DMS 0801189.


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259

2 STEFAN GESCHKE

2. Coinitialities of compact spaces

By Stone duality, the cofinality of a Boolean algebra can be expressed in termsof inverse limit representations of the Stone space of the algebra. This yields thenotion of coinitiality of a topological space. We use the notation in [6] for inversesystems.

Definition 2.1. Let X be a topological space. The coinitiality ci(X) of X isthe least limit ordinal δ such X is the limit of an inverse system {Xα, π

βα, δ} whose

bonding maps πβα are onto and not 1-1, provided such an inverse system exists.

For every Boolean space X, i.e., if X is the Stone space of a Boolean algebraA, one can show that ci(X) = cof(A). The fundamental facts about the cardinalinvariant ci(X) are summarized in the following lemma. The easy proofs can befound in [7].

Lemma 2.2. (a) If X is an infinite closed subspace of a compact space Y , thenci(Y ) ≤ ci(X).

(b) For every infinite compact space X, ci(X) ≤ cf(w(X)) where w(X) denotesthe weight of X.

(c) For every infinite compact space X, ci(X) ≤ 2ℵ0 .

Countable coinitiality can be characterized in terms of double sequences.

Definition 2.3. Let X be a compact space and let (xn)n∈ω be a discretesequence in X. Then (xn)n∈ω is a double sequence if for all free ultrafilters p overω, the p-limits of (x2n)n∈ω and (x2n+1)n∈ω are the same.

Theorem 2.4 (See [7]). Let X be an infinite compact space. Then ci(X) = ℵ0

if and only if X contains a double sequence.

Since direct limits are often easier to handle than inverse limits, it is sometimesconvenient to compute instead of the coinitiality of a space X the cofinality of theC∗-algebra C(X) of continuous functions from X to C.

Definition 2.5. Let A be an infinite dimensional C∗-algebra. Then the cofi-nality cof(A) of A is the least limit ordinal δ such that there is a strictly increasingchain (Aα)α<δ of closed ∗-subalgebras of A such that

⋃α<δ Aα is dense in A.

Observe that since a C∗-algebra A is a metric space, if (Aα)α<δ is an increasingchain of closed ∗-subalgebras of A and δ is an ordinal of uncountable cofinality, then⋃

α<δ Aα is a closed ∗-subalgebra of A. Thus, if cf(δ) > ℵ0 and⋃

α<δ Aα is densein A, then

⋃α<δ Aα is actually equal to A.

Theorem 2.6 (See [7]). For every infinite compact spaceX, ci(X) = cof(C(X)).

3. Dyadic families

Definition 3.1. Let X be a topological space. A family (Siα)α∈J,i∈2 of closed

subsets of X is dyadic if for each α ∈ J , S0α and S1

α are disjoint and for all disjointfinite sets E,F ⊆ J ,

⋂{S0

α : α ∈ E} ∩⋂

{S1α : α ∈ F} �= ∅.

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THE COINITIALITIES OF EFIMOV SPACES 3

Shapirovskiı showed that a compact space X maps onto Iω1 iff it has an un-countable dyadic family [11]. It is easy to show directly that a compact space hasan uncountable dyadic family iff it has a closed subspace that maps onto 2ω1 .

Adapting Koppelberg’s proof of cof(P(ω)) ≤ ℵ1 [9], we can show

Theorem 3.2. If a compact space X has an uncountable dyadic family, thenci(X) ≤ ℵ1.

Proof. Suppose X is compact and has an uncountable dyadic family. By theprevious remark, X has a closed subspace that maps onto 2ω1 . Since the coinitialityofX is bounded from above by the coinitialities of the infinite closed subspaces ofX,we may assume that X itself maps onto 2ω1 . Let f : X → 2ω1 be a continuous mapwitnessing this. Using Zorn’s lemma we may actually assume that f is irreducible,i.e., no proper closed subspace of X is mapped onto 2ω1 by f .

Let G(2ω1) be the Gleason cover of 2ω1 , i.e., the Stone space of the completionro(Frω1) of the free Boolean algebra with ℵ1 generators. Let g : G(2ω1) → 2ω1

be the Stone dual of the embedding from Frω1 into its completion. G(2ω1) isprojective in the category of compact spaces and hence there is a continuous maph : G(2ω1) → X such that

G(2ω1)

g��

��

��

h �� X

f

��

2ω1

commutes. Since f is irreducible, h is onto.Let C = C(G(2ω1)),

B = {c ∈ C : c = b ◦ h for some b ∈ C(X)}

and

A = {c ∈ C : c = a ◦ g for some a ∈ C(2ω1)}.Then A and B are closed ∗-subalgebras of C that are isomorphic to C(2ω1) andC(X), respectively. We say that B is the algebra of elements of C that factorthrough h and A is the algebra of elements of C that factor through g.

Now, whenever α < ω1, there are natural quotient maps πα : 2ω1 → 2α andρα : G(2ω1) → G(2α). Here ρα is the Stone dual of the canonical embeddingfrom ro(Frα) into ro(Frω1). For α < ω1 let gα : G(2α) → 2α be the dual of theembedding of Frα into ro(Frα). The diagram

G(2ω1)

g

��

ρα �� G(2α)

gα

��

2ω1πα

�� 2α

commutes.For each α < ω1 let Cα be the algebra of elements of C that factor through ρα

and let Aα be the algebra of elements of C that factor through πα ◦ g. Note thatAα = Cα ∩ A. Since ro(Frω1) =

⋃α<ω1

ro(Frα), G(2ω1) is the inverse limit of the

G(2α), α < ω1. It follows that⋃

α<ω1Cα is dense in C and thus, by the remark

after Definition 2.5,⋃

α<ω1Cα = C.

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4 STEFAN GESCHKE

For every α < ω1 let Bα = B∩Cα. Since C is the union of the Cα,⋃

α<ω1Bα =

B. Since Aα = A ∩Cα = A∩Bα and the sequence (Aα)α<ω1is strictly increasing,

the sequence (Bα)α<ω1is strictly increasing. This shows that cof(B) ≤ ℵ1. Hence

ci(X) ≤ ℵ1 �

4. Simple extensions

In this section we assume ♦ to construct two different Efimov spaces, one ofcoinitiality ℵ1 and one of coinitiality ℵ0. This in particular shows that for compactspaces X being of uncountable coinitiality does not imply that X contains a copyof βω or an uncountable dyadic family.

Remark 4.1. It is well known that a compact space X contains a copy of βω

if and only if it has a closed subspace that maps onto 22ℵ0. Under CH this reduces

to a subspace that maps onto 2ℵ1 . Hence under CH we have that a compact spaceX contains a copy of βω if and only if it has an uncountable dyadic family.

Definition 4.2. Let X be a compact space. Then p : Y → X is a simpleextension of X if Y is compact, p is continuous and onto, there is exactly one pointx0 ∈ X whose preimage with respect to p is not a singleton, and the preimage ofany point x ∈ X has at most two elements. In other words, exactly one point issplit when passing from X to Y , and it is split into exactly 2 points.

X is simplistic [1] if X is the limit of a continuous inverse system {Xα, πβα, δ}

where each πα+1α : Xα+1 → Xα is a simple extension and X0 is a singleton.

The Stone dual of a simple extension is a minimal extension, where a Booleanalgebra A is a minimal extension of a proper subalgebra B if there is no subalgebraof A that is strictly between B and A, i.e., if A is a minimal proper superalgebraof B.

Lemma 4.3 (Koppelberg [9]). If X is simplistic, then X has no uncountabledyadic family.

For a topological proof of this lemma see [2].

Theorem 4.4. Assume ♦. Then there is a zero-dimensional Efimov space Xof coinitiality ℵ1 without isolated points.

Proof. For concreteness, we construct X as a subspace of 2ω1 . X will be thelimit of a continuous inverse system {Xα, π

βα, ω1} where each Xα is a subspace of 2α

and each πβα, α < β < ω1, is the restriction of the projection from 2β to 2α to Xβ.

For limit ordinals δ < ω we let Xδ be the limit of the inverse system {Xα, πβα, δ}.

Note that Xδ can be naturally considered as a subset of 2δ. For α < ω1, we willchoose Xα+1 ⊆ 2α+1 so that πα+1

α : Xα+1 → Xα is a simple extension. By Lemma4.3 this implies that X has no uncountable dyadic family and therefore contains nocopy of βω.

For all α < ω we choose Xα as a closed subspace of 2α such that for all α < ω,pα+1α : Xα+1 → Xα is a simple extension and the inverse limit Xω ⊆ 2ω of the Xα,

α < ω, has no isolated points. Since we will never introduce isolated points duringthe remaining construction, all the Xα, ω ≤ α < ω1 will be homeomorphic to 2ω

and and the target space X will have no isolated points.By ♦, CH holds and thus (2<ω1)ω is of size ℵ1. Hence, using some suitable

coding, from the ♦-sequence we can obtain a sequence (xαn)n<ω,α<ω1

such that for

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α < ω1, (xαn)n∈ω is a sequence in 2α and for every sequence (yn)n∈ω in 2ω1 ,

{α < ω1 : (yn � α)n∈ω = (xαn)n∈ω}

is stationary in ω1.Suppose for some α < ω1 we have defined Xα. We define Xα+1 ⊆ 2α+1.First assume that (xα

n)n∈ω is not a double sequence in Xα. Choose any pointx0 ∈ Xα and a strictly descreasing sequence (An)n∈ω of clopen subsets of Xα suchthat A0 = Xα and

⋂n∈ω An = {x0}. Let

A = {x0} ∪⋃

n∈ω

A2n \A2n+1

andB = {x0} ∪

⋃

n∈ω

A2n+1 \A2n+2.

The crucial property of A and B is that both sets are closed, union up to Xα andhave intersection {x0}.

If (xαn)n∈ω is a double sequence in Xα, we choose x0 to be an accumulation

point of (xαn)n∈ω. This is possible since Xα is compact. Since (xα

n)n∈ω is a doublesequence, x0 is an accumulation point of (xα

2n)n∈ω. SinceXα is first countable, thereis a strictly increasing sequence (ni)i∈ω of natural numbers such that (xα

2ni)i∈ω

converges to x0. Since (xαn)n∈ω is a double sequence, (xα

2ni+1)i∈ω converges to x0

as well.Now let (An)n∈ω be a strictly descreasing sequence of clopen subsets of Xα such

that A0 = Xα and⋂

n∈ω An = {x0}. We can thin out the sequence (ni)i∈ω to astrictly increasing sequence (mi)i∈ω such that for some strictly increasing sequence(ki)i∈ω of natural numbers we have that for all i ∈ ω, x2mi

, x2mi+1 ∈ Akiand for

all j < i, x2mi, x2mi+1 �∈ Akj

. We can assume that k0 = 0.Now for each i ∈ ω we choose a clopen set Bi ⊆ Aki

\Aki+1 such that x2mi∈ Bi

and x2mi+1 �∈ Bi. Let

A = Xα \⋃

i∈ω

Bi

and

B = {x0} ∪⋃

n∈ω

Bi.

A and B are nonempty closed sets that union up to Xα and have intersection {x0}.In either case let Xα+1 consist of all functions x ∈ 2α such that (x � α ∈ B and

x(α) = 0) or (x � α ∈ A and x(α) = 1).Note that Xα+1 is a simple extension of Xα since only the point x0 has two

preimages in Xα+1. It is easily checked that Xα+1 is a closed subspace of 2α+1

without isolated points. This finishes the definition of the inverse system whoselimit is our target space X.

As already indicated previously, X has no isolated points. By Lemma 4.3,the space has no uncountable dyadic family and therefore contains no copy of βω.We show that X also has no double sequences, from which it follows that X is ofcoinitiality ℵ1.

Suppose (yn)n∈ω is a double sequence in X. Then for some α < ω1, (yn � α)n∈ω

is discrete and hence a double sequence. For every β < ω1 with β ≥ α it holds that(yn � β)n∈ω is a double sequence. By the choice of (xγ

n)n∈ω,γ∈ω1, there is β ≥ α

such that (yn � β)n∈ω = (xβn)n∈ω. When we constructed Xβ+1, a strictly increasing

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6 STEFAN GESCHKE

sequence (ni)i∈ω was chosen such that (xβ2ni

)i∈ω and (xβ2ni+1)i∈ω converge to a

point x0 ∈ Xβ. Since (yn � β)n∈ω = (xβn)n∈ω is discrete, x0 is not one of the

elements of the sequence yn � β.The point x0 has two preimages in Xβ+1, namely x0

0 and x01. All the

other points of Xβ only have a single preimage. In particular, each yn � β has onlya single preimage in Xβ+1, namely yn � β + 1. Xβ+1 is the union of two disjointclopen sets C0 = {x ∈ Xβ+1 : x(β) = 0} and C1 = {x ∈ Xβ+1 : x(β) = 1}. By the

construction of Xβ+1, for each i ∈ ω the preimage of xβ2ni

in Xβ+1, i.e., y2ni� β+1,

is an element of C0 and the preimage of xβ2ni+1, i.e., y2ni+1 � β + 1, is an element

of C1.Since C0 and C1 are closed, the sets {y2ni

� β + 1 : i ∈ ω} and {y2ni+1 �β + 1 : i ∈ ω} have disjoint closures. In particular, (yn � β + 1)n∈ω is not a doublesequence. A contradiction. �

Theorem 4.5. Assume ♦. Then there is a zero-dimensional Efimov space Xof coinitiality ℵ1 with infinitely many isolated points.

Proof. For each α < ω choose Xα as a closed subspace of 2α such that eachpα+1α : Xα+1 → Xα is a simple extension and such that the inverse limit Xω ⊆ 2ω

of the Xα, α < ω, is homeomorphic to the disjoint union of 2ω and the one-pointcompactification of the discrete space ω.

We continue the construction of the sequence (Xα)α<ω1as in the proof of

Theorem 4.4. We have to make sure that we never split any of the original isolatedpoints of Xω. This is possible since no discrete sequence accumulates at an isolatedpoint and since we always have non-isolated points at our disposal that can be splitat stage α ≥ ω. This construction yields a zero-dimensional Efimov space with aninfinite set of isolated points. �

5. Collapsing coinitialities

If a compact space X has an infinite set of isolated points, we can easily turnit into a space of countable coinitiality by splitting isolated points.

Let X be a compact space with infinitely many isolated points. Let (xn)n∈ω

be a 1-1 sequence of isolated points in X. Let Y be the space obtained fromX by splitting each xn into two distinct points y2n and y2n+1. In other words,Y = X \ {xn : n ∈ ω} ∪ {yn : n ∈ ω} where {yn : n ∈ ω} is disjoint from X. Thetopology on Y is generated by the sets {yn}, n ∈ ω, and

O \ {xn : n ∈ ω} ∪ {ym : m = 2n or m = 2n+ 1

for some n ∈ ω with xn ∈ O},O an open subset of X.

Theorem 5.1. The space Y defined above is compact and cof(Y ) = ℵ0.

Proof. It is easily checked that (yn)n∈ω is a double sequence in Y . �Corollary 5.2. Assume ♦. Then there is a zero-dimensional Efimov space

X of countable coinitiality.

Proof. By Theorem 4.5 there is a zero-dimensional Efimov space X with aninfinite set of isolated points. From X we construct a compact space Y of countablecofinality as in Theorem 5.1. It is easily checked that Y is zero-dimensional and

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does not contain a copy of βω. Now assume that Y contains a nontrivial convergentsequence. In this case one of the three sets {y2n : n ∈ ω}, {y2n+1 : n ∈ ω} andY \ {yn : n ∈ ω} contains a nontrivial convergent sequence, possibly converging toa point in Y that lies outside the set itself. If one of the first two sets contains sucha sequence, then so does {xn : n ∈ ω}, a contradiction. If Y \ {yn : n ∈ ω} containssuch a sequence, then so does X \ {xn : n ∈ ω}, again contradicting the choice ofX. �

6. Discussion

The author had conjectured previously that the absence of double sequencesin an infinite compact space implies the existence of an uncountable dyadic family.This conjecture is refuted by Theorem 4.4. It is very likely that Efimov spaces ofcountable and uncountable cofinality can be constructed assuming just CH, usingthe method developed in [3].

The main question, whether it is consistent that there is a compact spaceof coinitiality > ℵ1, remains wide open. We conclude with two less ambitiousproblems.

Problem 6.1. Characterize compact spaces of coinitiality ≤ ℵ1.

Problem 6.2. Is it consistent that every compact space of uncountable coini-tiality contains an uncountable dyadic family? What happens under PFA?

Observe that the last problem is a weakening of Efimov’s problem whether itis consistent that every infinite compact space space has a convergent sequence orcontains a copy of βω.

References

1. A. Dow, Lecture in Erice, Sicily (2008)2. A. Dow, Efimov spaces and the splitting number, Topology Proc. 29 (2005), no. 1, 105–1133. A. Dow, R. Pichardo-Mendoza, Efimov spaces, CH, and simple extensions, Topology Proc. 33(2009), 277–283

4. V. V. Fedorcuk, Completely closed mappings, and the compatibility of certain general topologytheorems with the axioms of set theory, Mat. Sb. (N.S.) 99 (141) (1976), no. 1, 3–33, 135

5. E. van Douwen, Cardinal functions on Boolean spaces, in Handbook of Boolean Algebras,J. Donald Monk (ed.) with the cooperation of Robert Bonnet (1989), Volume 2, 417–467

6. R. Engelking, General Topology, Sigma Series in Pure Mathematics Volume 6, HeldermannVerlag Berlin 1989

7. S. Geschke, The coinitiality of a compact space, Topology Proceedings 30 (2006), no. 1, 237–2508. W. Just, P. Koszmider, Remarks on cofinalities and homomorphism types of Boolean algebras,Algebra Univers., Vol 28 (1991), No. 1, 138–149

9. S. Koppelberg, Boolean algebras as unions of chains of subalgebras, Algebra Universalis, Vol. 7

(1977), 195–20310. P. Koszmider, The Consistency of ¬CH+pa ≤ ω1, Algebra Universalis Vol. 27 (1990), 80–8711. B. Sapirovskiı, Maps onto Tychonoff cubes, Russian Math. Surveys, (1980) vol. 35, no. 3,145-156

Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany


265265


Uniforming n-place Functions on Well Founded Trees

Esther Gruenhut and Saharon Shelah

Abstract. In this paper the Erdos-Rado theorem is generalized to the class ofwell founded trees. We define an equivalence relation on the class ds(∞)<ℵ0

( finite sequences of decreasing sequences of ordinals) with ℵ0 equivalenceclasses, and for n < ω a notion of n-end-uniformity for a colouring of ds(∞)<ℵ0

with μ colours. We then show that for every ordinal α, n < ω and cardinal

μ there is an ordinal λ so that for any colouring c of T = ds(λ)<ℵ0 with μcolours, T contains S isomorphic to ds(α) so that c�S<ℵ0 is n-end uniform.For c with domain Tn this is equivalent to finding S ⊆ T isomorphic to ds(α)so that c�Sn depends only on the equivalence class of the defined relation, soin particular T → (ds(α))nμ,ℵ0

. We also draw a conclusion on colourings of

n-tuples from a scattered linear order.

0. Introduction

This paper deals with a Ramsey-type theorem for scaterred order types. Wededicate this section to some general background. A Ramsey-type theorem beginswith a target element ϕ and a fixed number of colors, μ. The statement assertsthat there exists another element ψ (of the same type) so that for every coloring ofψ by μ colors, one can find a monochromatic ϕ-copy included in ψ.The simplest example is the class of infinite cardinals, and coloring functions definedon singletons. For instance, μ+ → (μ+)1μ holds for every infinite cardinal μ. It

means that for any coloring c : μ+ → μ there exists a copy of μ+ (namely, a subsetof μ+ whose cardinality is μ+) which is monochromatic under c.This simple version works for order types as well. Given any order type θ (this isthe target), and a fixed number of colors μ, one can find an order type ψ so thatψ → (θ)1μ (i.e., for every coloring c : ψ → μ there exits a monochromatic copy of θin ψ).We concentrate, throughout the paper, in the interesting class of scaterred ordertypes. Let us start with the following:

Definition 0.1. Scaterred order types.

(1) η is the order type of the set of rational numbers (Q, <)(2) For two order types ϕ, ψ we say that ϕ ≤ ψ iff there is an order preserving

embedding of ϕ into ψ

Research supported by the United States-Israel Binational Science Foundation (Grant no.2002323). Publication 909 in Shelah’s archive.AMS 2000 classification 03E05, 05C15.Key words: Set Theory, partition relation, well founded trees, scattered order types.

1



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267

2 ESTHER GRUENHUT AND SAHARON SHELAH

(3) An order type ϕ is scattered when ¬(ϕ ≤ η)

The investigation of scaterred order types goes back to Hausdorff. This definitionis a “negative” one. Hausdorff proved in [3] that the class of scaterred order typesis characterized by a simple “positive” closure property. This class is the smallestclass which contains 0, 1 and is closed under well ordered and reverse well orderedsums. In fact, as a consequence of Hausdorff’s proof we get that every linear ordreris a dense sum of scattered ordered types (see as well [5]).

We shall use the followin notation:

Notation 0.2. The Erdos-Rado arrows.

(1) ψ → (ϕ)�μ means that for every set S such that otp(S,<) = ψ and each

coloring c : [S]� → μ, there is an ordinal i < μ and a subset T ⊆ S so thatotp(T,<) = ϕ and c � [T ]� = {i}

(2) ψ � (ϕ)�μ means that the statment ψ → (ϕ)�μ does not hold

It is easy to show that if � = 1 (i.e., the colorings are defined on singletons)and μ is finite, then ψ → (ϕ)�μ holds in the class of scattered order types. Tryingto generalize it, we encounter with two problems. First, infinite amount of colorsposes a limitation (in the case of scattered order types), even when using just ℵ0

colors. Second, dealing with �-tuples with � > 1 becomes much more complicated.For the first problem, ψ � (ϕ)1ω is exemplified by ϕ = 1+(ω∗+ω)+(ω∗+ω)2+ . . .(recall that if θ = otp(S,<) then θ∗ is otp(S,>)). For the second problem, ψ �(ω∗ + ω)22, so we fail even when trying to use pairs.Nevertheless, one can still prove positive results for infinitely many colors and �-tuples, even when dealing with scattered order types. Aiming to these results, weneed again a bit of notation:

Notation 0.3. Square brackets.

(1) ψ → [ϕ]�μ means that for every set S such that otp(S,<) = ψ and each

coloring c : [S]� → μ, there is an ordinal i < μ and a subset T ⊆ S so thatotp(T,<) = ϕ and i /∈ c � [T ]�

(2) ψ → [ϕ]�λ,μ means that for every set S such that otp(S,<) = ψ and each

coloring c : [S]� → λ, there is a subset X ⊆ λ, |X| = μ and a subsetT ⊆ {x ∈ S : c(x) ∈ X} such that otp(T,<) = ϕ

The former property in the above definition is a property of omitting a color,the latter property is the main concern of this paper. Notice that if ψ → [ϕ]�λ,μ and

κ ≤ μ, then ψ → [ϕ]�λ,κ. Consequently, we may succeed even with infinite number

of colors and colorings of �-tuples, if we decrease κ. In particular, ψ → [ϕ]�λ,1 is

equivalent to ψ → (ϕ)�λ.In the general case (with no restriction to scattered order types) we can get bothpositive and negative results. For example, ψ → [ϕ]�μ,2 was proved by Shelah in [6],for every infinite μ and any natural number �. On the other hand, it is consistentto have an order type θ of cardinality ℵ1, such that ψ � [θ]2ℵ1

as shown by Hajnaland Komjath in [2].Under these considerations, we seek for ZFC theorems in the class of scatteredorder types. It was proved in [4] that ψ → [ϕ]1μ,ℵ0

for such types. We generalize

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UNIFORMING n-PLACE FUNCTIONS ON WELL FOUNDED TREES 3

it, to yield the relation ψ → [ϕ]�μ,ℵ0for every � ∈ ω. Notice that ψ � (ϕ)1ℵ0

, so thesubscript μ,ℵ0 is well motivated.

1. Some Definitions and Notation

This paper is a natural continuation of [4] in which Shelah and Komjath provethat for any scattered order type ϕ and cardinal μ there exists a scattered ordertype ψ such that ψ → [ϕ]1μ,ℵ0

. This was proved by a theorem on colourings of well

founded trees. By Hausdorff’s characterization (see [3] and [5] and the introductionabove ) every scattered order type can be embedded in a well founded tree, so wecan deduce a natural generalization of their theorem to the n-ary case, i.e for everyscattered order type ϕ, n < ω, and cardinal μ there is a scattered order type ψsuch that ψ → [ϕ]nμ,ℵ0

.We start with a few definitions.

Definition 1.1. For an ordinal α we define ds(α) = {η : η a decreasing se-quence of ordinals < α}. By ds(∞) we mean the class of decreasing sequences ofordinals.

We say T ⊆ ds(∞) is a tree when T is non-empty and closed under initialsegments. T, S will denote trees. For S ⊆ T ⊆ ds(∞) we say that S is a subtree ofT if it is also a tree. We use the following notation:

Notation 1.2. (1) For η, ν ∈ ds(∞) by η ∩ ν we mean η�� where � ismaximal such that η�� = ν��.

(2) For η ∈ ds(∞) and a tree T ⊂ ds(∞) we define

η�T = {ρ : ρ � η ∨ (∃ν ∈ T )(ρ = η�ν)}Note that for η ∈ ds(∞\{〈〉}) and {〈〉} � T ⊆ ds(∞) if η(lg(η)−1) > sup{ρ(0) :

ρ ∈ T} then η�T ⊆ ds(∞).

Definition 1.3. We define the following four binary relations on ds(∞):

(1) Let <1�x be the two place relation on ds(∞) defined by η <1

�x ν iff one ofthe following: (∃�)(η(�) < ν(�) and η�� = ν��) or η ν.

(2) Let <2�x be the two place relation on ds(∞) defined by η <2

�x ν iff one ofthe following: (∃�)(η(�) < ν(�) and η�� = ν��) or ν η.

(3) <∗�x=<1

�x ∩ <2�x.

(4) Let <3 be the two place relation on ds(∞) defined by η <3 ν iff one ofthe following holds: η ν or for the maximal � such that η�� = ν�� if � iseven then η(�) < ν(�) and if � is odd then η(�) > ν(�).

It is easily verified that <1�x, <

2�x and <3 are complete orders of ds(∞), and

therefore <∗�x is a partial order. The following remark refers to their order types

defined by <1�x, <

2�x and <3 on ds(∞) or ds(α).

Observation 1.4. (1) <1�x, <

2�x are well orderings for ds(∞).

(2) (ds(α), <3) is a scattered linear order type for every ordinal α.(3) Every scattered linear order type can be embedded in (ds(α), <3) for some

ordinal α.

Proof. (1) Let ∅ �= A ⊆ ds(∞), we define by induction on n < ω anelement an in the following manner a0 = min{η(0) : η ∈ A}, assumea0, · · · , an−1 have been chosen so that 〈ak : k < n〉 ∈ ds(∞) and for every

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η ∈ A 〈ak : k < n〉 ≤2�x η�n (if lg(η) ≤ n then η�n = η). Now choose

an = min{η(n) : η ∈ A ∧ η�n = 〈ak : k < n〉}, if that set isn’t empty.As the sequence derived in the above manner is a decreasing sequence ofordinals it is finite, say a0, · · · an−1 have been defined and an cannot bedefined, we will show that a = 〈ak : k < n〉 is the minimal element of Awith respect to <2

�x. By the definition of the sequence there is an η ∈ Aso that η�n = a, if lg(η) > n then we could have defined an, so η = aand in particular a ∈ A, and for every η ∈ A\{a} we have a <2

�x η. Letn∗ = min{m : a�m ∈ A} so a�n∗ is the <1

�x- minimal element in A.(2) The proof is by induction on α. Assume that (ds(β), <3) is a scattered

linear order type for every β < α, and assume towards contradictionthat Q can be embedded in (ds(α), <3), q �→ ηq. Let C = {� : (∃p, q ∈Q)(ηp(�) �= ηq(�))}, � = minC and Γ = {β : (∃q ∈ Q)(ηq(�) = β)}.Without loss of generality � is even and for β0 = minΓ, β1 = minΓ\{β0}there are q0 < q1 ∈ Q so that ηqi(�) = βi, i = 0, 1. Now (q0, q1) = B0∪B1

where Bi = {p ∈ (q0, q1) : ηp(�) = βi}. For some i ∈ {0, 1} the setBi contains an interval of Q and is embedded in (ηqi�(�+ 1)�ds(βi), <

3)but this would imply that Q can be embedded in (ds(βi), <

3) which is acontradiction to the induction hypothesis.

(3) By Hausdorff’s characterization it is enough to show for ordinals α andβ that both Aα,β = (ds(α), <3)× β and Aα,β∗ = (ds(α), <3)× β∗ can beembedded in (ds(α + β · 2 + 1), <3). The embedding is given as follows,for (η, γ) ∈ Aα,β we have (η, γ) �→ 〈α + β + γ + 1, α + β〉�η, and for(η, γ) ∈ Aα,β∗ we have (η, γ) �→ 〈α+ β · 2, α+ β + γ〉�η.

�

Definition 1.5. For trees T1, T2 ⊂ ds(∞), f : T1 → T2 is an embedding of T1

into T2 if f preserves level, and <1�x (or equivalently, <2

�x, <∗�x or <3).

Observation 1.6. For trees T1, T2 ⊂ ds(∞), if f : T1 → T2 preserves leveland then in order to determine whether f is an embedding it is enough to checkfor η ∈ T1 and ordinals γ1 < γ2 such that νi = η�〈γi〉 ∈ T1 (i = 1, 2) thatf(ν1) <

∗�x f(ν2).

As T ⊆ ds(∞) is well founded, i.e there are no infinite branches, it is naturalto define a rank function. in the following definition rkT,μ isn’t the standard rankfunction but for μ = 1 we get a similar definition to the usual definition of a rankon a well founded tree.

Definition 1.7. For a tree T ⊂ ds(∞) and cardinal μ define rkT,μ(η) :ds(∞) → {−1} ∪Ord by induction on α as follows:

(a) rkT,μ(η) ≥ 0 iff η ∈ T .(b) rkT,μ(η) ≥ α+ 1 iff μ ≤ |{γ : η�〈γ〉 ∈ T ∧ rkT,μ(η

�〈γ〉) ≥ α}|.(c) rkT,μ(η) ≥ δ limit iff (∀α < δ)(rkT,μ(η) ≥ α).

We say that rkT,μ(η) = α iff rkT,μ(η) ≥ α but rkT,μ(η) � α+ 1.Denote rkT,μ(T ) = rkT,μ(〈〉), and rkT (η) = rkT,1(η).

Definition 1.8. For a tree T ⊂ ds(∞), η ∈ T and cardinals μ, λ we define the

reduced rank rkλT,μ(η) = min{λ, rkT,μ(η)}.

We first note a few properties of the rank function.

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Observation 1.9. For η ∈ T ⊂ ds(∞) and an ordinal α we have:

(1) For cardinals μ ≤ μ′ we have rkT,μ(η) ≥ rkT,μ′(η), and in particularrkT (η) ≥ rkT,μ(η)

(2) rkT (η) = ∪{rkT (η�〈γ〉) + 1 : η�〈γ〉 ∈ T}.(3) rkds(α)(〈〉) = α.(4) If rkT,μ(η) ≥ α, μ ≥ α then we can embed η�ds(α) into T , so that ρ �→ ρ

for ρ � η.

Proof. 3 The proof is by induction on α.For α = 0 this is obvious. Assume correctness for every β < α. ds(α) =⋃

β<α

{〈β〉�ν : ν ∈ ds(β)}. For every β < α, ν ∈ ds(β) we have rkds(α)(〈β〉�ν) =

rkds(β)(ν), therefore (the last equality is due to the induction hypothesis):

∪{rkds(α)(〈β〉�ν) + 1 : ν ∈ ds(β)} = ∪{rkds(β)(ν) + 1 : ν ∈ ds(β)}= rk(ds(β))= β

We therefore have rk(ds(α)) = ∪{β + 1 : β < α} = α4 The proof is by induction on α.For α = 0 there is nothing to prove.Assume correctness for every β < α, and rkT,μ(η) ≥ α, α ≤ μ. Forβ < α let Cβ = {γ : rkT,μ(η

�〈γ〉) ≥ β}, so |Cβ| ≥ μ and Cβ ⊆ Cβ′ forβ′ < β < α. By induction on β < α we can choose an increasing sequenceof ordinals γβ such that γβ = minΓβ where Γβ = {γ ∈ Cβ : (∀β′ <β)(γ > γβ′)}. Assume towards contradiction that Γβ is empty, and letC ′

β = 〈γβ′ : β′ < β〉 ∩ Cβ. For every γ ∈ Cβ\C ′β (and there is such γ as

|Cβ | ≥ μ whereas |C ′β| ≤ |β| < μ) as γ /∈ Γβ then there is β′ < β such that

γ < γβ′ , assume β′ is minimal with this property, but that contradicts thechoice of γβ′ .By the induction hypothesis for every β < α there is ϕβ which embeds(η�〈γβ〉)�ds(β) in T so that ϕβ�{ρ : ρ � η�〈γβ〉} = Id. We now defineϕα : η�ds(α) → T in the following manner, if ρ � η then ϕα(ρ) = ρ, elseρ = η�ν for some ν ∈ ds(α), so there is β < α such that ν = 〈β〉�ν1 withν1 ∈ ds(β), and we define

ϕα(ρ) = ϕβ(η�〈γβ〉�ν1).

ϕα obviously preserves level.For ρ1 ρ2 in η�ds(α) if ρ1 � η then obviously ϕα(ρ1) ϕα(ρ2), andotherwise for some β < α we have ρi = η�〈β〉�νi, i ∈ {1, 2}, ν1 ν2 ∈ds(β), and as ϕβ is an embedding we have:

ϕα(ρ1) = ϕβ(η�〈γβ〉�ν1) ϕβ(η

�〈γβ〉�ν2) = ϕα(ρ2).

For ρ ∈ η�ds(α), γ1 < γ2 ordinals such that for i = 1, 2 ρi = ρ�〈γi〉 ∈η�ds(α), necessarily η � ρ and there are β1 ≤ β2 < α, νi ∈ ds(βi) sothat ρi = η�〈βi〉�νi. If β1 = β2 = β then ν1 <∗

�x ν2, and as ϕβ is anembedding,

ϕα(ρ1) = ϕβ(η�〈γβ〉�ν1) <∗

�x ϕβ(η�〈γβ〉�ν2) = ϕα(ρ2)

On the other hand, if β1 �= β2 then ϕα(ρi)(lg(η)) = γβi, and as γβ1

< γβ2,

also in this case ϕα(ρ1) <∗�x ϕα(ρ2).

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By Observation 1.6 ϕα is an embedding, and by definition ϕα�{ρ : ρ �η} = Id.

�

The following theorem was proved By Komjath and Shelah in [4]:

Theorem 1.10. Assume α is an ordinal and μ a cardinal. Set λ = (|α|μℵ0)+,

and let F : ds(λ+) → μ. Then there is an embedding ϕ : ds(α) → ds(λ+) and afunction c : ω → μ such that for every η ∈ ds(α) of length n+ 1

F (ϕ(η)) = c(n).

In what follows we will generalize the above theorem, in the process we will useinfinitary logics. For the readers’ convenience we include the following definitions.

Definition 1.11. (1) For infinite cardinals κ, λ, and a vocabulary τ con-sisting of a list of relation and function symbols and their ‘arity’ which isfinite, the infinitary language Lκ,λ for τ is defined in a similar manner tofirst order logic. The first subscript, κ, indicates that formulas have < κfree variables and that we can join together < κ formulas by

∧or

∨, the

second subscript, λ, indicates that we can put < λ quantifiers together ina row.

(2) Given a structure B for τ we say that A is an Lκ,λ-elementary submodel(or substructure), and denote A ≺κ,λ B or A ≺Lκ,λ

B, if A is a substruc-ture of B in the regular manner, and for any Lκ,λ formula ϕ with γ freevariables and a ∈ γ |A| we have

B |= ϕ(a) ⇔ A |= ϕ(a).

The Tarski-Vaught condition for a substructure A of B to be an elemen-tary submodel is that for any Lκ,λ-formula ϕ with parameters a ⊆ A wehave

B |= ∃xϕ(xa) ⇒ A |= ∃xϕ(xa).(3) A set X is transitive if for every x ∈ X we have x ⊆ X.(4) For every set X there exists a minimal transitive set, which is denoted by

TC(X), such that X ⊆ TC(X).(5) For an infinite regular cardinal κ we define

H(κ) = {X : |TC(X)| < κ}.Remark 1.12. In this paper the main use of infinitary logic will be in the

following manner:

(1) τ will consist of the two binary relations ∈ and <∗, so |Lκ+,κ+(τ )| = 2κ.(2) If κ′ ≤ κ, λ′ ≤ λ and A ≺κ,λ B then also A ≺κ′,λ′ B.(3) ≺κ,λ is a transitive relation.(4) For an infinite cardinal μ let κ = μ+, λ = 2μ, so κ is regular and λ<κ = λ.

Recall that for a structure B and X ⊆ ‖B‖ such that |X| + τ ≤ λ ≤B there is an elementary Lκ,κ submodel A of B of cardinality λ whichincludes X.For further reference on this point see [1].

(5) If A ≺κ,κ B and x is definable in B over A (i.e with parameters in A) byan Lκ,κ-formula, then it is also definable in A by the same formula. Inparticular if A ≺κ,κ B and X ⊆ |A|, |X| < κ then X ∈ |A|.

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Definition 1.13. We say that two finite sequence 〈η� : � < n〉, 〈ν� : � < n〉 aresimilar when:

(a) lg(η�) = lg(ν�) for � < n.(b) lg(η� ∩ ηm) = lg(ν� ∩ νm) for �,m < n.(c) (η� <

2�x ηm) ≡ (ν� <

2�x νm) for �,m < n (equivalently, we could use <1

�x).

Observation 1.14. (1) Similarity is an equivalence relation and the num-ber of equivalence classes of finite sequences is ℵ0.

(2) 〈η1, . . . , ηk, ν′〉, 〈η1, . . . , ηk, ν′′〉 are similar if(a) η1 <2

�x η2 <2�x . . . <2

�x ηk(b) ηk <2

�x ν′

(c) ηk <2�x ν′′

(d) lg(ν′) = lg(ν′′)(e) lg(ν′ ∩ ηk) = lg(ν′′ ∩ ηk)

Proof. (1) Similarity is obviously an equivalence relation.The equivalence class of a finite sequence of ds(∞) is determined by itslength n, the lengths 〈ni : i < n〉 of its elements, the lengths 〈ni,j : i, j <n〉 of their intersections, and a permutation of n (the order of the elementsaccording to <1

�x). Therefore for each n < ω there are ℵ0 equivalenceclasses of sequences of length n, and so the number of equivalence classesof finite sequences of ds(∞) is ℵ0.

(2) We need to show that lg(ν′ ∩ ηi) = lg(ν′′ ∩ ηi) for every 0 < i < k.ηk <2

�x ν′ and ηk <2�x ν′′. If ν′ ηk then we also have lg(ν′′ ∩ ηk) =

lg(ν′∩ηk) = lg(ν′) = lg(ν′′) so ν′′ ηk, and ν′ = ν′′. In this case obviouslythe required sequences are similar, so we can assume that there is � suchthat ηk�� = ν′�� and ν′(�) > ηk(�). By the same reasoning as above wededuce that ηk�� = ν′′�� and ν′′(�) �= ηk(�) so necessarily ν′′(�) > ηk(�).

�

The last term we will need before moving on to the main theorem is that ofuniformity.

Definition 1.15. Let T ⊆ ds(∞) be a tree, c : [T ]<ℵ0 → C. We identifyu ∈ [T ]<ℵ0 with the <2

�x-increasing sequence listing it.

(1) We say T is c-uniform if for any similar u1, u2 in [T ]<ℵ0 we have c(u1) =c(u2).

(2) We say T is c-end-uniform (or end-uniform for c) whenif η1 <2

�x η2 <2�x . . . <2

�x ηk <2�x ρ′, ρ′′ are in T and lg(ρ′) = lg(ρ′′), lg(ηk ∩

ρ′) = lg(ηk ∩ ρ′′) (equivalently 〈η1 . . . ηk, ρ′〉, 〈η1 . . . ηk, ρ′′〉 are similar-see1.4(3))then c(〈η1 . . . ηk, ρ′〉) = c(〈η1, . . . , ηk, ρ′′〉).

(3) We say T is c-n-end-uniform (or n-end-uniform for c) when for k < ω,ηi, ρ

′j , ρ

′′j ∈ ds(∞) (0 < i ≤ k, 0 < j ≤ n) such that

η1 <2�x< η2 <2

�x . . . <2�x ηk <2

�x ρ′1 <2�x . . . <2

�x ρ′n

η1 <2�x< η2 <2

�x . . . <2�x ηk <2

�x ρ′′1 <2�x< . . . < ρ′′n

if those two sequences are similar then

c(〈η1 . . . , ρ′1 . . .〉) = c(〈η1 . . . ρ′′1 . . .〉).

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2. Uniforming n-place functions on T ⊂ ds(α)

We are now ready for the main theorem of this paper.

Main Claim 2.1. Given a tree S ⊆ ds(∞) and a cardinal μ we can find a treeT ⊆ ds(∞) such that

(∗)1 for every c : [T ]<ℵ0 → μ there is T ′ ⊆ T isomorphic to S such that c�T ′

is c-end-uniform.(∗)2 |T | < �|S|+(|S|+ μ).

Proof. We assume that |S|, μ are infinite cardinals since one of our main goalsis proving a statement of the form x → [y]nμ,ℵ0

, otherwise the bound on T has tobe slightly adjusted.For each η ∈ S let

αη = αS(η) = otp({ν ∈ S : ν <2�x η}, <2

�x),μη = �5αη+1(|S|+ μ),λη = �3(μη)

+.

Note that μ〈〉, λ〈〉 are the maximal ones, and let χ >> λ<>, and <∗χ be a well

ordering of H(χ) (see 1.11(5)). By definition, for every η, ν ∈ S such that η <2�x ν

we have μη < μν , and λη < λν in the following we examine the relation betweenμν and λη for η �= ν.

Observation 2.2. For η <2�x ν we have μν ≥ λ+

η .

Proof. Since αν ≥ αμ + 1 we have:μν = �5αν+1(|S|+ μ)

≥ �5(αη+1)+1(|S|+ μ)= �5(μη)≥ �3(μη)

++

= λ+η

�

Let T := ds(λ+〈〉), we will show that T is as required. Obviously T meets

requirement (∗)2, and let c : [T ]<ℵ0 → μ. Because of the many details in thefollowing construction we bring it as a separate lemma.

Lemma 2.3. For η ∈ S we can choose Mη, T∗η and νη,n ∈ T for n < ω with the

following properties:

(1) Mη is an Lμ+η ,μ+

η-elementary submodel of B = (H(χ),∈, <∗

χ).

(2) ‖Mη‖ = 2μη .(3) S, T, c ∈ Mη.(4) Mρ, νρ,n ∈ Mη for ρ <∗

�x η, n < ω.(5) Properties of T ∗

η :

(a) T ∗η = νη,lg(η)

�T ′ where T ′ is isomorphic to ds(22μη).

(b) If ν′, ν′′ ∈ T ∗η and are of the same length then they realize the same

Lμ+η ,μ+

η-type over Mη.

(6) Properties of the νη,n:(a) νη,n ∈ T is of length n.(b) νη,lg(η) ∈ Mη.(c) lg(η) = m < n ⇒ νη,n(m) /∈ Mη.

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(d) νη,n ∈ T ∗η , and for n ≥ lg(η) has at least μη immediate successors in

T ∗η .

(7) If η = η1�〈α〉, then

(a) Mη, T∗η , νη,n ∈ Mη1

for n < ω.(b) νη1,n, νη,n realize the same Lμ+

η ,μ+η-type over {Mρ, νρ,n : n < ω, ρ <∗

�x

η}.(c) νη1,n = νη,n for n ≤ lg (η1).(d) νη,n <∗

�x νη1,n for n = lg(η).(e) νη,lg(η) = νη,lgη1

�〈γ〉 for some γ.(f) If η′ = η1

�〈α′〉 with α′ < α then νη′,lg(η′) <∗�x νη,lg(η).

Proof. We show a construction for such a choice by induction on <1�x, yes,

<1�x not <2

�x.As the induction is on <1

�x the base of the induction is the case η = 〈〉. Firstchoose M〈〉 ≺L

μ+〈〉,μ

+〈〉

B of cardinality 2μ〈〉 , so that S, T, c ∈ M〈〉 (this can be done,

see Remark 1.12). The number of Lμ+〈〉,μ

+〈〉

formulas ϕ(x, a) where a ⊆ μ+〈〉>M〈〉

(sequences of length < μ+〈〉 in M〈〉) is ≤ (2μ〈〉)μ〈〉 = 2μ〈〉 hence the number of

Lμ+〈〉,μ

+〈〉-types over M〈〉 is at most μ′ = 22

μ〈〉, so we color T = ds(λ+

〈〉) by ≤ μ′

colors, c〈〉 : T → μ′, so that for ρ ∈ T its color, c〈〉(ρ), codes the Lμ+〈〉,μ

+〈〉-type which

ρ realizes in B over M〈〉. As

((�2(μ〈〉))μ′ℵ0

)+ = �3(μ〈〉)+ = λ〈〉

by Theorem 1.10 there is an embedding of ds(�2(μ〈〉)) in T , and define T ∗〈〉 to be its

image, so that types of sequences from T ∗〈〉 depend only on their length. We choose

representatives 〈ν〈〉,n : 0 < n < ω〉 from each level larger than 0 so that for n > 0ν〈〉,n and has at least μ〈〉 immediate successors in T ∗

〈〉 and satisfies 6(c). The latter

can be done by cardinality considerations, ‖M〈〉‖ = 2μ〈〉 , while the cardinality oflevels in T ∗

η〈〉 is �2(μ〈〉). We let ν〈〉,0 = 〈〉.It is easily verified that for η = 〈〉 all the requirements of the construction are met.We now show the induction step.Assume η = η1

�〈α1〉, lg(η1) = r, and that we have defined for η1 (and below by<1

�x) and we define for η.

�1 Let Aη = {Mρ, νρ,n : n < ω, ρ <∗�x η}.

For any ρ <∗�x η if ρ = η1

�〈α〉 for some α < α1 then from requirement (7)(a) ofthe construction for ρ we have Mρ ∈ Mη1

, and also for all n < ω νρ,n ∈ Mη1, else

ρ <∗�x η1 therefore from requirement (4) of the construction for η1 we have for all

n < ω νρ,n ∈ Mη1, and Mρ ∈ Mη1

. So Aη ⊆ Mη1, and |Aη| ≤ μη1

, so Aη is definableby an Lμ+

η1,μ+

η1-formula with parameters in Mη1

, so we have:

�2 Aη ⊆ Mη1, |Aη| ≤ μη ≤ μη1

, therefore Aη ∈ Mη1.

For every n < ω let

�3 ϕn(x) = ϕμη1,n(x) =

∧( the Lμ+

η ,μ+η− type which νη1,n realizes over Aη)

And let

�4 Tϕ = {ρ ∈ T : B |= ϕlg(ρ)(ρ)}.

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As the cardinality of the Lμ+η ,μ+

η-type of any ν ∈ B over Aη is at most 2μη which is

less than μη1, for every n < ω we have that ϕn is an Lμ+

η1,μ+

η1-formula and therefore

Tϕ is definable in Mμη1by an Lμ+

η1,μ+

η1-formula, namely

ρ ∈ Tϕ ↔(ρ ∈ T ∧

( ∨

n<ω

(lg(ρ) = n ∧ ϕn(ρ))))

So

�5 Tϕ ∈ Mη1and for every n < ω we obviously have νη1,n ∈ Tϕ.

Recall that for all n < ω νη1,n ∈ T ∗η1, so for any ρ ∈ T ∗

η1of length n, we have that

ρ realizes the same Lμ+η1

,μ+η1-type over Mη1

as νη1,n so in particular they realize

the same Lμ+η ,μ+

η-type over Aη, so ρ ∈ Tϕ. For m ≥ n νη1,n, νη1,m�n are of the

same length, so in particular ϕm(x) � ϕn(x�n). If ρ ∈ Tϕ, lgρ = m so B |= ϕm(ρ)therefore B |= ϕn(ρ�n) and therefore also ρ�n ∈ Tϕ. We summarize:

�6 Tϕ is a subtree of T and T ∗η1

⊆ Tϕ.

The following point is a crucial one, we show that:

�7 rkTϕ,μη1(νη1,n) > μη1

for every n such that lg(η1) ≤ n < ω .

Assume toward contradiction that rkTϕ,μη1(νη1,m) ≤ μη1

for some lg(η1) ≤ m < ω,and define for each n such that m ≤ n < ω :

γn = rkTϕ,μη1(νη1,n) and γ∗

n = rkμη1

Tϕ,μη1(νη1,n)

(see Definitions 1.7 and 1.8). We now prove by induction on n ≥ m that γn ≤ μη1,

i.e γn = γ∗n. For n = m this is our assumption, and assume that it is known for n.

The following can be expressed by Lμ+η1

,μ+η1-formulas with parameters in Mη1

:

ψ1 : ‘x has rkμη1

Tϕ,μη1(x) = γn’

ψ2 : ‘x has at least μη1immediate successors y in Tϕ with rk

μη1

Tϕ,μη1(y) ≥ γ∗

n+1’

We have B |= ψ1(νη1,n), and since T ∗η1

⊂ Tϕ (see �6) we also have B |= ψ2(νη1,n).By the induction hypothesis for η1 we have νη1,n, νη1,n+1�n ∈ T ∗

η1and as they are the

same length realize the same Lμ+η1

,μ+η1-type over Mη1

, so B |= ψ1∧ψ2(νη1,n+1�n), orin more detail, we have that rk

μη1

Tϕ,μη1(νη1,n+1�n) = γn, i.e rkTϕ,μη1

(νη1,n+1�n) = γn,

and νη1,n+1�n has at least μη1immediate successors in Tϕ with reduced rank γ∗

n+1,so by the definition of rank (Definition 1.7) we have γn > γ∗

n+1. By the inductionhypothesis γn ≤ μη1

, therefore also γ∗n+1 = γn+1. In particular we can deduce

that γn+1 < γn, so having carried out the induction we have an infinite decreasingsequence of ordinals which is a contradiction.Recall that lg(η1) = r so lg(η) = r + 1,

�8 Define νη,� = νη1,� for � ≤ r.

By 2.2 μη1≥ λ+

η , by �7 rkTϕ,μη1(νη1,r) > μη1

therefore rkTϕ,μη1(νη1,r) > λ+

η so

by definition there are ν ∈ SucT (νη1,r) ∩ Tϕ satisfying rkTϕ,μη1(ν) ≥ λ+

η , defining

νη,r+1 to be one such ν which is minimal with respect to <1�x (this is equivalent to

demanding that ν(r) is minimal) can be done by an Lμ+η1

,μ+η1

formula. We therefore

conclude:

�9 We can choose νη,r+1 ∈ SucT (νη1,r) ∩ Tϕ ∩Mη1such that

(i) rkTϕ,μη1(νη,r+1) ≥ λ+

η .

(ii) νη,r+1 is minimal under (i) in <1�x.

276276


As νη,lg(η) ∈ Mη1and νη1,lg(η)(lg(η1)) /∈ Mη1

, we have:

�10 νη,lg(η) <1�x νη1,lg(η), notice that as they are the same length <1

�x⇒<∗�x.

Now for any ρ = η1�〈α〉 ∈ S where α < α1 we have that ρ <∗

�x η and thereforeνρ,r+1 ∈ Aη (see �1). νη,lg(η), νη1,lg(η) realize the same Lμ+

η ,μ+η-type over Aη, and

by requirement (7)(d) of the construction for ρ (lg(ρ) = lg(η)) we have νρ,lg(η) <1�x

νη1,lg(η) so also νρ,lg(η) <1�x νη,lg(η) and as above, as they are the same length

<1�x⇒<∗

�x, and we therefore conclude that:

�11 If ρ = η1�〈α〉 ∈ S where α < α1 then νρ,lg(η) <

∗�x νη,lg(η).

Since |{S, t, c, νηlg(η)} ∪ Aη| < 2μη by Remark 1.12 we can choose Mη so that

�12 Mη ≺Lμ+η ,μ

+η

Mη1, and therefore also Mη ≺L

μ+η ,μ

+η

B, of cardinality 2μη

and {S, t, c, νηlg(η)} ∪ Aη ⊆ Mη.

By the same remark we can conclude that

�13 Mη ∈ Mη1.

Lastly we choose T ∗η and νη,m for m > lg(η).

We have already commented that rkTϕ,μη1(νη,lg(η)) > λ+

η , so from Observation 1.9

we can embed νη,lg(η)�ds(λ+

η ) into Tϕ so that ρ �→ ρ for ρ � νη,lg(η), and denoteone such embedding by ψ, without loss of generality ψ ∈ Mη1

.

The number of Lμ+η ,μ+

η-types over Mη is at most μ′ = 22

μη. We color ds(λ+

η ) in ≤ μ′

colors, the color of ρ ∈ ds(λ+η ) is determined by the Lμ+

η ,μ+η-type which ψ(νη,lg(η)

�ρ)

realizes over Mη, call this coloring cη. As ((�2(μη))μ′ℵ0

)+ = �3(μη)+ = λη, we can

use 1.10 to get an embedding θ of ds(�2(μη)) into ds(λ+η ) so that for ρ ∈ ds(�2(μη))

the Lμ+η ,μ+

η-type that νη,n+1

�θ(ρ) realizes overMη depends only on its length. Since

the set X of Lμ+η ,μ+

η-types over Mη is in Mη1

of cardinality at most μ′ < μη1we have

X ⊂ Mη1, also ds(λ+

η ) ∈ Mη1so cη ∈ Mη1

and therefore without loss of generalityθ ∈ Mη1

. We define

�14 T ∗η = νη,lg(η)

�θ(ds(�2(μη))

).

T ∗η ∈ Mη1

and meets requirement (5) of the construction. We will now choose rep-resentatives 〈ρm : 0 < m < ω〉 from each level of ds(�2(μη)) so that νη,n+1

�θ(ρm)has at least μη immediate successors in T ∗

η and νη,n+1�θ(ρm)(lg(η)) /∈ Mη1

, sincethe existence of such representatives in B can be expressed by an Lμ+

η1,μ+

η1-formula

with parameters in Mη1so without loss of generality ρm ∈ Mη1

and define

�15 νη,lg(η)+m = νη,n+1�θ(ρm).

T ∗η is a subtree of Tϕ therefore ρ ∈ T ∗

η realizes the same Lμ+η ,μ+

ηtype over Aη as

νη1,lg(ρ). The νη,n for n > lg(η) were chosen to satisfy (6)(c)-(d) so in particularthey are in Tϕ, and therefore realize the same Lμ+

η ,μ+η-type over Aη as νη1,n. By

the induction hypothesis we have already constructed for η1 so for all n we havelg(νη,n) = lg(νη1,n) = n so also (6)(a) is satisfied. Requirements (1)-(4) and (6)(b)of the construction are taken care of by �12. �7-�11, �13 and �15 guaranteerequirement (7). �

All that is left in order to complete the proof of the claim is to show that {νη,lg(η) :η ∈ S} is end-uniform with respect to c.Let η1 <2

�x η2 <2�x . . . <2

�x ηk <2�x ρ′, ρ′′, be as in 1.15(2); without loss of generality

277277


ρ′ <∗�x ρ′′. Let t = lg(ρ′ ∩ ρ′′), μ′ = μ+

ρ′ and A = {νρ,lgρ : ρ <∗�x ρ′�(t+ 1)}.

We first show that for every i ≤ k ηi <∗�x ρ′�(t + 1) so that νηi.lg(ηi) ∈ A. As

ηi <2�x ρ′ and lg(ηi ∩ ρ′′) = lg(ηi ∩ ρ′) so ρ′ � ηi, therefore there is �i such that

ηi��i = ρ′��i and ηi(�i) < ρ′(�i), but then ηi��i = ρ′′��i i.e ρ′��i = ρ′′��i so �i ≤ t(and ηi(�i) < ρ′′(�i)) and ηi <

∗�x ρ′�(t+ 1).

We now prove by induction on � ∈ [t, lg(ρ′)] that νρ′��,lgρ′ and νρ′�t,lgρ′ realize thesame Lμ′,μ′-type over A. For � = t this is obvious. Let us assume correctness for �and prove for �+1. For every n < ω by (7)(b) of the construction νρ′��,n, νρ′�(�+1),n

realize the same Lμ+

ρ′�(�+1),μ+

ρ′�(�+1)

-type over {Mρ, νρ,n : ρ <∗�x ρ′�(� + 1)} and

in particular over A, for if ρ <∗�x ρ′�(t + 1) then also ρ <∗

�x ρ′�(� + 1). Soνρ′��,lgρ′ , νρ′�(�+1),lgρ realize the same Lμ+

ρ′�(�+1),μ+

ρ′�(�+1)

-type so also the same Lμ′,μ′-

type over A, and from the induction hypothesis νρ′�t,lgρ′ and νρ′��,lgρ′ realize thesame Lμ′,μ′-type over A. Similarly we show for ρ′′, so νρ′,lgρ′ and νρ′′,lgρ′′ realizethe same Lμ+

η1,μ+

η1-type over A.

From the above we can deduce that in particular

c(〈νη1,lg(η1), . . . , νηk,lg(ηk), νρ′,lg(ρ′)〉) = c(〈νη1,lg(η1), . . . , νηk,lg(ηk), νρ′′,lg(ρ′′)〉).�

Conclusion 2.4. Given a tree S ⊆ ds(∞) and n(∗) < ω and μ we can find atree T ⊆ ds(∞) such that:

(∗)1 For every c : [T ]<ℵ0 → μ there is S′ ⊆ T isomorphic to S such that S′ isn(∗)-end-uniform for c.

(∗)2 In particular, for every c : [T ]n(∗) → μ is S′ ⊆ T isomorphic to S such thatc�S′ depends only on the equivalence classes of the equivalence relationdefined in 1.13.

(∗)3 |T | < �1,n(∗)(|S|, μ) (see Definition 2.5 below).

Proof. Let S, μ be as above. Since for |S|, μ ≥ ℵ0 we have that �1,n(∗)(|S|, μℵ0) =

�1,n(∗)(|S|, μ), replacing μ with μℵ0 gives the same bound, and we can therefore

assume that μ = μℵ0 .Let 〈hn : n < ω〉 be the equivalence classes of the similarity relationship on finitesequences of ds(∞) (see 1.14(1)), and let f : ω(μ ∪ {−1}) → μ be one-to-one andonto.We construct by induction a sequence 〈Tn : n < ω〉 so that T0 = S, and for everyn > 0:

(a) |Tn| < �1,n(|S|, μ)(b) Tn−1, Tn, μ correspond to S, T, μ in Theorem 2.1.(c) For every c : [Tn]

<ℵ0 → μ there is S′ ⊆ Tn isomorphic to S such that S′

is n-end-uniform for c.

By Theorem 2.1 we can obviously construct such a sequence satisfying clauses(a), (b), We will show by induction on n that for this sequence also clause (c) holds.For n = 1 this is Theorem 2.1. Assume correctness for n and let c : [Tn+1]

<ℵ0 → μ.By (b) there is T ′ ⊆ Tn+1 isomorphic to Tn so that T ′ is end-uniform for c. Letϕ : Tn → T ′ be an isomorphism and let d : [T ′]<ℵ0 → ω(μ ∪ {−1}) as follows: forρ = 〈ρ1 . . . ρk〉 where ρ1 <2

�x ρ2 <2�x . . . <2

�x ρk and m < ω

d(ρ)(m) =

{c(ρ�〈η〉) if ρ�〈η〉 ∈ hm for some η-1 otherwise

278278


d is well defined as T ′ is end-uniform for c, and by defining ϕ(ρ1, . . . ρk) = (ϕ(ρ1), . . . ϕ(ρk))for ρ1, . . . ρk ∈ Tn we have f ◦ d ◦ ϕ : [Tn]

<ℵ0 → μ, so by the induction hypothesisthere is T ′′ ⊆ Tn isomorphic to S so that T ′′ is n-end-uniform for f ◦ d ◦ ϕ. Weclaim that S′ = ϕ(T ′′) is isomorphic to S and that S′ is n + 1-end-uniform for c.As T ′′ is isomorphic to S and ϕ is an isomorphism S′ is obviously isomorphic to S.Let the following sequences in S′ be similar,

η1 <2�x< η2 <2

�x . . . <2�x ηk <2

�x ρ′1 <2�x . . . <2

�x ρ′n+1

η1 <2�x< η2 <2

�x . . . <2�x ηk <2

�x ρ′′1 <2�x< . . . < ρ′′n+1

So in T ′′ the following sequences are similar:

ϕ−1(η1 . . . ρ′1 . . . ρ

′n) = (ϕ−1(η1)ϕ

−1(ρ′1) . . . ϕ−1(ρ′n))

ϕ−1(η1 . . . ρ′′1 . . . ρ

′′n) = (ϕ−1(η1)ϕ

−1(ρ′′1) . . . ϕ−1(ρ′′n))

so f ◦ d ◦ϕ(ϕ−1(η1 . . . ηk, ρ′1 . . . ρ

′n)) = f ◦ d ◦ ϕ(ϕ−1(η1 . . . ηk, ρ

′′1 . . . ρ

′′n)). Therefore

we have f(d(η1 . . . ηk, ρ′1 . . . ρ

′n)) = f(d(η1 . . . ηk, ρ

′′1 . . . ρ

′′n)), and as f is one-to-one,

d(η1 . . . ηk, ρ′1 . . . ρ

′n) = d(η1 . . . ηk, ρ

′′1 . . . ρ

′′n), and therefore c(η1 . . . ηk, ρ

′1 . . . ρ

′n+1) =

c(η1 . . . ηk, ρ′′1 . . . ρ

′′n+1), and (∗)1-(∗)3 are easily verified. �

Definition 2.5. For cardinals λ ≥ ℵ0 and μ define �1,α(λ, μ) by induction onα. �1,0(λ, μ) = �0(λ) = λ, �1,α+1(λ, μ) = ��1,α(λ,μ)+(�1,α(λ, μ) + μ), and for alimit ordinal α �1,α(λ, μ) =

∑

β<α

�1,β(λ, μ).

We end with a conclusion for scattered order types.

Conclusion 2.6. For a scattered order type ϕ, a cardinal μ and n < ω, thereis a scattered order type ψ so that ψ → [ϕ]nμ,ℵ0

.

Proof. Given a scattered order type ϕ, a cardinal μ and n < ω by Observation1.4(3) we can embed ϕ in (ds(α), <3) for some ordinal α. By Conclusion 2.4(∗)2above there is an ordinal λ and a tree T ⊂ ds(λ) so that for every coloring c :Tn → μ there is a subtree S ⊆ T isomorphic to ds(α) so that c�S depends only onthe equivalence class of similarity. Noting the above Observation, as (T,<3) is ascattered order, and as there are only ℵ0 equivalence classes, we are done. �

References

1. M. A. Dickman, Larger infinitary languages, Model Theoretic Logics (J. Barwise and S. Fefer-

man, eds.), Perspectives in Mathematical Logic, Springer-Verlag, New York Berlin HeidelbergTokyo, 1985, pp. 317–364.

2. Andras Hajnal and Peter Komjath, A strongly non-Ramsey order type, Combinatorica 17,no.3 (1997), 363–367.

3. F. Hausdorff, Grundzuge einer Theorie der geordneten Mengen, Math. Ann. 65 (1908), no. 4,435–505. MR MR1511478

4. Peter Komjath and Saharon Shelah, A partition theorem for scattered order types, Combina-torics Probability and Computing 12 (2003, no.5-6), 621–626, Special issue on Ramsey theory.math.LO/0212022.

5. Joseph G. Rosenstein, Linear orderings, Pure and Applied Mathematics, vol. 98, Aca-demic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982. MR MR662564(84m:06001)

6. Saharon Shelah, Consistency of positive partition theorems for graphs and models, Set theoryand its applications (Toronto, ON, 1987), Lecture Notes in Mathematics, vol. 1401, Springer,Berlin-New York, 1989, ed. Steprans, J. and Watson, S., pp. 167–193.

279279


Institute of Mathematics The Hebrew University of Jerusalem, Jerusalem 91904,

Israel. e-mail: [email protected]

Institute of Mathematics The Hebrew University of Jerusalem, Jerusalem 91904,

Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08854,

USA e-mail: [email protected]

280280


A Classical Proof of the Kanovei-Zapletal Canonization

Benjamin D. Miller

Abstract. We give a classical proof of the Kanovei-Zapletal canonization ofBorel equivalence relations on Polish spaces [5, 6].

1. Introduction

A canonization theorem for a class M of structures is a result asserting thatfor some small subclass N ⊆ M, some class X of large sets, and every structureM ∈ M, there exist N ∈ N and X ∈ X such that M � X = N � X.

Here we consider such theorems in the context of descriptive set theory. A well-known example is the following straightforward corollary of Mycielski’s theorem onmeager subsets of the plane (see §8 of [7]):

Theorem 1 (Galvin). Suppose that X is a perfect Polish space and E is anequivalence relation on X which has the Baire property. Then there is a perfect setB ⊆ X such that E � B ∈ {Δ(B), B ×B}, where Δ(B) = {(x, x) | x ∈ B}.

It is natural to ask whether there are analogous theorems if we consider evenlarger sets. One must of course be careful here, as in the presence of the axiomof choice, perfect subsets of Polish spaces are as large as they come. Fortunately,work in descriptive set theory over the past two decades has provided us with anatural successor of the continuum among the definable cardinals.

Suppose that X is a standard Borel space. A Borel equivalence relation F onX is smooth if there is a Borel function ϕ : X → 2ω such that

∀x0, x1 ∈ X (x0Fx1 ⇐⇒ ϕ(x0) = ϕ(x1)).

Suppose that B ⊆ X is a Borel set. We say that B is F -smooth if F � B is smooth.Otherwise, we say that B is F -non-smooth.

Suppose now that X is a Polish space and F is a Borel equivalence relation onX which is not smooth. Theorem 1 implies that every perfect subset of X containsan F -smooth perfect set, and the Harrington-Kechris-Louveau dichotomy theorem[4] implies that every F -non-smooth Borel subset of X contains an F -non-smooth

2010 Mathematics Subject Classification. Primary 03E15; Secondary 03E05.Key words and phrases. Canonization, Glimm-Effros dichotomy.I would like to thank the organizers of the 2009 Boise Extravaganza in Set Theory for sup-

porting my visit to Boise and encouraging me to submit this paper.


1



281



281

2 BENJAMIN D. MILLER

perfect set. In particular, it follows that the condition of being F -non-smooth isstrictly stronger than the condition of containing a perfect set.

One natural attempt at strengthening Theorem 1 is to fix a Borel equivalencerelation F on X which is not smooth, and to augment the conclusion of the theo-rem by asking that the set B is F -non-smooth. Unfortunately, this version of theresult cannot possibly hold in the special case that E = F , as the equivalence rela-tions Δ(B) and B×B are themselves smooth. The Kanovei-Zapletal canonizationtheorem [5, 6] asserts that this is the only counterexample to the stronger result:

Theorem 2 (Kanovei-Zapletal). Suppose that X is a Polish space, E and Fare Borel equivalence relations on X, and F is not smooth. Then there is an F -non-smooth Borel set B ⊆ X such that E � B ∈ {Δ(B), F � B,B ×B}.

The original proof of this result used effective descriptive set theory and forcing.The goal of this note is to point out that, at least in the special case when X = 2ω

and F is the equivalence relation given by

xE0y ⇐⇒ ∃m ∈ ω∀n ∈ ω \m (x(n) = y(n)),

it can also be seen as a corollary of two essentially well-known facts using purelyclassical methods, and the full theorem can then be obtained via an application ofthe Harrington-Kechris-Louveau dichotomy theorem [4].

2. Preliminaries

Suppose that X and Y are sets. A homomorphism from a set R ⊆ X ×X to aset S ⊆ Y × Y is a function ϕ : X → Y such that

∀x0, x1 ∈ X ((x0, x1) ∈ R =⇒ (ϕ(x0), ϕ(x1)) ∈ S).

A homomorphism from a sequence (Ri)i∈I of subsets of X×X to a sequence (Si)i∈I

of subsets of Y × Y is a function ϕ : X → Y which is a homomorphism from Ri toSi for all i ∈ I.

The following Mycielski-style fact is implicit in many arguments involving Borelequivalence relations and Baire category:

Proposition 3. Suppose that R ⊆ 2ω × 2ω is meager. Then there is a contin-uous homomorphism from (Δ(2ω)c, Ec

0, E0) to (Δ(2ω)c, Rc, E0).

Proof. Fix a decreasing sequence of dense open sets Un ⊆ Δ(2ω)c with theproperty that R ∩

⋂n∈ω Un = ∅. We will recursively construct natural numbers

kn ∈ ω and functions un : 2n → 2kn such that:

(1) ∀n ∈ ω (kn < kn+1).(2) ∀i ∈ 2∀n ∈ ω∀s ∈ 2n (un(s) � un+1(s

�i)).(3) ∀i ∈ 2∀n ∈ ω∀s, t ∈ 2n (Nun+1(s�i) ×Nun+1(t�(1−i)) ⊆ Un).(4) ∀i, j ∈ 2∀n ∈ ω∀s, t ∈ 2n

(i = j ⇐⇒ un+1(s�i) � [kn, kn+1) = un(t

�j) � [kn, kn+1)).

We begin by setting k0 = 0 and u0(∅) = ∅. Suppose now that we have foundkn ∈ ω and un : 2

n → 2kn . By a straightforward recursion of finite length, we canfind k ∈ ω and distinct sequences v0, v1 ∈ 2k such thatNun(s)�vi×Nun(t)�v1−i

⊆ Un

for all i ∈ 2 and s, t ∈ 2n. Set kn+1 = kn + k and un+1(s�i) = un(s)

�vi for i ∈ 2and s ∈ 2n. This completes the recursive construction.

282282

THE KANOVEI-ZAPLETAL CANONIZATION 3

Conditions (1) and (2) ensure that we obtain a continuous function π : 2ω → 2ω

by setting π(x) = limn→ω un(x � n), condition (3) implies that π is a homomor-phism from (Δ(2ω)c, Ec

0) to (Δ(2ω)c, Rc), and condition (4) ensures that π is ahomomorphism from E0 to E0.

We say that an equivalence relation is countable if all of its equivalence classesare countable. The following fact is a straightforward generalization of the Glimm-Effros dichotomy theorem [1, 3]:

Theorem 4. Suppose that X is a Polish space, E ⊆ F are countable Bor-el equivalence relations on X, and E is not smooth. Then there is a continuoushomomorphism from (Δ(2ω)c, Ec

0, E0) to (Δ(X)c, F c, E).

Proof. The orbit equivalence relation associated with a group G of permuta-tions of X is given by

x0EXG x1 ⇐⇒ ∃g ∈ G (g · x0 = x1).

In [2], Feldman-Moore established that every countable Borel equivalence relationon a Polish space is the orbit equivalence relation associated with a countablegroup of Borel automorphisms (see also Theorem 1.3 of [8]). This trivially impliesthat there are countable groups G ≤ H of Borel automorphisms of X such thatE = EX

G and F = EXH . Fix an increasing sequence of finite symmetric sets Hn ⊆ H

containing 1H such that H =⋃

n∈ω Hn. By standard change of topology results(see §13 of [7]), we can assume that X is a zero-dimensional Polish metric spaceand H is a group of homeomorphisms of X.

We will recursively construct clopen sets Un ⊆ X and homeomorphisms gn ∈ Gsuch that for all n ∈ ω, the following conditions are satisfied:

(1) Un is E-non-smooth.(2) Un+1 ⊆ Un ∩ g−1

n (Un).

(3) ∀s ∈ 2n+1 (diam(gs(Un+1)) ≤ 1/(n+ 1)), where gs =∏

i∈|s| gs(i)i .

(4) ∀h ∈ Hn∀s, t ∈ 2n (hgs(Un+1) ∩ gtgn(Un+1) = ∅).We begin by setting U0 = X.

Suppose now that n ∈ ω and we have already found Un ⊆ X and gm ∈ G forall m ∈ n. For each g ∈ G, define an open set Vg ⊆ X by

Vg = {x ∈ Un ∩ g−1(Un) | ∀h ∈ Hn∀s, t ∈ 2n (hgs · x = gtg · x)}.Set C = Un \

⋃g∈G Vg, and observe that if (x, y) ∈ E � C, then there exists g ∈ G

such that g · x = y, so the fact that x /∈ Vg ensures the existence of h ∈ Hn and

s, t ∈ 2n such that hgs · x = gtg · x = gt · y, thus y = g−1t hgs · x. As there are only

finitely many possible values of g−1t hgs, it follows that C intersects each equivalence

class of E in a finite set, and is therefore E-smooth. As Un = C ∪⋃

g∈G Vg and Un

is E-non-smooth, there exists gn ∈ G such that Vgn is E-non-smooth.Our topological assumptions ensure that Vgn is the union of countably many

clopen sets U ⊆ X which satisfy the following conditions:

(a) ∀s ∈ 2n+1 (diam(gs(U)) ≤ 1/(n+ 1)).(b) ∀h ∈ Hn∀s, t ∈ 2n (hgs(U) ∩ gtgn(U) = ∅).

Let Un+1 be any such E-non-smooth set. This completes the recursive construction.Conditions (2) and (3) ensure that for all x ∈ 2ω, the clopen sets of the form

gx�n(Un), for n ∈ ω, are decreasing and of vanishing diameter. We therefore obtain

283283

4 BENJAMIN D. MILLER

a continuous function π : 2ω → X by setting

π(x) = the unique element of⋂

n∈ω

gx�n(Un).

Lemma 5. If k ∈ ω, s ∈ 2k, and x ∈ 2ω, then π(s�x) = gs · π(0k�x).Proof of lemma. Simply observe that

{π(s�x)} =⋂

n∈ω

gs�(x�n)(Uk+n)

=⋂

n∈ω

gsg0k�(x�n)(Uk+n)

= gs

(⋂

n∈ω

g0k�(x�n)(Uk+n)

)

= gs({π(0k�x)}),thus π(s�x) = gs · π(0k�x).

Lemma 6. If n ∈ ω, x, y ∈ 2ω, and x(n) = y(n), then π(y) /∈ Hn · π(x).Proof of lemma. By reversing the roles of x and y if necessary, we can as-

sume that x(n) = 0 and y(n) = 1. Suppose, towards a contradiction, that thereexists h ∈ Hn with π(y) = h · π(x). Set s = x � n and t = y � n. Lemma 5 ensuresthat the points x′ = g−1

s · π(x) = g−1s h−1 · π(y) and y′ = g−1

n g−1t · π(y) are both in

Un+1, thus π(y) ∈ hgs(Un+1) ∩ gtgn(Un+1), which contradicts condition (4).

Lemma 5 implies that π is a homomorphism from E0 to E, and Lemma 6ensures that π is a homomorphism from (Δ(2ω)c, Ec

0) to (Δ(X)c, F c).

3. Canonization

With the preliminaries in hand, we can now establish our primary results:

Theorem 7 (Kanovei-Zapletal). Suppose that E is a Borel equivalence relationon 2ω. Then there is an E0-non-smooth Borel set B ⊆ 2ω such that E � B ∈{Δ(B), E0 � B,B ×B}.

Proof. A straightforward Baire category argument shows that all non-meagerBorel subsets of 2ω are E0-non-smooth. We use this fact freely throughout the proof.

If there exists x ∈ 2ω such that [x]E is non-meager, then the set B = [x]E is asdesired, since E � B = B×B. Otherwise, the Kuratowski-Ulam theorem (see §8 of[7]) implies that E is meager, so Proposition 3 ensures the existence of a continuoushomomorphism ϕ : 2ω → 2ω from (Δ(2ω)c, Ec

0, E0) to (Δ(2ω)c, (E ∪ E0)c, E0). Set

F = (ϕ× ϕ)−1(E), noting that F ⊆ E0.If F is smooth, then there is a Borel transversal A ⊆ 2ω of F , in which case

A is non-meager and F � A = Δ(A), so the set B = ϕ(A) is as desired, since E �B = Δ(B). If F is not smooth, then Theorem 4 gives a continuous homomorphismψ : 2ω → 2ω from (Δ(2ω)c, Ec

0, E0) to (Δ(X)c, Ec0, F ), so the set B = ϕ ◦ ψ(2ω) is

as desired, since E � B = E0 � B.

Theorem 8 (Kanovei-Zapletal). Suppose that X is a Polish space, E and Fare Borel equivalence relations on X, and F is not smooth. Then there is an F -non-smooth Borel set B ⊆ X such that E � B ∈ {Δ(B), F � B,B ×B}.

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THE KANOVEI-ZAPLETAL CANONIZATION 5

Proof. The Harrington-Kechris-Louveau dichotomy theorem [4] yields a con-tinuous homomorphism π : 2ω → X from (Δ(2ω)c, Ec

0, E0) to (Δ(X)c, F c, F ). SetE′ = (π × π)−1(E) and F ′ = (π × π)−1(F ) = E0. By Theorem 7, there is anF ′-non-smooth set B′ ⊆ 2ω such that E′ � B′ ∈ {Δ(B′), F ′ � B′, B′×B′}, in whichcase the set B = π(B′) is as desired.

References

1. Edward G. Effros, Transformation groups and C∗-algebras, Ann. of Math. (2) 81 (1965), 38–55.MR MR0174987 (30 #5175)

2. Jacob Feldman and Calvin C. Moore, Ergodic equivalence relations, cohomology, and vonNeumann algebras. I, Trans. Amer. Math. Soc. 234 (1977), no. 2, 289–324. MR MR0578656(58 #28261a)

3. James Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961),124–138. MR MR0136681 (25 #146)

4. L. A. Harrington, A. S. Kechris, and A. Louveau, A Glimm-Effros dichotomy for Borel equiv-alence relations, J. Amer. Math. Soc. 3 (1990), no. 4, 903–928. MR MR1057041 (91h:28023)

5. Vladimir Kanovei, Canonization of Borel equivalence relations on large sets, Euler and moderncombinatorics, international conference (St. Petersburg, Russia), Euler International Mathe-matical Institute, June 2007, pp. 12–13.

6. Vladimir Kanovei and Jindrich Zapletal, Canonizing Borel equivalence relations on Polishspaces, Preprint, 2007.

7. Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol.156, Springer-Verlag, New York, 1995. MR MR1321597 (96e:03057)

8. Alexander S. Kechris and Benjamin D. Miller, Topics in orbit equivalence, Lecture Notes inMathematics, vol. 1852, Springer-Verlag, Berlin, 2004. MR MR2095154 (2005f:37010)

Benjamin D. Miller, 8159 Constitution Road, Las Cruces, New Mexico 88007

E-mail address: [email protected]: http://glimmeffros.googlepages.com

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Lords of the iteration

Andrzej Ros�lanowski and Saharon Shelah

Abstract. We introduce several properties of forcing notions which implythat their λ–support iterations are λ–proper. Our methods and techniquesrefine those studied in [RS01], [RS07], [RS05] and [RS], covering some newforcing notions (though the exact relation of the new properties to the old onesremains undecided).

0. Introduction

Since the beginning of 1980s it has been known that the theory of proper forcingdoes not admit naive generalization to the context of larger cardinals and iterationswith larger supports. The evidence of that was given already in Shelah [She82](see [She98, Appendix 3.6(2)]). It seems that the first steps towards developing thetheory of forcing iterated with uncountable supports were done in Shelah [She03a],[She03b], but the properties introduced there were aimed at situations when wedo not want to add new subsets of λ (corresponding to the case of no new realsin CS iterations of proper forcing notions). Later Ros�lanowski and Shelah [RS01]introduced an iterable property called properness over semi-diamonds and thenEisworth [Eis03] proposed an iterable relative of it. These properties work nicelyfor λ–support iterations (where λ = λ<λ is essentially arbitrary) and forcings addingnew subsets of λ, but the price to pay is that many natural examples are not covered.If we restrict ourselves to inaccessible λ, then the properties given by Ros�lanowskiand Shelah [RS07, RS05, RS] may occur useful. Those papers give both iterationtheorems and new examples of forcing notions for which the theorems apply.

In the present paper we further advance the theory and we give results ap-plicable to both the case of inaccessible λ as well as those working for successorcardinals. The tools developed here may be treated as yet another step towardscomparing and contrasting the structure of λλ with that of ωω. That line of research

1991 Mathematics Subject Classification. Primary 03E40; Secondary:03E35.Key words and phrases. iterated forcing, λ–support, iteration theorems,The first author would like to thank the Hebrew University of Jerusalem and the Lady

Davis Fellowship Trust for awarding him with Schonbrunn Visiting Professorship under whichthis research was carried out.

Both authors acknowledge support from the United States-Israel Binational Science Founda-tion (Grant no. 2002323). This is publication 888 of the second author.


1



287



287

2 ANDRZEJ ROS�LANOWSKI AND SAHARON SHELAH

already has received some attention in the literature (see e.g., Cummings and She-lah [CS95], Shelah and Spasojevic [SS02] or Zapletal [Zap97]). Also with betteriteration theorems one may hope for further generalizations of Ros�lanowski andShelah [RS99] to the context of uncountable cardinals. (Initial steps in the latterdirection were presented in Ros�lanowski and Shelah [RS07].) However, while we dogive some examples of forcing notions to which our properties apply, we concentrateon the development of the theory of forcing leaving the real applications for furtherinvestigations. The need for the development of such general theory was indirectlystated by Hytinnen and Rautila in [HR01], where they commented:

Our proof is longer than the one in [MS93] partly because weare not able to utilize the general theory of proper forcing, espe-cially the iteration lemma, but we have to prove everything “fromscratch”.

We believe that the present paper brings us substantially closer to the right generaliteration theorems for iterations with uncountable supports.

In the first section we introduce D�–parameters (which will play an importantrole in our definitions) and a slight generalization of the B–bounding property from[RS05]. We also define a canonical example for testing usefulness of our iteration

theorems: the forcing QEE in which conditions are complete λ–trees in which along

each λ-branch the set of splittings forms a set from a filter E (and the splitting atν is into a set from a filter Eν on λ). The main result of the first section (Theorem

1.10) says that we may iterate with λ–supports forcing notions QEE , provided λ is

inaccessible and E is always the same and has some additional properties.If we want to iterate forcing notions like QE

E but with different E on eachcoordinate (when the result of the first section is not applicable), we may decideto use very orthogonal filters. Section 2 presents an iteration theorem 2.7 which istailored for such situation. Also here we need the assumption that λ is inaccessible.

The following section introduces B–noble forcing notions and the iteration the-orem 3.3 for them. The main gain here is that it allows us to iterate (with λ–

supports) forcing notions like QEE even if λ is not inaccessible. The fourth section

gives more examples of forcing notions and shows a possible application. In Corol-lary 4.5 we substantially improve a result from [RS05] showing that dominatingnumbers associated with different filters may be distinct even if λ is a successor.

The fifth section shows that some of closely related forcing notions may havedifferent properties. Section 6 presents yet another property that is useful in λ–support iterations (for inaccessible λ): reasonably merry forcing notions. Thisproperty has the flavour of putting together being B–bounding (of [RS05]) withbeing fuzzy proper (of [RS07]). We also give an example of a forcing notion whichis reasonably merry but which was not covered by earlier properties. We concludethe paper with a section listing open problems.

This research is a natural continuation of papers mentioned earlier ([She03a],[She03b], [RS01], [RS07], [RS05] and [RS]). All our iteration proofs are basedon trees of conditions and the arguments are similar to those from the earlier works.While we tried to make this presentation self-contained, the reader familiar withthe previous papers will definitely find the proofs presented here easier to follow(as several technical aspects do re-occur).

288288

LORDS OF THE ITERATION 3

0.1. Notation. Our notation is rather standard and compatible with that ofclassical textbooks (like Jech [Jec03]). In forcing we keep the older convention thata stronger condition is the larger one.

(1) Ordinal numbers will be denoted be the lower case initial letters of theGreek alphabet (α, β, γ, δ . . .) and also by i, j (with possible sub- andsuperscripts).

Cardinal numbers will be called κ, λ, μ; λ will be always assumedto be a regular uncountable cardinal such that λ<λ = λ (we mayforget to mention this).

Also, χ will denote a sufficiently large regular cardinal; H(χ) is thefamily of all sets hereditarily of size less than χ. Moreover, we fix a wellordering <∗

χ of H(χ).(2) We will consider several games of two players. One player will be called

Generic or Complete or just COM , and we will refer to this player as“she”. Her opponent will be called Antigeneric or Incomplete or just INCand will be referred to as “he”.

(3) For a forcing notion P, almost all P–names for objects in the extensionvia P will be denoted with a tilde below (e.g., τ

˜, X˜). There will be some

exceptions to this rule, however. ΓP will stand for the canonical P–namefor the generic filter in P. Also some (names for) normal filters generatedin the extension from objects in the ground model will be denoted by D,DP or D[P].

The weakest element of P will be denoted by ∅P (and we will alwaysassume that there is one, and that there is no other condition equivalent toit). All forcing notions under considerations are assumed to be atomless.

By “λ–support iterations” we mean iterations in which domains ofconditions are of size ≤ λ. However, on some occasions we will pretendthat conditions in a λ–support iteration Q = 〈Pζ ,Q

˜ζ : ζ < ζ∗〉 are total

functions on ζ∗ and for p ∈ lim(Q) and α ∈ ζ∗ \ dom(p) we will letp(α) = ∅

˜Q

˜α .

(4) By “a sequence” we mean “a function defined on a set of ordinals” (so thedomain of a sequence does not have to be an ordinal). For two sequencesη, ν we write ν�η whenever ν is a proper initial segment of η, and ν � ηwhen either ν�η or ν = η. The length of a sequence η is the order typeof its domain and it is denoted by lh(η).

(5) A tree is a �–downward closed set of sequences. A complete λ–tree is atree T ⊆ <λλ such that every �-chain of size less than λ has an �-boundin T and for each η ∈ T there is ν ∈ T such that η�ν.

Let T be a λ–tree. For η ∈ T we let

succT (η) = {α < λ : η�〈α〉 ∈ T} and (T )η = {ν ∈ T : ν�η or η � ν}.We also let root(T ) be the shortest η ∈ T such that |succT (η)| > 1 andlimλ(T ) = {η ∈ λλ : (∀α < λ)(η�α ∈ T )}.

0.2. Background on trees of conditions.

Definition 0.1. Let P be a forcing notion.

(1) For a condition r ∈ P, let �λ0 (P, r) be the following game of two players,

Complete and Incomplete:

289289


the game lasts at most λ moves and during a play theplayers construct a sequence 〈(pi, qi) : i < λ〉 of pairs ofconditions from P in such a way that (∀j < i < λ)(r ≤pj ≤ qj ≤ pi) and at the stage i < λ of the game, firstIncomplete chooses pi and then Complete chooses qi.

Complete wins if and only if for every i < λ there are legal moves for bothplayers.

(2) We say that the forcing notion P is strategically (<λ)–complete if Completehas a winning strategy in the game �λ

0 (P, r) for each condition r ∈ P.(3) Let N ≺ (H(χ),∈, <∗

χ) be a model such that <λN ⊆ N , |N | = λ andP ∈ N . We say that a condition p ∈ P is (N,P)–generic in the standardsense (or just: (N,P)–generic) if for every P–name τ

˜∈ N for an ordinal

we have p �“ τ˜∈ N ”.

(4) P is λ–proper in the standard sense (or just: λ–proper) if there is x ∈ H(χ)such that for every model N ≺ (H(χ),∈, <∗

χ) satisfying

<λN ⊆ N, |N | = λ and P, x ∈ N,

and every condition q ∈ N ∩ P there is an (N,P)–generic condition p ∈ Pstronger than q.

Remark 0.2. Let us recall that if P is either strategically (<λ+)–complete orλ+–cc, then P is λ–proper. Also, if P is λ–proper then

• λ+ is not collapsed in forcing by P, moreover• for every set of ordinals A ∈ VP of size λ there is a set A+ ∈ V of size λsuch that A ⊆ A+.

Definition 0.3 (Compare [RS07, Def. A.1.7], see also [RS05, Def. 2.2]).

(1) Let γ be an ordinal, ∅ �= w ⊆ γ. A (w, 1)γ–tree is a pair T = (T, rk) suchthat

• rk : T −→ w ∪ {γ},• if t ∈ T and rk(t) = ε, then t is a sequence 〈(t)ζ : ζ ∈ w ∩ ε〉,• (T,�) is a tree with root 〈〉 and• if t ∈ T , then there is t′ ∈ T such that t � t′ and rk(t′) = γ.

(2) If, additionally, T = (T, rk) is such that every chain in T has a �–upperbound in T , we will call it a standard (w, 1)γ–tree

We will keep the convention that T xy is (T x

y , rkxy).

(3) Let Q = 〈Pi,Q˜

i : i < γ〉 be a λ–support iteration. A tree of conditions in

Q is a system p = 〈pt : t ∈ T 〉 such that• (T, rk) is a (w, 1)γ–tree for some w ⊆ γ,• pt ∈ Prk(t) for t ∈ T , and• if s, t ∈ T , s�t, then ps = pt�rk(s).

If, additionally, (T, rk) is a standard tree, then p is called a standard treeof conditions.

(4) Let p0, p1 be trees of conditions in Q, pi = 〈pit : t ∈ T 〉. We write p0 ≤ p1

whenever for each t ∈ T we have p0t ≤ p1t .

Note that our standard trees and trees of conditions are a special case of that[RS07, Def. A.1.7] when α = 1.

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Proposition 0.4. Assume that Q = 〈Pi,Q˜

i : i < γ〉 is a λ–support iterationsuch that for all i < γ we have

�Pi“ Q˜

i is strategically (<λ)–complete ”.

Suppose that p = 〈pt : t ∈ T 〉 is a tree of conditions in Q, |T | < λ, and I ⊆ Pγ isopen dense. Then there is a tree of conditions q = 〈qt : t ∈ T 〉 such that p ≤ q and(∀t ∈ T )(rk(t) = γ ⇒ qt ∈ I).

Proof. This is essentially [RS07, Proposition A.1.9] and the proof there ap-plies here without changes. �

1. D�–parameters

In this section we introduce D�–parameters and we use them to get a possibleslight improvement of [RS05, Theorem 3.1] (in Theorem 1.10). We also define our

canonical testing forcing QEE to which this result can be applied.

Definition 1.1. (1) A pre-D�–parameter on λ is a triple p = (P , S,D) =(Pp, Sp, Dp) such that

• D is a proper uniform normal filter on λ, S ∈ D,• P = 〈Pδ : δ ∈ S〉 and Pδ ∈ [δδ]<λ for each δ ∈ S.

(2) For a function f ∈ λλ and a pre-D�–parameter p = (P , S,D) we let

setp(f) = {δ ∈ S : f�δ ∈ Pδ}.(3) We say that a pre-D�–parameter p = (P , S,D) is a D�–parameter on λ if

setp(f) ∈ D for every f ∈ λλ.

Example 1.2. (1) If λ is strongly inaccessible, D is the filter generatedby club subsets of λ and Pδ = δδ, P = 〈Pδ : δ < λ〉, then (P , λ,D) is aD�–parameter on λ.

(2) ♦+λ is a statement asserting existence of a D�–parameter with the filter

generated by clubs of λ.(3) ♦λ implies the existence of a D�–parameter (P , S,D) such that |Pδ| = |δ|.(4) For more instances of the existence of D�–parameters we refer the reader

to Shelah [She00, §3].Definition 1.3. Let p be a pre-D�–parameter on λ and Q be a forcing notion

not collapsing λ. In VQ we define

• Dp[Q] = Dp[Q] is the normal filter generated by Dp ∪ {setp(f) : f ∈ λλ},• p[Q] = (Pp, Sp, Dp[Q]).

Remark 1.4. If Q is a strategically (<λ)–complete forcing notion and D isa (proper) normal filter on λ, then in VQ the normal filter on λ generated byD ∩V is also a proper filter. Abusing notation, we will denote this filter by D (orDQ). The filter Dp[Q] can be larger, but it is still a proper filter, provided p is aD�–parameter.

Lemma 1.5. Assume that p = (P , S,D) is a D�–parameter on λ and Q is astrategically (<λ)–complete forcing notion. Then �Q ∅ /∈ Dp[Q]. Consequently,�Q“ p[Q] is a D�–parameter on λ”.

Proof. Assume that p ∈ Q and A˜

δ is a Q–name for an element of D ∩V andf˜δ is a Q–name for an element of λλ (for δ < λ). Using the strategic completeness

of Q build a sequence 〈pδ, Aδ, fδ : δ < λ〉 such that for each δ < λ:

291291


(i) pδ ∈ Q, p ≤ p0 ≤ pα ≤ pδ for α < δ,(ii) Aδ ∈ D ∩V, fδ ∈ λλ and(iii) pδ �Q“ A

˜δ = Aδ and f

˜α�δ = fα�δ for all α ≤ δ ”.

Since p is a D�–parameter, we know that B = �δ<λ

Aδ ∩ �δ<λ

setp(fδ) ∈ D. Let

δ ∈ B. Then

pδ �Q “ δ ∈ �α<λ

A˜

α and f˜α�δ = fα�δ ∈ Pδ for all α < δ ”,

so pδ �Q “ δ ∈ �α<λ

A˜

α ∩ �α<λ

setp(f˜α) ”. �

Lemma 1.6. Assume that λ<λ = λ, p = (P , S,D) is a D�–parameter on λ, Qis a strategically (<λ)–complete forcing notion and N ≺ (H(χ),∈, <∗

χ) is such that

p ∈ N , |N | = λ and <λN ⊆ N . Let 〈Nδ : δ < λ〉 be an increasing continuoussequence of elementary submodels of N such that p ∈ N0, δ ⊆ Nδ, Pδ ⊆ Nδ+1,〈Nε : ε ≤ δ〉 ∈ Nδ+1 and |Nδ| < λ (for δ < λ). Then

�Q “(∀A ⊆ N

)({δ < λ : A ∩Nδ ∈ Nδ+1

}∈ D[Q]

)”.

Proof. We may find an increasing continuous sequence 〈αδ : δ < λ〉 ⊆ λ anda bijection f : N −→ λ such that f [Nδ] = αδ and f�Nδ ∈ Nδ+1 (for δ < λ). ForA ⊆ N let ϕA : λ −→ 2 be such that ϕA(α) = 1 if and only if f−1(α) ∈ A. Plainly,if δ = αδ and ϕA�δ ∈ Pδ, then A ∩Nδ ∈ Nδ+1. �

Let Cλ0 be a forcing notion consisting of all pairs (α, f) such that α < λ and

f ∈∏

β<α

(β +1) ordered by the extension (so (α, f) ≤ (α′, f ′) if and only if f ⊆ f ′).

Thus it is a (<λ)–complete forcing notion which is an incarnation of the λ–Cohenforcing notion.

Proposition 1.7. Assume λ is strongly inaccessible. If p = (P , S,D) is a

D�–parameter on λ such that (∀δ ∈ S)(|Pδ| ≤ |δ|), then �C

λ0“ DC

λ0 �= D[Cλ

0 ] ”.

Proof. Let f˜be the canonical Cλ

0–name for the generic function in∏

α<λ

(α+1),

so (α, f) �C

λ0f ⊆ f

˜. Plainly, �

Cλ0setp(f

˜) ∈ D[Cλ

0 ] and we are going to argue

that �C

λ0λ \ setp(f

˜) ∈

(DC

λ0

)+. To this end, suppose that p ∈ Cλ

0 and A˜

δ is a

Cλ0–name for an element of D ∩ V (for δ < λ). By induction on ξ < λ choose

〈αξ, Bξ, pξ : ξ < λ〉 so that

(α) 〈αξ : ξ < λ〉 is an increasing continuous sequence of ordinals below λ,(β) Bξ ∈ D, pξ = 〈pξσ : σ ∈

∏

ζ<ξ

(αζ + 1)〉 ⊆ Cλ0 , p

0〈〉 = p = (α0, f

0〈〉),

(γ) if σ ∈∏

ζ<ξ

(αζ + 1), then pξσ = (αξ, fξσ) and fξ

σ(αζ) = σ(ζ) for ζ < ξ,

(δ) if ξ < ξ′, σ′ ∈∏

ζ<ξ′(αζ + 1) and σ = σ′�ξ, then pξσ ≤ pξ

′

σ′ ,

(ε) if ξ < λ is limit, σ ∈∏

ζ<ξ

(αζ + 1), then (αξ = sup(αζ : ζ < ξ) and)

fξσ =

⋃

ζ<ξ

fζσ�ζ , and

(ζ) pξ+1σ � Bξ ⊆ A

˜ξ for every σ ∈

∏

ζ≤ξ

(αζ + 1).

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(Remember that λ is inaccessible, so∣∣∏

ζ<ξ

(αζ + 1)∣∣ < λ for each ξ.) Next, consider

the set B = �ξ<λ

Bξ ∈ D. Let δ ∈ B∩S be a limit ordinal. Since |Pδ| ≤ |δ| <∏

ξ<δ

|αξ|,

we may pick σ ∈∏

ξ<δ

(αξ+1) such that fδσ /∈ Pδ. Then pδσ � δ ∈ �

ξ<λA˜

ξ \setp(f˜). �

Definition 1.8. Let p = (P , S,D) be a D�–parameter on λ, Q be a strategi-cally (<λ)–complete forcing notion.

(1) For a condition p ∈ Q we define a game �rbBp (p,Q) between two players,

Generic and Antigeneric, as follows. A play of �rbBp (p,Q) lasts λ steps

and during a play a sequence⟨Iα, 〈pαt , qαt : t ∈ Iα〉 : α < λ

⟩

is constructed. Suppose that the players have arrived to a stage α < λ ofthe game. Now,(ℵ)α first Generic chooses a set Iα of cardinality < λ and a system 〈pαt :

t ∈ Iα〉 of conditions from Q,1

(�)α then Antigeneric answers by picking a system 〈qαt : t ∈ Iα〉 of condi-tions from Q such that (∀t ∈ Iα)(p

αt ≤ qαt ).

At the end, Generic wins the play⟨Iα, 〈pαt , qαt : t ∈ Iα〉 : α < λ

⟩of

�rbBp (p,Q) if and only if

(�)prbB there is a condition p∗ ∈ Q stronger than p and such that

p∗ �Q “{α < λ :

(∃t ∈ Iα

)(qαt ∈ ΓQ

)}∈ D[Q] ”.

(2) A forcing notion Q is reasonably B–bounding over p if for any p ∈ Q,Generic has a winning strategy in the game �rbB

p (p,Q).

Remark 1.9. The notion introduced in 1.8 is almost the same as the one of[RS05, Definition 3.1(2),(5)]. The difference is that in (�)prbB we use the filter D[Q]and not DQ = D, so potentially we have a weaker property here. We do not know,however, if there exists a forcing notion which is reasonably B–bounding over pand not reasonably B–bounding over D. (See Problem 7.1.)

In a similar fashion we may also modify the property of being nicely doubleb–bounding (see [RS, Definition 2.9(2),(4)]) and get the parallel iteration theorem.

Theorem 1.10. Assume that

(1) λ is a strongly inaccessible cardinal and p is a D�–parameter on λ,(2) Q = 〈Pα,Q

˜α : α < γ〉 is a λ–support iteration,

(3) for every α < λ, �Pα“ Q˜

α is reasonably B–bounding over p[Pα] ”.

Then

(a) Pγ = lim(Q) is λ–proper,(b) if τ

˜is a Pγ–name for a function from λ to V, p ∈ Pγ , then there are

q ≥ p and 〈Aξ : ξ < λ〉 such that (∀ξ < λ)(|Aξ| < λ) and

q � “ {ξ < λ : τ˜(ξ) ∈ Aξ} ∈ Dp[Pγ ] ”.

Proof. The proof is essentially the same as that of [RS05, Theorem 3.1] witha small modification at the end (in Claim 3.1 there); compare with the proof ofTheorem 2.7 here and specifically with 2.7.1. �

1Note that no relation between pαt and pβs for β < α is required to hold.

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Definition 1.11. Let E = 〈Eν : ν ∈ <λλ〉 be a system of (<λ)–completenon-principal filters on λ and let E be a normal filter on λ. We define a forcingnotion QE

E as follows.

A condition p in QEE is a complete λ–tree p ⊆ <λλ such that

• for every ν ∈ p, either |succp(ν)| = 1 or succp(ν) ∈ Eν , and• for every η ∈ limλ(p) the set {α < λ : succp(η�α) ∈ Eη�α} belongs to E.

The order ≤=≤Q

EEis the reverse inclusion: p ≤ q if and only if (p, q ∈ QE

E and )

q ⊆ p.

Proposition 1.12. Assume that E, E are as in 1.11. Let p = (P , S,D) be aD�–parameter on λ such that λ \ S ∈ E.

(1) QEE is a (<λ)–complete forcing notion of size 2λ.

(2) QEE is reasonably B–bounding over p.

(3) If λ is strongly inaccessible and (∀δ ∈ S)(|Pδ| ≤ |δ|), then �Q

EE

DQEE �=

D[QEE ].

Proof. (1) Should be clear.

(2) Let p ∈ QEE . We are going to describe a strategy st for Generic in �rbB

p (p,QEE).

In the course of the play, Generic constructs aside a sequence 〈Tξ : ξ < λ〉 so that

if⟨Iξ, 〈pξt , q

ξt : t ∈ Iξ〉 : ξ < λ

⟩is the sequence formed by the innings of the two

players, then the following conditions are satisfied.

(a) Tξ ∈ QEE and if ξ < ζ < λ then p = T0 ⊇ Tξ ⊇ Tζ and Tζ ∩ ξλ = Tξ ∩ ξλ.

(b) If ζ < λ is limit, then Tζ =⋂

ξ<ζ

Tξ.

(c) If ξ ∈ S then

• Iξ = Pξ ∩ Tξ and pξt = (Tξ)t for t ∈ Iξ,

• Tξ+1 =⋃{qξt : t ∈ Iξ} ∪

⋃{(Tξ)ν : ν ∈ ξλ ∩ Tξ \ Iξ

}.

(d) If ξ /∈ S, then Iξ = ∅ and Tξ+1 = Tξ.

Conditions (a)–(d) fully describe the strategy st. Let us argue that it is a winning

strategy and to this end suppose that⟨Iξ, 〈pξt , q

ξt : t ∈ Iξ〉 : ξ < λ

⟩is a play of

�rbBp (p,QE

E) in which Generic uses st and constructs aside the sequence 〈Tξ : ξ < λ〉so that (a)–(d) are satisfied. Put p∗ =

⋂

ξ<λ

Tξ ⊆ <λλ. It follows from (a)+(b) that

p∗ is a complete λ–tree and for each ν ∈ p∗ either |succp∗(ν)| = 1 or succp∗(ν) ∈ Eν .

Suppose now that η ∈ limλ(p∗) and for ξ < λ let Bξ

def= {α < λ : succTξ

(η�α) ∈Eη�α}. Since η ∈ limλ(Tξ) for each ξ < λ, we know that Bξ ∈ E. Let

B = �ξ<λ

Bξ ∩ {ξ < λ : ξ is limit and ξ /∈ S}.

It follows from our assumptions that B ∈ E. For each α ∈ B we know thatsuccTξ

(η�α) ∈ Eη�α for ξ < α and Tα =⋂

ξ<α

Tξ, so succTα(η�α) ∈ Eη�α. More-

over, Tβ ∩ α+1λ = Tα ∩ α+1λ for all β > α (remember (a)+(d)) and consequently

succp∗(η�α) = succTα(η�α) ∈ Eη�α. Thus we have shown that p∗ ∈ QE

E .

Let W˜

be a QEE–name given by �

QEE

W˜

=⋃{root(p) : p ∈ Γ

QEE}. It should

be clear that �Q

EEW˜

∈ λλ and thus �Q

EEsetp(W

˜) ∈ D[QE

E ]. Plainly, if ξ ∈ S and

294294


t ∈ ξλ ∩ p∗, then (p∗)t ≥ qξt and hence

p∗ �Q

EE“ if ξ ∈ setp(W

˜), then W

˜�ξ ∈ Iξ and qξW

˜�ξ ∈ Γ

QEE”,

so Generic won the play.

(3) We are going to show that �Q

EE

setp(W˜) /∈ DQ

EE . To this end suppose that

p ∈ QEE and A

˜ξ (for ξ < λ) are QE

E–names for elements of D. Let st be the winning

strategy of Generic in �rbBp (p,QE

E) described in part (2) above. Consider a play⟨Iξ, 〈pξt , q

ξt : t ∈ Iξ〉 : ξ < λ

⟩of �rbB

p (p,QEE) in which

(∗)1 Generic follows st and constructs aside a sequence 〈Tξ : ξ < λ〉,(∗)2 Antigeneric plays so that at a stage ξ ∈ S he picks a set Bξ ∈ D and

conditions qξt ≥ pξt (for t ∈ Iξ) such that(∀t ∈ Iξ

)(qξt � Bξ ⊆

⋂

ζ≤ξ

A˜

ζ

).

Let p∗ =⋂

ξ<λ

Tξ be the condition determined at the end of part (2) and let B =

�ξ<λ

Bξ. Choose an increasing continuous sequence 〈γξ : ξ < λ〉 ⊆ λ and a complete

λ–tree T ⊆ p∗ such that for every ξ < λ we have

(∗)3 if ν ∈ T ∩ γξλ, then |{ρ ∈ T ∩ γξ+1λ : ν�ρ}| = |γξ| and(∗)4 if ρ ∈ T ∩ γξ+1λ, then ρ�α ∈ Pα for some α ∈ (γξ, γξ+1) ∩ S.

(The choice can be done by induction on ξ; remember that p is a D�–parameterand λ is assumed to be inaccessible.) Pick a limit ordinal ξ ∈ B ∩ S such thatξ = γξ. Since |T ∩ ξλ| > |γξ|, we may choose ν ∈ T ∩ ξλ \ Pξ. Put q = (p∗)ν . Thenq ≥ p∗ ≥ p and q �

QEEξ ∈ �

ξ<λA˜

ξ \ setp(W˜) (remember (∗)2 + (∗)4). �

2. Iterations with lords

Theorem 1.10 can be used for λ-support iteration of forcing notions QEE when

on each coordinate we have the same filter E. But if we want to use different filterson various coordinates we have serious problems. However, if we move to the otherextreme: having very orthogonal filters we may use a different approach to arguethat the limit of the iteration is λ–proper.

Definition 2.1. (1) A forcing notion with λ–complete (κ, μ)–purity is atriple (Q,≤,≤pr) such that ≤,≤pr are transitive reflexive (binary) rela-tions on Q such that(a) ≤pr ⊆ ≤,(b) both (Q,≤) and (Q,≤pr) are strategically (<λ)–complete,(c) for every p ∈ Q and a (Q,≤)–name τ

˜for an ordinal below κ, there

are a set A of size less than μ and a condition q ∈ Q such that p ≤pr qand q forces (in (Q,≤)) that “τ

˜∈ A”.

(2) If (Q,≤,≤pr) is a forcing notion with λ–complete (κ, μ)–purity for everyκ, then we say that it has λ–complete (∗, μ)–purity.

(3) If (Q,≤,≤pr) is a forcing notion with λ–complete (κ, μ)–purity, then allour forcing terms (like “forces”, “name” etc) refer to (Q,≤). The relation≤pr has an auxiliary character only and if we want to refer to it we add“purely” (so “q is stronger than p” means p ≤ q, and “q is purely strongerthan p” means that p ≤pr q).

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Definition 2.2. Let Q = (Q,≤,≤pr) be a forcing notion with λ–complete(∗, λ+)–purity, p = (P , S,D) be a D�–parameter on λ, U be a normal filter on λand μ = 〈μα : α < λ〉 be a sequence of cardinals below λ.

(1) For a condition p ∈ Q we define a game �prU,p,μ(p,Q) between two players,

Generic and Antigeneric, as follows. A play of �prU,p,μ(p,Q) lasts λ steps

and during a play a sequence⟨�α, 〈pαt , qαt : t ∈ μα〉 : α < λ

⟩

is constructed. So suppose that the players have arrived to a stage α < λof the game. Now,

(ℵ)prα first Antigeneric pics �α ∈ {0, 1}.(�)prα After this, Generic chooses a system 〈pαt : t ∈ μα〉 of paiwise incom-

patible conditions from Q, andprα(ג) Antigeneric answers with a system of conditions qαt ∈ Q (for t ∈ μα)

such that for each t ∈ μα:• pαt ≤ qαt , and• if �α = 1, then pαt ≤pr q

αt .

At the end, Generic wins the play⟨�α, 〈pαt , qαt : t ∈ μα〉 : α < λ

⟩

if and only if either {α < λ : �α = 1} /∈ U , or(�)ppr there is a condition p∗ ∈ Q stronger than p and such that

p∗ �Q “{α < λ :

(∃t ∈ μα

)(qαt ∈ ΓQ

)} ∈ D[Q] ”.

(2) We say that the forcing notion Q (with λ–complete (∗, λ+)–purity) ispurely B∗–bounding over U ,p, μ if for any p ∈ Q, Generic has a winningstrategy in the game �pr

U,p,μ(p,Q).

Remark 2.3. Note that in the definition of the game �prU,p,μ(p,Q) the size

of the index set used at stage α is declared to be μα (while in the related game�rbBp (p,Q) we required just |Iα| < λ). The reason for this is that otherwise in the

proof of the iteration theorem for the current case we could have problems withdeciding the size of the set Iα; compare clause (∗)4 of the proof of Theorem 2.7.

Observation 2.4. Assume E, E are as in 1.11. For p, q ∈ QEE let p ≤pr q

mean that p ≤ q and root(p) = root(q). Then

(1) (QEE ,≤,≤pr) is a forcing notion with λ–complete (∗, λ+)–purity,

(2) if, additionally, each Eν (for ν ∈ <λλ) is an ultrafilter on λ, then (QEE ,≤

,≤pr) has (κ, 2)–purity for every κ < λ.

Proposition 2.5. Assume that E, E are as in 1.11, p = (P , S,D) is a D�–parameter on λ and μ = 〈μα : α < λ〉 is a sequence of non-zero cardinals below

λ such that (∀α ∈ S)(|Pα| ≤ μα). Then (QEE ,≤,≤pr) is purely B∗–bounding over

E,p, μ.

Proof. Let p ∈ QEE and let st be the strategy described in the proof of 1.12(2)

with a small modification that we start the construction with ξ0 = lh(root(p)) + 1(so Tξ0 = p and the first ξ0 steps of the play are not relevant). Then we also replaceclauses (c)+(d) there by

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(cd) If ξ ≥ ξ0 then

• Iξ ⊆ ξλ ∩ Tξ is of size μξ, and pξt = (Tξ)t for t ∈ Iξ, and• if ξ ∈ S then Pξ ∩ Tξ ⊆ Iξ, and

• Tξ+1 =⋃{qξt : t ∈ Iξ} ∪

⋃{(Tξ)ν : ν ∈ ξλ ∩ Tξ \ Iξ

}.

(So, in particular, Antigeneric’s choice of �ξ has no influence on the answers byGeneric.) We are going to show that st is a winning strategy for Generic in

�prE,p,μ(p,Q

EE). To this end suppose that

⟨�ξ, 〈pξt , q

ξt : t ∈ Iξ〉 : ξ < λ

⟩is a play

of �prE,p,μ(p,Q

EE) in which Generic follows st (we identify Iξ with |Iξ| = μξ) and

〈Tξ : ξ < λ〉 is the sequence of side objects constructed in the course of the play.Assume A = {ξ < λ : �ξ = 1} ∈ E (otherwise Generic wins by default). Like in

1.12(2), put p∗ =⋂

ξ<λ

Tξ. To argue that p∗ ∈ QEE we note that if η ∈ limλ(p

∗) and

δ ∈ �ξ<λ

{α < λ : succTξ

(η�α) ∈ Eη�α}∩A ∩ {ξ < λ : ξ > ξ0 is limit },

then

succp∗(η�δ) ={

succTδ(η�δ) if η�δ /∈ Iδ

succqδη�δ(η�δ) if η�δ ∈ Iδ

∈ Eη�δ.

Exactly as in 1.12(2) we justify that p∗ witnesses (�)ppr. �

Lemma 2.6. Assume that

(1) λ is strongly inaccessible,(2) Q = 〈Pα,Q

˜α : α < γ〉 is a λ–support iteration, w ⊆ γ, |w| < λ, α0 ∈ w,

(3) for every α < γ,

�Pα“ Q˜

α = (Q˜

α,≤,≤pr) is a forcing notion with λ–complete (∗, λ+)–purity ”,

(4) Pα0is λ–proper,

(5) T = (T, rk) is a standard (w, 1)γ–tree, |T | < λ,(6) p = 〈pt : t ∈ T 〉 is a standard tree of conditions in Pγ , and(7) τ

˜is a Pγ–name for an ordinal.

Then there are a set A of size λ and a standard tree of conditions q = 〈qt : t ∈T 〉 ⊆ Pγ such that

(a)(∀t ∈ T

)(rk(t) = γ ⇒ qt � τ

˜∈ A

), and

(b) p ≤ q and if t ∈ T , rk(t) > α0 then qt�α0�Pα0

pt(α0) ≤pr qt(α0).

Proof. Let us start with the following observation.

Claim 2.6.1. If p ∈ Pγ then there are a set A0 of size λ and a condition q ≥ psuch that q �Pγ

τ˜∈ A0 and q�α0 �Pα0

p(α0) ≤pr q(α0).

Proof of the Claim. Let us look at Pγ as the result of 3 stage compositionPα0

∗ Q˜

α0∗ P˜(α0+1),γ , where P

˜(α0+1),γ is a Pα0+1–name for the following forcing

notion. The set of conditions in P˜(α0+1),γ is {r�(α0, γ) : r ∈ Pγ} (so it belongs to

V); the order of P˜(α0+1),γ is such that if Gα0+1 ⊆ Pα0+1 is generic over V, then

V[Gα0+1] |= “ r ≤P˜(α0+1),γ [Gα0

] s if and only if

there is q ∈ Gα0+1 such that q�r ≤Pγq�s ”.

Now, pick a Pα0+1–name (r˜, α˜) such that

p�(α0 + 1) �Pα0+1“ p�(α0, γ) ≤ r

ãnd r

˜� α˜= τ˜”

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and then choose a Pα0–name A

˜∗ for a subset of P

˜(α0+1),γ × ON and a Pα0

–nameq(α0) for a condition in Q

˜α0

such that

p�α0 �Pα0“ p(α0) ≤pr q(α0) and |A

˜∗| = λ and

q(α0) �Q

˜α0

(∃(s, β) ∈ A

˜∗)(r

˜= s & α

˜= β

)”.

Since Pα0is λ–proper, we may choose a set A+ ⊆ P

˜(α0+1),γ × ON of size λ and a

condition q�α0 ≥ p�α0 such that q�α0 � A˜

∗ ⊆ A+. Then

q�(α0 + 1) �Pα0+1

(∃(s, β) ∈ A+

)(r˜= s & α

˜= β

).

Put A = {β : (∃s)((s, β) ∈ A+)}. Now we may easily define q�(α0, γ) so thatdom

(q�(α0, γ)

)=

⋃{dom(s) : (∃β)((s, β) ∈ A+)} and

q�(α0 + 1) �Pα0+1“ r˜≤P

˜(α0+1),γ

q�(α0, γ) and α˜∈ A ”. �

Fix an enumeration 〈tζ : ζ ≤ ζ∗〉 of {t ∈ T : rk(t) = ζ} (so ζ∗ < λ). For eachα ∈ γ \ {α0} fix a Pα–name st

˜0α for a winning strategy of Complete in the game

�λ0

((Q˜

α,≤), ∅˜Q

˜α

)such that as long as Incomplete plays ∅

˜Q

˜α, Complete answers

with ∅˜Q

˜α

as well. Let st˜

ζpr be the <∗

χ–first Pα0–name for a winning strategy of

Complete in �λ0

((Q˜

α0,≤pr), ptζ (α0)

)(for ζ ≤ ζ∗). Note that if ζ, ζ ′ ≤ ζ∗ and

tζ�(α0 + 1) = tζ′�(α0 + 1), then st˜

ζpr = st

˜ζ′pr.

By induction on ζ ≤ ζ∗ we choose a sequence 〈pζ , qζ , Aζ : ζ ≤ ζ∗〉 so that thefollowing demands are satisfied.

(i) pζ = 〈pζt : t ∈ T 〉, qζ = 〈qζt : t ∈ T 〉 are standard trees of conditions, Aζ isa set of ordinals of size λ.

(ii) If ε < ζ ≤ ζ∗, then p ≤ pε ≤ qε ≤ pζ and Aε ⊆ Aζ .

(iii) pζtζ �Pγτ˜∈ Aζ .

(iv) If α ∈ γ \ {α0}, ζ, ξ ≤ ζ∗, then

qξtζ�α �Pα“ 〈pεtζ�α(α), q

εtζ�α(α) : ε ≤ ξ〉 is a result of a play of �λ

0

((Q˜

α,≤), ∅˜Q

˜α

)

in which Complete uses st˜

0α ”.

(v) If ζ, ξ ≤ ζ∗, then

qξtζ�α0�Pα0

“ 〈pεtζ�α0(α0), q

εtζ�α0

(α0) : ε ≤ ξ〉 is a result of a play

of �λ0

((Q˜

α0,≤pr), ptζ (α0)

)in which Complete uses st

˜ζpr ”.

Suppose that we have determined pε, qε, Aε for ε < ζ ≤ ζ∗. First we choosep′ = 〈p′t : t ∈ T 〉 ⊆ Pγ . If ξ = 0 then we set p′ = p. Otherwise we choose p′ so thatfor t ∈ T we have:

(vi) dom(p′t) =⋃

ε<ζ

dom(qεt ), and

(vii) if α ∈ dom(p′t) \ {α0}, then p′(α) is the <∗χ–first Pα–name for a condition

in Q˜

α such that p′t�α �Pα(∀ε < ζ)(qεt (α) ≤ p′t(α)), and

(viii) p′(α0) is the <∗χ–first Pα0

–name for a condition in Q˜

α0such that

p′t�α0 �Pα0(∀ε < ζ)(qεt (α0) ≤pr p

′t(α0)).

The choice is possible by (iv)+(v), and since we pick “the <∗χ–first names” we easily

see that p′ is a standard tree of conditions. Now we use 2.6.1 to find a set Aζ of

size λ and a condition pζtζ ∈ Pγ such that⋃

ε<ζ

Aε ⊆ Aζ , p′tζ ≤ pζtζ , pζtζ�α0 �Pα0p′tζ (α0) ≤pr p

ζtζ (α0) and pζtζ �Pγ

τ˜∈ Aζ .

298298


Next, for each t ∈ T we let pζt ∈ Prk(t) be such that

(ix) if s = t ∩ tζ , then

pζt �rk(s) = pζtζ�rk(s) and pζt �[rk(s), rk(t)) = p′t�[rk(s), rk(t)).

Clearly, pζ = 〈pζt : t ∈ T 〉 is a standard tree of conditions satisfying the relevant

parts of the demands in (ii)–(v). Now we choose a tree of conditions qζ = 〈qζt : t ∈T 〉 so that the requirements of (iv)+(v) hold (for this we proceed like in (vi)–(viii)above).

After the construction is carried out we note that qζ∗and Aζ∗

are as requiredin the assertion of the lemma. �


(1) λ is strongly inaccessible, μ = 〈μα : α < λ〉 is a sequence of cardinalsbelow λ, p = (P , S,D) is a D�–parameter on λ, and

(2) Q = 〈Pα,Q˜

α : α < γ〉 is a λ–support iteration,(3) U

˜α is a Pα–name for a normal filter on λ (for α < γ),

(4) Aα,β ⊆ λ is such that �PαAα,β ∈ U

˜α and �Pβ

λ \ Aα,β ∈ U˜β (for

α < β < γ), and(5) for every α < γ,

�Pα“ Q˜

α is purely B∗–bounding over U˜α,p[Pα], μ ”.

Then Pγ = lim(Q) is λ–proper.

Proof. The arguments follow closely the lines of the arguments for [RS05,Thm. 3.1, 3.2] and [RS, Thm. 2.12]. The proof is by induction on γ, so assumethat we know also that each Pα is λ–proper for α < λ.

Let N ≺ (H(χ),∈, <∗χ) be such that <λN ⊆ N , |N | = λ and Q, 〈Aα,β : α <

β < γ〉,p, . . . ∈ N . Let p ∈ N ∩ Pγ and 〈τ˜δ : δ < λ〉 list all Pγ–names for ordinals

from N . For each ξ ∈ N ∩ γ fix a Pξ–name st˜

0ξ ∈ N for a winning strategy of

Complete in �λ0 (Q˜

ξ, ∅˜Q

˜ξ) such that it instructs Complete to play ∅

˜Q

˜ξas long as her

opponent plays ∅˜Q

˜ξ.

By induction on δ < λ we will choose

(⊗)aδ Tδ, pδ∗, qδ∗, r−δ , rδ, wδ, Zδ, αδ, and

(⊗)bδ �δ,ξ, p˜δ,ξ, q

˜δ,ξ and st

˜ξ for ξ ∈ N ∩ γ,

so that the following demands are satisfied.

(∗)0 All objects listed in (⊗)aδ + (⊗)bδ belong to N . After stage δ < λ of theconstruction, the objects in (⊗)aδ are known as well as those in (⊗)bδ forξ ∈ wδ.

(∗)1 r−δ , rδ ∈ Pγ , r−0 (0) = r0(0) = p(0), wδ ⊆ γ, |wδ| = |δ| + 1,

⋃

α<λ

dom(rα) =⋃

α<λ

wα = N ∩ γ, w0 = {0}, wδ ⊆ wδ+1 and if δ is limit then wδ =⋃

α<δ

wα.

(∗)2 For each α < δ < λ we have (∀ξ ∈ wα+1)(rα(ξ) = rδ(ξ)) and p ≤ r−α ≤rα ≤ r−δ ≤ rδ.

(∗)3 If ξ ∈ (γ \ wδ) ∩N , then

rδ�ξ � “ the sequence 〈r−α (ξ), rα(ξ) : α ≤ δ〉 is a legal partial play of�λ0

(Q˜

ξ, ∅˜Q

˜ξ

)in which Complete follows st

˜0ξ ”

299299


and if ξ ∈ wδ+1 \ wδ, then st˜

ξ ∈ N is a Pξ–name for a winning strategyof Generic in �pr

U˜

ξ,p,μ(rδ(ξ),Q

˜ξ). (And st0 ∈ N is a winning strategy of

Generic in �prU˜

0,p,μ(p(0),Q0).)

(∗)4 Tδ = (Tδ, rkδ) is a standard (wδ, 1)γ–tree, Tδ =

⋃

α≤γ

∏

ξ∈wδ∩α

μδ.

(∗)5 pδ∗ = 〈pδ∗,t : t ∈ Tδ〉 and qδ∗ = 〈qδ∗,t : t ∈ Tδ〉 are standard trees of conditions,

pδ∗ ≤ qδ∗.(∗)6 For t ∈ Tδ we have that dom(pδ∗,t) =

(dom(p)∪

⋃

α<δ

dom(rα)∪wδ

)∩ rkδ(t)

and for each ξ ∈ dom(pδ∗,t) \ wδ:

pδ∗,t�ξ �Pξ“ if the set {rα(ξ) : α < δ} ∪ {p(ξ)} has an upper bound in Q

˜ξ,

then pδ∗,t(ξ) is such an upper bound ”.

(∗)7 For ξ ∈ N ∩ γ, �δ,ξ ∈ {0, 1} and p˜δ,ξ, q

˜δ,ξ are Pξ–names for sequences of

conditions in Q˜

ξ of length μδ.(∗)8 If either ξ = 0 = β or ξ ∈ wβ+1 \ wβ , β < λ, then

�Pξ“ 〈�α,ξ, p

˜α,ξ, q

˜α,ξ : α < λ〉 is a play of �pr

U˜

ξ,p,μ(rβ(ξ),Q

˜ξ)

in which Generic uses st˜

ξ ”.

(∗)9 αδ ∈ wδ (αδ will be called the lord of stage δ) and(∀β ∈ wδ \ {αδ}

)(δ /∈

⋂{λ \Aξ,β : ξ ∈ wδ ∩ β

}∩⋂{

Aβ,ξ : ξ ∈ wδ \ (β + 1)})

.

(∗)10 �δ,ξ = 0 for ξ ∈ N ∩ γ \ {αδ} and �δ,αδ= 1.

(∗)11 If t ∈ Tδ, rkδ(t) = ξ < γ, then for each ε < μδ

qδ∗,t �Pξ“ p˜δ,ξ(ε) = pδ∗,t〈ε〉(ξ) and q

˜δ,ξ(ε) = qδ∗,t〈ε〉(ξ) ”.

(∗)12 If t0, t1 ∈ Tδ, rkδ(t0) = rkδ(t1) and ξ ∈ wδ ∩ rkδ(t0), t0�ξ = t1�ξ but(t0)ξ�=

(t1)ξ, then

pδ∗,t0�ξ �Pξ“ the conditions pδ∗,t0(ξ), p

δ∗,t1(ξ) are incompatible ”.

(∗)13 Zδ is a set of ordinals, |Zδ| = λ and for each t ∈ Tδ with rkδ(t) = γ wehave qδ∗,t �Pγ

(∀α ≤ δ

)(τ˜α ∈ Zδ

).

(∗)14 dom(r−δ ) = dom(rδ) =⋃

t∈Tδ

dom(qδ∗,t)∪dom(p) and if t ∈ Tδ, ξ ∈ dom(rδ)∩

rkδ(t) \ wδ, and qδ∗,t�ξ ≤ q ∈ Pξ, rδ�ξ ≤ q, then

q �Pξ“ if the set {rα(ξ) : α < δ} ∪ {qδ∗,t(ξ), p(ξ)} has an upper bound in Q

˜ξ,

then r−δ (ξ) is such an upper bound ”.

First we fix an increasing continuous sequence 〈wα : α < λ〉 of subsets of N ∩ γsuch that the relevant demands in (∗)1 are satisfied. Now, suppose that we havearrived to a stage δ < λ of the construction and all objects listed in (⊗)aα andrelevant cases of (⊗)bα (see (∗)0) have been determined for α < δ.

To ensure (∗)0, all choices below are made in N (e.g., each time we choose anobject with some properties, we pick the <∗

χ–first such object).If δ is a successor ordinal and ξ ∈ wδ \wδ−1, then we let st

˜ξ ∈ N be a Pξ–name

for a winning strategy of Generic in �prU˜

ξ,p,μ(rδ−1(ξ),Q

˜ξ). We also put �α,ξ = 0 for

all α < δ and we pick p˜α,ξ, q

˜α,ξ (for α < δ) so that the suitable parts of (∗)7 + (∗)8

at ξ are satisfied.

300300


Clause (∗)4 fully describes Tδ. Now we choose the lord of stage δ. If for someβ ∈ wδ we have

δ ∈⋂{

λ \Aξ,β : ξ ∈ wδ ∩ β}∩⋂{

Aβ,ξ : ξ ∈ wδ \ (β + 1)},

then αδ is equal to this β (note that there is at most one β ∈ wδ with the requiredproperty). Otherwise we let αδ = 0. Then we put �δ,αδ

= 1 and �δ,ξ = 0 for allξ ∈ wδ \ {αδ}.

Next, for each ξ ∈ wδ we choose a Pξ–name p˜δ,ξ such that

�Pξ“ p˜δ,ξ = 〈p

˜δ,ξ(ε) : ε < μδ〉 is given to Generic by st

˜ξ

as an answer to 〈�α,ξ, p˜α,ξ, q

˜α,ξ : α < δ〉�〈�δ,ξ〉 ”.

(Note that �Pξ“ conditions p

˜δ,ξ(ε0), p

˜δ,ξ(ε1) are incompatible” whenever ε0 < ε1 <

μδ and ξ ∈ wδ.) After this we may choose a tree of conditions pδ∗ = 〈pδ∗,t : t ∈ Tδ〉such that for each t ∈ Tδ:

• dom(pδ∗,t) =(dom(p) ∪

⋃

α<δ

dom(rα) ∪ wδ

)∩ rkδ(t) and

• for ξ ∈ dom(pδ∗,t) \ wδ, pδ∗,t(ξ) is the <∗

χ–first Pξ–name for a condition inQ˜

ξ such that

pδ∗,t�ξ �Pξ“ if the set {rα(ξ) : α < δ} ∪ {p(ξ)} has an upper bound in Q

˜ξ,

then pδ∗,t(ξ) is such an upper bound ”,

• pδ∗,t(ξ) = p˜α,ξ

((t)ξ

)for ξ ∈ dom(pδ∗,t) ∩ wδ.

Using Lemma 2.6 we may pick a tree of conditions qδ∗ = 〈qδ∗,t : t ∈ Tδ〉 and a set Zδ

of ordinals such that

• pδ∗ ≤ qδ∗, |Zδ| = λ,• if t ∈ Tδ, rkδ(t) = γ then qδ∗,t �Pγ

(∀α ≤ δ)(τ˜α ∈ Zδ),

• if t ∈ Tδ, rkδ(t) > αδ then qδ∗,t�αδ�Pαδ

pδ∗,t(αδ) ≤pr qδ∗,t(αδ).

Note that if ξ ∈ wδ and ε0 < ε1 < μδ, t ∈ Tδ, rkδ(t) = ξ, then

qδ∗,t �Pξ“ the conditions qδ∗,t〈ε0〉(ξ), q

δ∗,t〈ε〉(ξ) are incompatible ”.

Hence we have no problems with finding Pξ–names q˜δ,ξ (for ξ ∈ wδ) such that

• �Pξ“ q˜δ,ξ = 〈q

˜δ,ξ(ε) : ε < μδ〉 is a sequence of conditions in Q

˜ξ ”,

• �Pξ“ (∀ε < μδ)(p

˜δ,ξ(ε) ≤ q

˜δ,ξ(ε)) ” and �Pαδ

“ (∀ε < μδ)(p˜δ,αδ

(ε) ≤pr

q˜δ,αδ

(ε)) ”,

• if t ∈ Tδ, rkδ(t) > ξ, then qδ∗,t�ξ �Pξqδ∗,t(ξ) = q

˜δ,ξ

((t)ξ

).

Now we define r−δ , rδ ∈ Pγ so that

dom(r−δ ) = dom(rδ) =⋃

t∈Tδ

dom(qδ∗,t) ∪ dom(p)

and

• if ξ ∈ wα+1, α < δ, then r−δ (ξ) = rδ(ξ) = rα(ξ),

• if ξ ∈ dom(r−δ ) \ wδ, then r−δ (ξ) is the <∗χ–first Pξ–name for an element

of Q˜

ξ such that

r−δ �ξ �Pξ“ r−δ (ξ) is an upper bound of {rα(ξ) : α < δ} ∪ {p(ξ)} andif t ∈ Tδ, rkδ(t) > ξ, and qδ∗,t�ξ ∈ ΓPξ

and the set{rα(ξ) : α < δ} ∪ {qδ∗,t(ξ), p(ξ)} has an upper bound in Q

˜ξ,

then r−δ (ξ) is such an upper bound ”,

301301


and rδ(ξ) is the <∗χ–first Pξ–name for an element of Q

˜ξ such that

rδ�ξ �Pξ“ rδ(ξ) is given to Complete by st

˜0ξ as the answer to

〈r−α (ξ), rα(ξ) : α < δ〉�〈r−δ (ξ)〉 ”Note that by a straightforward induction on ξ ∈ dom(rδ) one easily applies (∗)3from previous stages to show that r−δ , rδ are well defined and rδ ≥ r−δ ≥ rα, p for

α < δ. If δ = 0 we also stipulate r−0 (0) = r0(0) = p(0).This completes the description of stage δ of our construction. One easily verifies

that the demands (∗)0–(∗)14 are satisfied.After completing all λ stages of the construction, for each ξ ∈ N ∩ γ we look

at the sequence 〈�δ,ξ, p˜δ,ξ, q

˜δ,ξ : δ < λ〉. For δ < λ such that ξ ∈ wδ let

Bξδ =

⋂{λ \Aζ,ξ : ζ ∈ wδ ∩ ξ

}∩⋂{

Aξ,ζ : ζ ∈ wδ \ (ξ + 1)},

and for δ < λ such that ξ /∈ wδ put Bξδ = λ. It follows from our assumptions that

�Pξ(∀δ < λ)(Bξ

δ ∈ U˜ξ) and thus also �Pξ

�α<λ

Bξα ∈ U

˜ξ. Note that if δ is a limit

ordinal, ξ ∈ wδ and δ ∈ �α<λ

Bξα, then also δ ∈ Bξ

δ and hence ξ = αδ (remember

(∗)9) and �δ,ξ = 1 (by (∗)10). Consequently for each ξ ∈ N ∩ γ

�Pξ“ {δ < λ : �δ,ξ = 1} ∈ U

˜ξ ”.

Therefore, for every ξ ∈ N ∩ γ we may pick a Pξ–name q(ξ) for a condition in Q˜

ξ

such that

• if ξ ∈ wβ+1 \ wβ , β < λ (or ξ = 0 = β), then

�Pξ“ q(ξ) ≥ rβ(ξ) and q(ξ) �Q

˜ξ

{δ<λ :

(∃ε<μδ

)(q˜δ,ξ(ε) ∈ ΓQ

˜ξ

)}∈ D[Pξ+1] ”.

This determines a condition q ∈ Pγ (with dom(q) = N ∩γ) and easily (∀β < λ)(p ≤rβ ≤ q) (remember (∗)2). For each ξ ∈ N ∩ γ fix Pξ+1–names C

˜

ξi , f˜

ξi (for i < λ)

such that

q�(ξ + 1) �Pξ+1“(∀i < λ

)(C˜

ξi ∈ D ∩V & f

˜

ξi ∈ λλ

)and

(∀δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(f

˜

ξi ))(∃ε < μδ

)(q˜δ,ξ(ε) ∈ ΓQ

˜ξ

)”.

Claim 2.7.1. For each limit ordinal δ < λ, the condition q forces (in Pγ) that

“(∀ξ ∈ wδ

)(δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(f

˜

ξi ))

⇒(∃t ∈ Tδ

)(rkδ(t) = γ & qδ∗,t ∈ ΓPγ

)”.

Proof of the Claim. The proof is essentially the same as that for [RS05,Claim 3.1], however for the sake of completeness we will present it fully. Supposethat r ≥ q and a limit ordinal δ < λ are such that

(�)a r �Pγ“(∀ξ ∈ wδ

)(δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(f

˜

ξi ))”.

For each ζ < γ fix a Pζ–name st˜

∗ζ for a winning strategy of Complete in �λ

0 (Q˜

ζ , ∅˜Q

˜ζ)

such that as long as Incomplete plays ∅˜Q

˜ζ, Complete answers with ∅

˜Q

˜ζas well.

We are going to show that there is t ∈ Tδ such that rkδ(t) = γ and theconditions qδ∗,t and r are compatible. Let 〈εα : α ≤ α∗〉 = wδ∪{γ} be the increasingenumeration. By induction on α ≤ α∗ we will choose conditions r∗α, r

∗∗α ∈ Pεα and

t = 〈(t)εα : α < α∗〉 ∈ Tδ such that letting tα◦ = 〈(t)εβ : β < α〉 ∈ Tδ we have

(�)b qδ∗,tα◦ ≤ r∗α and r�εα ≤ r∗α,

302302


(�)c for every β < α and ζ < εα,

r∗∗β �Pζ“ 〈r∗β′(ζ), r∗∗β′ (ζ) : β′ < β〉 is a legal partial play of �λ

0 (Q˜

ζ , ∅˜Q

˜ζ)

in which Complete uses her winning strategy st˜

∗ζ ”.

Suppose that α ≤ α∗ is a limit ordinal and we have already defined tα◦ = 〈(t)εβ :β < α〉 and 〈r∗β, r∗∗β : β < α〉. Let ξ = sup(εβ : β < α). It follows from (�)c that

we may find a condition r∗α ∈ Pεα such that r∗α�ξ ∈ Pξ is stronger than all r∗∗β (for

β < α) and also r∗α�[ξ, εα) = r�[ξ, εα). Clearly r�εα ≤ r∗α and also qδ∗,tα◦ �ξ ≤ r∗α�ξ(remember (�)b for β < α). Now by induction on ζ ∈ [ξ, εα) we argue thatqδ∗,tα◦ �ζ ≤ r∗α�ζ. So suppose that ξ ≤ ζ < εα and we know qδ∗,tα◦ �ζ ≤ r∗α�ζ. It

follows from (∗)3+(∗)5+(∗)6 that r∗α�ζ � (∀i < δ)(ri(ζ) ≤ pδ∗,tα◦ (ζ) ≤ qδ∗,tα◦ (ζ)) and

therefore we may use (∗)14 to conclude that

r∗α�ζ �Pζqδ∗,tα◦ (ζ) ≤ rδ(ζ) ≤ q(ζ) ≤ r(ζ) = r∗α(ζ).

Finally we let r∗∗α ∈ Pεα be a condition such that for each ζ < εα

r∗∗α �ζ �Pζ“ r∗∗α (ζ) is given to Generic by st

˜∗ζ as the answer to

〈r∗β(ζ), r∗∗β (ζ) : β < ζ〉�〈r∗α(ζ)〉 ”.Now suppose that α = β + 1 ≤ α∗ and we have already defined r∗β, r

∗∗β ∈ Pεβ and

tβ◦ ∈ Tδ. It follows from the choice of q and (�)a + (�)b + (∗)11 that

r∗∗β �Pεβ“ r(εβ) �Q

˜εβ

(∃ε < μδ

)(qδ∗,tβ◦〈ε〉(εβ) ∈ ΓQ

˜εβ

)”.

Therefore we may choose ε = (t)εβ < μδ (thus defining tα◦ ) and a condition r∗α ∈ Pεα

such that

• r∗∗β ≤ r∗α�εβ and

r∗α�εβ �Pεβ“ r∗α(εβ) ≥ r(εβ) & r∗α(εβ) ≥ qδ∗,tα◦ (εβ) ”,

• r∗α�(εβ , εα) = r�(εβ , εα).Exactly like in the limit case we argue that r∗α has the desired properties and thenin the same manner as there we define r∗∗α .

We finish the proof of the claim noting that t = tα∗

◦ ∈ Tδ and the condition r∗α∗

are such that r∗α∗ ≥ r and r∗α∗ ≥ qδ∗,t. �

Let us use 2.7.1 to argue that q is (N,Pγ)–generic. To this end suppose τ˜∈ N

is a Pγ–name for an ordinal, say τ˜

= τ˜α, α < λ, and let q′ ≥ q. Since Pγ is

strategically (<λ)–complete we may build an increasing sequence 〈q′i : i < λ〉 of

conditions above q′ and a sequence 〈Cξi , f

ξi : ξ ∈ N∩γ, i < λ〉 such that Cξ

i ∈ D∩V,

fξi ∈ λλ and for each ξ ∈ wi

q′i �(∀j ≤ i

)(C˜

ξj = Cξ

j & f˜

ξj�i = fξ

j �i).

The set{δ < λ :

(∀ξ ∈ wδ

)(∀j < δ

)(δ ∈ Cξ

j ∩ setp(fξj ))}

is in D, so we may choosea limit ordinal δ > α such that for each ξ ∈ wδ we have

δ ∈ �i<λ

Cξi ∩ �

i<λsetp(fξ

i ).

Then q′δ �(∀ξ ∈ wδ

)(δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(f

˜

ξi ))and therefore, by 2.7.1,

q′δ �(∃t ∈ Tδ

)(rkδ(t) = γ & qδ∗,t ∈ ΓPγ

).

303303


Using (∗)13 we conclude q′δ � τ˜∈ Zδ and hence q′δ � τ

˜∈ N . �

Remark 2.8. Naturally, we want to apply Theorem 2.7 to γ = λ++ in amodel where 2λ = λ+, so one may ask if the assumptions (3) + (4) of 2.7 can besatisfied in such a universe. But they are not so unusual: suppose that we startwith V |= λ<λ = λ & 2λ = λ+. Consider the following forcing notion Cλ

λ+ .

A condition p ∈ Cλλ+ is a function p : dom(p) −→ 2 such that dom(p) ⊆ λ+ × λ

and |dom(p)| < λ.The order is the inclusion.Plainly, Cλ

λ+ is a (<λ)–complete λ+–cc forcing notion of size λ+. Suppose now

that G ⊆ Cλλ+ is generic over V and let us work in V[G]. Put f =

⋃G (so

f : λ+ × λ −→ 2) and for α < λ+ and i < 2 let Aiα = {ξ < λ : f(α, ξ) = i}.

For a function h : λ+ −→ 2 let Uh be the normal filter generated by the family{A

h(α)α : α < λ+

}. One easily verifies that each Uh is a proper (normal) filter and

plainly if h, h′ : λ+ −→ 2 are distinct, say h(α) = 0, h′(α) = 1, then A0α ∈ Uh and

λ \A0α = A1

α ∈ Uh′ .

3. Noble iterations

The iteration theorems 1.10 and 2.7 have one common drawback: they assumethat λ is strongly inaccessible. In this section we introduce a property slightlystronger than being B–bounding over p and we show the corresponding iterationtheorem. The main gain is that the only assumption on λ is λ = λ<λ.

Definition 3.1. Let Q = (Q,≤) be a forcing notion and p = (P , λ,D) be aD�–parameter on λ.

(1) For a condition p ∈ Q we define a game �B+p (p,Q) between two players,

Generic and Antigeneric, as follows. A play of �B+p (p,Q) lasts λ steps

during which the players construct a sequence⟨fα,Xα, pα, qα : α < λ

⟩

such that(a) fα : α −→ Q and fβ ⊆ fα for β < α,(b) Xα ⊆ Pα and for every η ∈ Xα the sequence 〈fα

(η(ξ)

): ξ < α〉 ⊆ Q

has an upper bound in Q and if η0, η1 ∈ Xα are distinct, then forsome ξ < α the conditions fα

(η0(ξ)

), fα

(η1(ξ)

)are incompatible,

(c) pα = 〈pηα : η ∈ Xα〉 ⊆ Q is a system of conditions in Q such that(∀ξ < α

)(fα

(η(ξ)

)≤ pηα

)for η ∈ Xα,

(d) qα = 〈qηα : η ∈ Xα〉 ⊆ Q is a system of conditions in Q such that(∀η ∈ Xα)(p

ηα ≤ qηα)

The choices of the objects listed above are done so that at stage α < λ ofthe play:(ℵ)α first Generic picks a function fα : α −→ Q with the property de-

scribed in (a) above (so if α is limit, then fα =⋃

β<α

fβ). She also

chooses Xα, pα satisfying the demands of (b)+(c) (note that Xα couldbe empty).

(�)α Then Antigeneric decides a system qα as in (d).At the end, Generic wins the play

⟨fα,Xα, pα, qα : α < λ

⟩if and only if

(�)B+p there is a condition p∗ ∈ Q stronger than p and such that

p∗ �Q “{α < λ :

(∃η ∈ Xα

)(qηα ∈ ΓQ

)}∈ D[Q] ”.

304304


(2) A forcing notion Q is B–noble over p if it is strategically (<λ)–completeand Generic has a winning strategy in the game �B+

p (p,Q) for every p ∈ Q.

Note that in the above definition we assumed that P = 〈Pδ : δ < λ〉. This wascaused only to simplify the description of the game – if the domain of P is S ∈ D,then we may extend it to λ in some trivial way without changing the resultingproperties.

Observation 3.2. If p is a D�–parameter and a forcing notion Q is B–nobleover p, then Q is reasonably B–bounding over p.


(1) λ = λ<λ and p = (P , λ,D) is a D�–parameter on λ, and(2) Q = 〈Pξ,Q

˜ξ : ξ < γ〉 is a λ–support iteration such that for every ξ < γ,

�Pξ“ Q˜

ξ is B–noble over p[Pξ] ”.

Then

(a) Pγ = lim(Q) is λ–proper,(b) for each Pγ–name τ

˜for a function from λ to V and a condition p ∈ Pγ

there are q ∈ Pγ and 〈Aα : α < λ〉 such that |Aα| < λ (for α < λ) andq ≥ p and

q �Pγ“ {α < λ : τ

˜(α) ∈ Aα} ∈ D[Pγ ] ”.

Proof. (a) Assume that N ≺ (H(χ),∈, <∗χ) is such that <λN ⊆ N , |N | = λ

and Q,p, . . . ∈ N . Let p ∈ N ∩ Pγ . Choose N = 〈Nδ : δ < λ〉 and α = 〈αδ : δ < λ〉such that

• N is an increasing continuous sequence of elementary submodels of N ,• α is an increasing continuous sequence of ordinals below λ,• N =

⋃

δ<λ

Nδ, Q,p, p, . . . ∈ N0, δ ⊆ Nδ, Pδ ⊆ Nδ+1, N�(δ + 1) ∈ Nδ+1,

|Nδ| < λ and• αδ + otp(Nδ ∩ γ) + 888 < αδ+1, α�(δ + 2) ∈ Nδ+1.

Put wδ = Nδ ∩ γ and for each ξ < γ let st˜

0ξ be the <∗

χ–first Pξ–name for a winning

strategy of Complete in �λ0 (Q˜

ξ, ∅˜Q

˜ξ) such that it instructs Complete to play ∅

˜Q

˜ξas

long as her opponent plays ∅˜Q

˜ξ. We also assume that whenever possible, ∅

˜Q

˜ξis the

<∗χ–first name for the answer by st

˜0ξ to a particular sequence of names. Note that

〈st˜

0ξ : ξ < γ〉 ∈ N0.

By induction on δ < λ we will construct

(⊗)δ Tδ, pδ, qδ, r−δ , rδ and f˜δ,ξ,X

˜δ,ξ, p

˜δ,ξ, q

˜δ,ξ, st

˜ξ and pδ,ξ for ξ ∈ wδ

so that the following conditions (∗)0–(∗)16 are satisfied.

(∗)0 Objects listed in (⊗)δ form the <∗χ–first tuple with the properties de-

scribed in (∗)1–(∗)16 below. Consequently, the sequence

〈 objects listed in (⊗)ε : ε < δ〉is definable from N�αδ, α�δ, Q, p, p (in the language L(∈, <∗

χ)), so ifδ = αδ is limit, then this sequence belongs to Nδ+1. Also, objects listedin (⊗)δ are known after stage δ (and they all belong to N).

(∗)1 r−δ , rδ ∈ Pγ , wδ ⊆ dom(r−δ ) = dom(rδ) and r−0 (ξ) = r0(ξ) = p(ξ) forξ ∈ w0.

305305


(∗)2 For each ε < δ < λ we have (∀ξ ∈ wε+1)(rε(ξ) = rδ(ξ)) and p ≤ r−ε ≤rε ≤ r−δ ≤ rδ.

(∗)3 If ξ ∈ dom(rδ) \ wδ, then

rδ�ξ �Pξ“ the sequence 〈r−ε (ξ), rε(ξ) : ε ≤ δ〉 is a legal partial play of

�λ0

(Q˜

ξ, ∅˜Q

˜ξ


˜0ξ ”


ξ is the <∗χ–first Pξ–name for a winning

strategy of Generic in �B+p (rδ(ξ),Q

˜ξ). (And for ξ ∈ w0, st

˜ξ is the <∗

χ–

first Pξ–name for a winning strategy of Generic in �B+p (p(ξ),Q

˜ξ). Note

that st˜

ξ ∈ N0 for ξ ∈ w0 and st˜

ξ ∈ Nαδ+1for ξ ∈ wδ+1.)

(∗)4 Tδ = (Tδ, rkδ) is a (wδ, 1)γ–tree, Tδ ⊆

⋃

α≤γ

∏

ξ∈wδ∩α

(Pδ ∪ {∗}

). (Note that

we do not require here that Tδ is standard, so some chains in Tδ may haveno �–bounds.)

(∗)5 pδ = 〈pδt : t ∈ Tδ〉 and qδ = 〈qδt : t ∈ Tδ〉 are trees of conditions, pδ ≤ qδ.(∗)6 For t ∈ Tδ we have that dom(pδt ) ⊇

(dom(p) ∪

⋃

α<δ

dom(rα) ∪wδ

)∩ rkδ(t)

and for each ξ ∈ dom(pδt ) \ wδ:

pδt �ξ �Pξ“ if the set {rε(ξ) : ε < δ} ∪ {p(ξ)} has an upper bound in Q

˜ξ,

then pδt (ξ) is such an upper bound ”.

(∗)7 If ξ ∈ wβ+1 \ wβ, β < δ, then

�Pξ“ 〈f

˜ε,ξ,X

˜ε,ξ, p

˜ε,ξ, q

˜ε,ξ : ε < δ〉 is a partial play of

�B+p (rβ(ξ),Q

˜ξ) in which Generic uses st

˜ξ ”.

(∗)8 dom(r−δ ) = dom(rδ) =⋃

t∈Tδ

dom(qδt ) and if t ∈ Tδ, ξ ∈ dom(rδ)∩rkδ(t)\wδ,

and qδt �ξ ≤ q ∈ Pξ, rδ�ξ ≤ q, then

q �Pξ“ if the set {rα(ξ) : α < δ} ∪ {qδt (ξ), p(ξ)} has an upper bound in Q

˜ξ,


(∗)9 pδ,ξ = 〈pδ,ξt : t ∈ Tδ & rkδ(t) ≤ ξ〉 ⊆ Pξ is a tree of conditions (forξ ∈ wδ ∪ {γ}), pδ,γ = pδ.

(∗)10 If ζ, ξ ∈ wδ ∪ {γ}, ζ < ξ and t ∈ Tδ, rkδ(t) = ζ, then pδ,ζt ≤ pδ,ξt .

The demands (∗)11–(∗)16 formulated below are required only if δ = αδ is a limitordinal.

(∗)11 If t ∈ Tδ, rkδ(t) = ξ ∈ wδ and Xδt = {(s)ξ : t�s ∈ Tδ}, then

• either ∅ �= Xδt ⊆ Pδ and pδ,ξt �Pξ

“ X˜

δ,ξ = Xδt ”,

• or Xδt = {∗} and pδ,ξt �Pξ

“ X˜

δ,ξ = ∅ ”.(∗)12 If s ∈ Tδ, rkδ(s) = ζ, ξ ∈ wδ ∩ ζ and (s)ξ �= ∗, then

pδ,ζs �ξ �Pξ“ p˜δ,ξ((s)ξ) ≤ pδ,ζs (ξ) ”.

(∗)13 If ξ ∈ wδ ∪ {γ}, otp(wδ ∩ ξ) = ζ then• {t ∈ Tδ : rkδ(t) ≤ ξ} ⊆ Nδ+ζ+1, {t ∈ Tδ : rkδ(t) ≤ ξ}, 〈pδ,β : β ∈(wδ ∪ {γ}) ∩ (ξ + 1)〉 ∈ Nδ+ζ+2, and

• if ζ is limit, then 〈pδ,β : β ∈ wδ ∩ ξ〉 ∈ Nδ+ζ+1, and if t = 〈ti : i <i∗〉 ∈ Nδ+ζ+1 is a �–chain in {t ∈ Tδ : rkδ(t) < ξ} with sup(rkδ(ti) :i < i∗) = sup(wδ ∩ ξ), then t has a �–bound in Tδ.

306306


(∗)14 If t ∈ Tδ, ξ ∈ wδ ∩ rkδ(t) and (t)ξ �= ∗, thenqδt �ξ �Pξ

“ q˜δ,ξ((t)ξ) = qδt (ξ) ”.


(t1)ξ, then

pδt0�ξ �Pξ“ the conditions pδt0(ξ), p

δt1(ξ) are incompatible ”.

(∗)16 If τ˜∈ Nδ is a Pγ–name for an ordinal and t ∈ Tδ satisfies rkδ(t) = γ, then

the condition qδt forces a value to τ˜.

The rule (∗)0 (and conditions (∗)1–(∗)16) actually fully determines our objects, butwe should argue that at each stage there exist objects with properties listed in(∗)1–(∗)16.

Suppose we have arrived to a stage δ < λ of the construction and all objectslisted in (⊗)β for β < δ have been determined so that all relevant demands aresatisfied, in particular the sequence 〈 objects listed in (⊗)ε : ε < δ〉 is definablefrom N�αδ, α�δ, Q,p, p.

If δ is a successor ordinal and ξ ∈ wδ \ wδ−1, then we let st˜

ξ be the <∗χ–

first Pξ–name for a winning strategy of Generic in �B+p (rδ−1(ξ),Q

˜ξ). We also pick

the <∗χ–first sequence 〈f

˜ε,ξ,X

˜ε,ξ, p

˜ε,ξ, q

˜ε,ξ : ε < δ〉 so that (∗)7 is satisfied. Then

assuming that δ is not limit or δ �= αδ we may find objects listed in (⊗)δ so thatthe demands in (∗)1–(∗)10 are satisfied and |{t ∈ Tδ : rkδ(t) = γ}| = 1.

So suppose now that δ = αδ is a limit ordinal. For each ξ ∈ wδ we let f˜δ,ξ be

the <∗χ–first Pξ–name such that �Pξ

“f˜δ,ξ =

⋃

α<δ

f˜α,ξ”, and X

˜δ,ξ, p

˜δ,ξ be the <

∗χ–first

Pξ–names such that

�Pξ“ f˜δ,ξ,X

˜δ,ξ, p

˜δ,ξ are given to Generic by st

˜ξ

as the answer to 〈f˜ε,ξ,X

˜ε,ξ, p

˜ε,ξ, q

˜ε,ξ : ε < δ〉 ”.

Note that

〈 objects listed in (⊗)ε : ε < δ〉�〈f˜δ,ξ,X

˜δ,ξ, p

˜δ,ξ : ξ ∈ wδ〉 ∈ Nδ+1.

Now by induction on ξ ∈ wδ ∪{γ} we will choose {t ∈ Tδ : rkδ(t) ≤ ξ} and pδ,ξ andauxiliary objects p∗,ξ so that, in addition to demands (∗)9–(∗)13 we also have:

(∗)17 p∗,ξ = 〈p∗,ξt : t ∈ Tδ & rkδ(t) ≤ ξ〉 ⊆ Pξ is a tree of conditions, p∗,ξ ≤ pδ,ξ

and dom(p∗,ξt ) ⊇(dom(p) ∪

⋃

ε<δ

dom(rε) ∪ wδ

)∩ rkδ(t) whenever t ∈ Tδ,

rkδ(t) ≤ ξ, and

(∗)18 if ξ0 < ξ1 are from wδ ∪ {γ}, t ∈ Tδ, rkδ(t) = ξ0, then pδ,ξ0t ≤ p∗,ξ1t , and

(∗)19 if t ∈ Tδ, rkδ(t) = ξ, then dom(p∗,ξt ) = dom(pδ,ξt ) and for β ∈ dom(pδ,ξt )we have

pδ,ξt �β �Pβ“ the sequence 〈p∗,ζt�ζ(β), p

δ,ζt�ζ(β) : ζ ∈ (wδ ∪ {γ}) ∩ (ξ + 1)〉 is

a legal partial play of �λ0 (Q˜

β , ∅˜Q

˜β) in which Complete uses

the winning strategy st˜

0β ”.

To take care of clause (∗)13, each time we pick an object, we choose the <∗χ–first

one with the respective property.

Case 1: otp(wδ ∩ ξ) = ζ + 1 is a successor ordinal.Let ξ0 = max(wδ ∩ ξ) and suppose that we have defined T ∗ = {t ∈ Tδ : rkδ(t) ≤ ξ0}and p∗,ξ0 , pδ,ξ0 satisfying the relevant demands of (∗)9–(∗)19. Let t ∈ T ∗ be such

307307


that rkδ(t) = ξ0. It follows from (∗)11 that either pδ,ξ0t �“ X˜

δ,ξ0 = ∅ ” or pδ,ξ0t �“X˜

δ,ξ0 = Xδt ” for some non-empty set Xδ

t ⊆ Pδ. In the former case stipulateXt = {∗}. Note that necessarily Xδ

t ⊆ Nδ+1 and Xδt ∈ Nδ+ζ+2 (remember (∗)13).

We declare that

{t ∈ Tδ : rkδ(t) ≤ ξ} = T ∗ ∪ {t ∪ {(ξ0, a)} : t ∈ T ∗ & rkδ(t) = ξ0 & a ∈ Xδt }.

Plainly, |{t ∈ Tδ : rkδ(t) ≤ ξ}| < λ and even {t ∈ Tδ : rkδ(t) ≤ ξ} ⊆ Nδ+ζ+1

and {t ∈ Tδ : rkδ(t) ≤ ξ} ∈ Nδ+ζ+2 (again, by (∗)13). Choose a tree of conditionsp+ = 〈p+t : t ∈ Tδ & rkδ(t) ≤ ξ〉 ⊆ Pξ so that

• dom(p+t ) ⊇(dom(p) ∪

⋃

ε<δ

dom(rε) ∪ wδ

)∩ rkδ(t) for t ∈ Tδ, rkδ(t) ≤ ξ,

• if t ∈ Tδ, rkδ(t) < ξ then p+t = pδ,ξ0t ,• if t ∈ Tδ, rkδ(t) = ξ and (t)ξ0 �= ∗, then p+t (ξ0) is a Pξ0–name such that

pδ,ξ0t�ξ �Pξ0“ p˜δ,ξ0

((t)ξ0

)≤ p+t (ξ0) ”,

• if t ∈ Tδ, rkδ(t) = ξ and β ∈ dom(p+t ) \ (ξ0 + 1), then

p+t �β �Pβ“ if the set {rε(β) : ε < δ} ∪ {p(β)} has an upper bound in Q

˜β ,

then p+t (β) is such an upper bound ”.

(Note: p+ ∈ Nδ+ζ+2.) Next we may use Proposition 0.4 to pick a tree of conditions

p∗,ξ = 〈p∗,ξt : t ∈ Tδ & rkδ(t) ≤ ξ〉 such that p+ ≤ p∗,ξ and

• if ξ < γ, t ∈ Tδ, rkδ(t) = ξ, then either p∗,ξt � X˜

δ,ξ = ∅ or for some

non-empty set Xδt ⊆ Pδ we have p∗,ξt � X

˜δ,ξ = Xδ

t .

(Again, by our rule of picking “the <∗χ–first”, p

∗,ξ ∈ Nδ+ζ+2.) Then we choose a

tree of conditions pδ,ξ = 〈pδ,ξt : t ∈ Tδ & rkδ(t) ≤ ξ〉 so that p∗,ξ ≤ pδ,ξ and for

every t ∈ Tδ with rkδ(t) = ξ we have dom(pδ,ξt ) = dom(p∗,ξt ) and for β ∈ dom(pδ,ξt ),

pδ,ξt (β) is the <∗χ–first Pβ–name for a condition in Q

˜β such that

pδ,ξt �β �Pβ“ pδ,ξt (β) is given to Generic by st

˜0β as the answer to

〈p∗,εt�ε(β), pδ,εt�ε(β) : ε ∈ wδ ∩ ξ〉�〈p∗,ξt (β)〉 ”.

Note that, by the rule of picking “the <∗χ–first”, p

δ,ξ ∈ Nδ+ζ+2. It should be also

clear that p∗,ξ, pδ,ξ satisfy all the relevant demands stated in (∗)9–(∗)19.Case 2: otp(wδ ∩ ξ) = ζ is a limit ordinal.Suppose we have defined {t ∈ Tδ : rkδ(t) ≤ ε} and p∗,ε, pδ,ε for ε ∈ wδ ∩ ξ.By our rule of choosing “the <∗

χ–first objects”, we know that the sequence⟨{t ∈

Tδ : rkδ(t) ≤ ε}, p∗,ε, pδ,ε : ε ∈ wδ ∩ ξ⟩belongs to Nδ+ζ+1. We also know that

{t ∈ Tδ : rkδ(t) < ξ} ⊆ Nδ+ζ . Let T+ be the set of all limit branches in({t ∈

Tδ : rkδ(t) < ξ},�), so elements of T+ are sequences s = 〈(s)ε : ε ∈ wδ ∩ ξ〉 such

that s�ε = 〈(s)ε′ : ε′ ∈ wδ ∩ ε〉 ∈ {t ∈ Tδ : rkδ(t) ≤ ε} for ε ∈ wδ ∩ ξ. (Of course,T+ ∈ Nδ+ζ+1.) We put

{t ∈ Tδ : rkδ(t) ≤ ξ} = {t ∈ Tδ : rkδ(t) < ξ} ∪(T+ ∩Nδ+ζ+1

)∈ Nδ+ζ+2.

Due to (∗)19 at stages ε ∈ wδ ∩ ξ, we may choose a tree of conditions p+ = 〈p+t :t ∈ Tδ & rkδ(t) ≤ ξ〉 ⊆ Pξ such that

• dom(p+t ) ⊇(dom(p) ∪

⋃

ε<δ

dom(rε) ∪ wδ

)∩ rkδ(t) for t ∈ Tδ, rkδ(t) ≤ ξ,

and

308308


• if t ∈ Tδ, rkδ(t) = ξ0 < ξ then dom(p+t ) ⊇ dom(pδ,ξ0t ) and for eachβ ∈ dom(p+t ) ∩ ξ0 we have

p+t �β �Pβ“ pδ,ξ0t (β) ≤ p+t (β) ”, and

• if t ∈ Tδ, rkδ(t) = ξ, sup(wδ ∩ ξ) ≤ β < ξ, β ∈ dom(p+t ), then

p+t �β �Pβ“ if the set {rε(β) : ε < δ} ∪ {p(β)} has an upper bound in Q

˜β ,

then p+t (β) is such an upper bound ”.

Then, like in the successor case, we may find a tree of conditions p∗,ξ = 〈p∗,ξt : t ∈Tδ & rkδ(t) ≤ ξ〉 such that p+ ≤ p∗,ξ and

• if ξ < γ, t ∈ Tδ, rkδ(t) = ξ, then either p∗,ξt � X˜

δ,ξ = ∅ or for some

non-empty set Xδt ⊆ Pδ we have p∗,ξt � X

˜δ,ξ = Xδ

t .

Also like in that case we choose pδ,ξ = 〈pδ,ξt : t ∈ Tδ & rkδ(t) ≤ ξ〉. Clearly, allrelevant demands in (∗)9–(∗)19 are satisfied.

The last stage of the above construction gives us a tree Tδ = {t ∈ Tδ : rkδ(t) ≤γ} and a tree of conditions pδ,γ = pδ = 〈pδt : t ∈ Tδ〉. Since Tδ ⊆ Nδ+otp(wδ)+1,we know that |Tδ| < λ so we may apply Proposition 0.4 to get a tree of conditionsqδ = 〈qδt : t ∈ Tδ〉 ≥ pδ such that (∗)16 is satisfied. Remembering (∗)15 + (∗)12, weeasily find Pξ–names q

˜δ,ξ (for ξ ∈ wδ) such that

• �Pξ“ q˜δ,ξ = 〈q

˜δ,ξ(η) : η ∈ X

˜δ,ξ〉 is a system of conditions in Q

˜ξ ”,

• �Pξ“ (∀η ∈ X

˜δ,ξ)(p

˜δ,ξ(η) ≤ q

˜δ,ξ(η)) ”, and

• if t ∈ Tδ, rkδ(t) > ξ, then qδt�ξ �Pξ“ X˜

δ,ξ �= ∅ ⇒ qδt (ξ) = q˜δ,ξ

((t)ξ

)”.

So then (∗)14 is satisfied. Now we define r−δ , rδ ∈ Pγ essentially by (∗)1–(∗)3 and(∗)8.

After completing all λ stages of the construction, for each ξ ∈ N ∩γ we look atthe sequence 〈f

˜α,ξ,X

˜α,ξ, p

˜α,ξ, q

˜α,ξ : α < λ〉. By (∗)7, it is a Pξ–name for a play of

�B+p (rβ(ξ),Q

˜ξ) (where ξ ∈ wβ+1 \ wβ) in which Generic uses her winning strategy

st˜

ξ. Therefore, for every ξ ∈ N ∩ γ we may pick a Pξ–name q(ξ) for a condition inQ˜

ξ such that

• if ξ ∈ wβ+1 \ wβ , β < λ (or ξ ∈ w0, β = 0), then

�Pξ“q(ξ) ≥ rβ(ξ) and q(ξ) �Q

˜ξ

{δ<λ :

(∃η ∈ X

˜δ,ξ

)(q˜δ,ξ(η) ∈ ΓQ

˜ξ

)}∈ Dp[Pξ+1]”.

This determines a condition q ∈ Pγ (with dom(q) = N ∩γ) and easily (∀β < λ)(p ≤rβ ≤ q) (remember (∗)2). For each ξ ∈ N ∩ γ fix Pξ+1–names C

˜

ξi , g˜

ξi (for i < λ)

such that

q�(ξ + 1) �Pξ+1“(∀i < λ

)(C˜

ξi ∈ D ∩V & g

˜

ξi ∈ λλ

)and

(∀δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(g

˜

ξi ))(∃η ∈ X

˜δ,ξ

)(q˜δ,ξ(η) ∈ ΓQ

˜ξ

)”.

Let B˜

be a Pγ–name for the set {δ < λ : ΓPγ∩Nδ ∈ Nδ+1}. It follows from Lemma

1.6 that �PγB˜

∈ Dp[Pγ ].

Claim 3.3.1. If αδ = δ is limit, then

q �Pγ“ if δ ∈ B

ãnd

(∀ξ ∈ wδ

)(δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(g

˜

ξi ))

then(∃t ∈ Tδ

)(rkδ(t) = γ & qδt ∈ ΓPγ

)”.

309309


Proof of the Claim. Suppose that δ = αδ is a limit ordinal and a conditionr ≥ q forces (in Pγ) that

(∗)a20 δ ∈ B˜, and

(∗)b20(∀ξ ∈ wδ

)(δ ∈ �

i<λC˜

ξi ∩ �

i<λsetp(g

˜

ξi )).

Passing to a stronger condition if necessary, we may also assume that

(∗)a21 if r′ ∈ Pγ ∩Nδ, then either r′ ≤ r or r′, r are incompatible in Pγ .

Let Hδ = {r′ ∈ Pγ ∩Nδ : r′ ≤ r}. It follows from (∗)a20 + (∗)a21 that

(∗)b21 r �“ ΓPγ∩Nδ = Hδ ” and Hδ ∈ Nδ+1.

By 3.1(1)(a)+ (∗)7 + (∗)0 we may choose a sequence τ˜= 〈τ

˜(ξ, α) : ξ ∈ wδ & α <

δ〉 ∈ Nδ+1 such that

• τ˜(ξ, α) is a Pξ–name for an element of Q

˜ξ, τ˜(ξ, α) ∈ Nδ,

• �Pξτ˜(ξ, α) = f

˜δ,ξ(α).

Next we choose a sequence t∗ = 〈(t∗)ξ : ξ ∈ wδ〉 ∈∏

ξ∈wδ

(Pδ ∪ {∗}

)so that for each

ξ ∈ wδ, (t∗)ξ is the <∗

χ–first member of Pδ ∪ {∗} satisfying:

(∗)22 if t = t∗�ξ = 〈(t∗)ε : ε ∈ wδ ∩ ξ〉 ∈ Tδ and

(i) for some non-empty set X ⊆ Pδ, pδ,ξt �Pξ

X = X˜

δ,ξ (remember (∗)11)and there is η ∈ X such that

(ii)(∀α < δ

)(τ˜(ξ, η(α)) ∈ {r′(ξ) : r′ ∈ Hδ}

),

then (t∗)ξ ∈ X and(∀α < δ

)(τ˜(ξ, (t∗)ξ(α)) ∈ {r′(ξ) : r′ ∈ Hδ}

).

Note that for every ξ ∈ wδ ∪ {γ} the sequence t∗�ξ is definable (in L(∈, <∗χ))

from p, τ˜, Hδ, wδ, ξ and 〈pδ,ε : ε ∈ wδ ∩ ξ〉. Consequently, if ξ ∈ wδ ∪ {γ} and

ζ = otp(wδ ∩ ξ), then t∗�ξ ∈ Nδ+ζ+1. Now, by induction on ξ ∈ wδ ∪ {γ} we aregoing to show that t∗�ξ ∈ Tδ and choose conditions r∗ξ , r

∗∗ξ ∈ Pξ such that

(∗)a23 qδt∗�ξ ≤ r∗ξ , r�ξ ≤ r∗ξ and if ε ∈ wδ ∩ ξ then r∗∗ε ≤ r∗ξ , and

(∗)b23 dom(r∗ξ ) = dom(r∗∗ξ ) and r∗ξ ≤ r∗∗ξ and for every β ∈ dom(r∗∗ξ )

r∗∗ξ �Pβ“ 〈r∗ε(β), r∗∗ε (β) : ε ∈ wδ ∩ (ξ + 1)〉 is a partial play of �λ

0 (Q˜

β , ∅˜Q

˜β)


0β ”.

Suppose that otp(wδ ∩ ξ) is a limit ordinal and for ε ∈ wδ ∩ ξ we know thatt∗�ε ∈ Tδ and we have defined r∗ε , r

∗∗ε . It follows from (∗)13 that t∗�ξ ∈ Tδ. Let

β = sup(wδ ∩ ξ) ≤ ξ. It follows from (∗)b23 that we may find a condition r∗ξ ∈ Pξ

such that r∗ξ�β is stronger than all r∗∗ε (for ε ∈ wδ ∩ ξ) and r∗ξ�[β, ξ) = r�[β, ξ).Clearly r�ξ ≤ r∗ξ and also qδt∗�ξ�β ≤ r∗ξ�β (remember (∗)a23 for ε ∈ wδ ∩ ξ). Now by

induction on α ∈ [β, ξ) we argue that qδt∗�ξ�α ≤ r∗ξ�α. So suppose that β ≤ α < ξ

and we know already that qδt∗�ξ�α ≤ r∗ξ�α. It follows from (∗)3 + (∗)5 + (∗)6 that

r∗ξ�α �Pα

(∀i < δ

)(ri(α) ≤ pδt∗�ξ(α) ≤ qδt∗�ξ(α)

)and therefore we may use (∗)8 to

conclude that

r∗ξ�α �Pαqδt∗�ξ(α) ≤ rδ(α) ≤ q(α) ≤ r(α) = r∗ξ (α),

as desired. Finally we define r∗∗ξ ∈ Pξ essentially by (∗)b23.Now suppose that otp(wδ ∩ ξ) is a successor ordinal and let ξ0 = max(wδ ∩ ξ).

Assume we know that t∗�ξ0 ∈ Tδ and that we have already defined r∗ξ0 , r∗∗ξ0

∈ Pξ0 .

310310


It follows from the choice of q and from (∗)b20 that

r∗∗ξ0 �Pξ0“ r(ξ0) �Q

˜ξ0

(∃η ∈ X

˜δ,ξ0

)(q˜δ,ξ0(η) ∈ ΓQ

˜ξ0

)”.

Thus we may choose r∗ ∈ Pξ0+1 and η ∈ Pδ such that r∗∗ξ0 ≤ r∗�ξ0, r∗�ξ0 � r(ξ0) ≤r∗(ξ0) and r∗�ξ0 �Pξ0

“ η ∈ X˜

δ,ξ0 & q˜δ,ξ0(η) ≤ r∗(ξ0) ”. Then r�(ξ0 + 1) ≤ r∗

and (by (∗)7, 3.1(1)(c,d)) r∗�ξ0 �Pξ0

(∀α < δ

)(τ˜(ξ0, η(α)) ≤ r∗(ξ0)

)and hence (by

(∗)a21) r�ξ0 � τ˜(ξ0, η(α)) ≤ r(ξ0) for all α < δ. Therefore

(∗)a24 for all α < δ, τ˜(ξ0, η(α)) ∈ {r′(ξ0) : r′ ∈ Hδ}.

Since pδ,ξ0t∗�ξ0 ≤ r∗�ξ0 (remember (∗)a23 for ξ0 and (∗)5) we may use (∗)11 to conclude

that for some non-empty set X ⊆ Pδ we got pδ,ξ0t∗�ξ0 �Pξ0X = X

˜δ,ξ0 and η ∈ X

satisfies (ii) of (∗)22. Hence (t∗)ξ0 ∈ X is such that

(∗)b24 for all α < δ, τ˜(ξ0, (t

∗)ξ0(α)) ∈ {r′(ξ0) : r′ ∈ Hδ},and in particular t∗�ξ ∈ Tδ (remember (∗)11). We claim that (t∗)ξ0 = η. If not,then by 3.1(1)(b) we have

r∗�ξ0 �Pξ0

(∃α < δ

)(τ˜(ξ0, (t

∗)ξ0(α)), τ˜(ξ0, η(α)) are incompatible in Q

˜ξ0

),

so we may pick α < δ and a condition r+ ∈ Pξ0 such that r∗�ξ0 ≤ r+ and

r+ �Pξ0“ τ˜(ξ0, (t

∗)ξ0(α)), τ˜(ξ0, η(α)) are incompatible in Q

˜ξ0 ”.

However, r+ �Pξ0“ τ˜(ξ0, (t

∗)ξ0(α)) ≤ r(ξ0) & τ˜(ξ0, η(α)) ≤ r(ξ0) ” (by (∗)a24 +

(∗)b24), a contradiction.Now we define r∗ξ ∈ Pξ so that r∗ξ�(ξ0 + 1) = r∗ and r∗ξ�(ξ0, ξ) = r�(ξ0, ξ). By

the above considerations and (∗)14 we know that qδt∗�ξ�(ξ0 + 1) ≤ r∗ = r∗ξ�(ξ0 + 1).

Exactly like in the case of limit otp(wδ ∩ ξ) we argue that qδt∗�ξ ≤ r∗ξ . Finally, we

choose r∗∗ξ ∈ Pξ by (∗)b23.The last stage γ of the inductive process described above shows that t∗ ∈ Tδ

and qδt∗ ≤ r∗γ , r ≤ r∗γ . Now the claim readily follows. �

We finish the proof of part (a) of the theorem exactly like in the proof of 2.7.

(b) Included in the proof of the first part. �

4. Examples and counterexamples

Let us note that our canonical test forcing QEE is B–noble:

Proposition 4.1. Assume that E, E are as in 1.11 and p = (P , S,D) is a

D�–parameter on λ such that λ \ S ∈ E. Then the forcing QEE is B–noble over p.

Proof. The proof is a small modification of that of 1.12(2). First we fix an

enumeration 〈να : α < λ〉 = <λλ (remember λ<λ = λ), and for α < λ let f(α) ∈ QEE

be a condition such that root(f(α)

)= να and

(∀ν ∈ f(α)

)(να � ν ⇒ succf(α)(ν) = λ

).

Let p ∈ QEE . Consider the following strategy st of Generic in �B+

p (p,QEE). In the

course of the play Generic is instructed to build aside a sequence 〈Tξ : ξ < λ〉 sothat if 〈fξ,Xξ, pξ, qξ : ξ < λ〉 is the sequence of the innings of the two players, thenthe following conditions (a)–(d) are satisfied.

(a) Tξ ∈ QEE and if ξ < ζ < λ then p = T0 ⊇ Tξ ⊇ Tζ and Tζ ∩ ξλ = Tξ ∩ ξλ,

311311


(b) if ξ < λ is limit, then Tξ =⋂

ζ<ξ

Tζ ,

(c) if ξ ∈ S, then• fξ = f�ξ and Xξ ⊆ Pξ is a maximal set (possibly empty) such that

(α) for each η ∈ Xξ the family{f(η(α)

): α < ξ

}∪ {Tξ} has an upper

bound in QEE and lh

(⋃{νη(α) : α < ξ}

)= ξ,

(β) if η0, η1 ∈ Xξ are distinct, then for some α < ξ the conditionsf(η0(α)

), f

(η1(α)

)are incompatible,

• for η ∈ Xξ the condition pξη is an upper bound to{f(η(α)

): α < ξ

}∪{Tξ},

• Tξ+1 =⋃{qξη : η ∈ Xξ} ∪

⋃{(Tξ)ν : ν ∈ ξλ ∩ Tξ and ν /∈ pξη for η ∈ Xξ

},

(d) if ξ /∈ S, then Xξ = ∅, fξ = f�ξ and Tξ+1 = Tξ.

After the play is over, Generic puts p∗ =⋂

ξ<λ

Tξ ⊆ <λλ. Almost exactly as in

the proof of 1.12(2), one checks that p∗ ∈ QEE is a condition witnessing (�)B+

p of3.1(1). �

Definition 4.2. Let E, E be as in 1.11. We define a forcing notion PEE as

follows.A condition p in PE

E is a complete λ–tree p ⊆ <λλ such that

• for every ν ∈ p, either |succp(ν)| = 1 or succp(ν) ∈ Eν , and• for some set A ∈ E we have

(∀ν ∈ p

)(lh(ν) ∈ A ⇒ succp(ν) ∈ Eν

).

The order ≤=≤PEEis the reverse inclusion: p ≤ q if and only if (p, q ∈ PE

E and )

q ⊆ p.

Proposition 4.3. Assume that E, E are as in 1.11 and p = (P , S,D) is a D�–

parameter on λ such that λ \ S ∈ E. Then PEE is a (<λ)-complete forcing notion

of size 2λ which is B-noble over p.

Proof. The arguments of 4.1 can be repeated here with almost no changes (a

slight modification is needed for the justification that p∗ ∈ PEE). �

We may use the forcing PEE to substantially improve [RS05, Corollary 5.1].

First, let us recall the following definition.

Definition 4.4. Let F be a filter on λ including all co-bounded subsets of λ,∅ /∈ F .

(1) We say that a family F ⊆ λλ is F–dominating whenever(∀g ∈ λλ

)(∃f ∈ F

)({α < λ : g(α) < f(α)} ∈ F

).

The F–dominating number dF is the minimal size of an F–dominatingfamily in λλ.

(2) We say that a family F ⊆ λλ is F–unbounded whenever(∀g ∈ λλ

)(∃f ∈ F

)({α < λ : g(α) < f(α)} ∈ (F)+

).

The F–unbounded number bF is the minimal size of an F–unboundedfamily in λλ.

312312


(3) If F is the filter of co-bounded subsets of λ, then the corresponding dom-inating/unbounded numbers are also denoted by dλ, bλ. If F is the filtergenerated by club subsets of λ, then the corresponding numbers are calleddcl, bcl.

Corollary 4.5. Assume λ = λ<λ, 2λ = λ+. Suppose that p = (P , S,D) is aD�–parameter on λ, and E is a normal filter on λ such that λ \S ∈ E. Then thereis a λ++–cc λ–proper forcing notion P such that

�P “ 2λ = λ++ = bEP = dEP = dλ & bλ = bD[P] = dD[P] = λ+ ”.

Proof. For ν ∈ <λλ let Eν be the filter generated by clubs of λ and letE = 〈Eν : ν ∈ <λλ〉. Let Q = 〈Pξ,Q

˜ξ : ξ < λ++〉 be a λ–support iteration such

that for every ξ < λ++, �Pξ“ Q˜

ξ = PEE ”. (Remember, we use the convention

that in VPξ the normal filter generated by E is also denoted by E etc.) LetP = Pλ++ = lim(Q).

It follows from 3.3(a)+4.3 that P is λ–proper. Using [RS, Theorem 2.2] (seealso Eisworth [Eis03, §3]) we see that P satisfies the λ++–cc, �P 2λ = λ++ and Pis (<λ)–complete. Thus, the forcing with P does not collapse cardinals and it alsofollows from 3.3(b) that

�P “ λλ ∩V is D[P]–dominating in λλ ”.

It is also easy to check, that for each ξ < λ++

�P “ λλ ∩VPξ is not E–unbounded in λλ ”

and hence we may easily conclude that �P“ bEP = 2λ ”. �Definition 4.6. Assume that

• λ is weakly inaccessible, λ<λ = λ,• H : λ −→ λ is such that |α|+ ≤ |H(α)| for each α < λ,• F is a normal filter on λ, F = 〈Fν : ν ∈

⋃

α<λ

∏

ξ<α

H(ξ)〉 where Fν is a

(<|α|+

)–complete filter on H(α) whenever ν ∈

∏

ξ<α

H(ξ), α < λ.

We define forcing notions QHF ,F

and PHF ,F

as follows.

(1) A condition p in QHF ,F

is a complete λ–tree p ⊆⋃

α<λ

∏

ξ<α

H(ξ) such that

(a) for every ν ∈ p, either |succp(ν)| = 1 or succp(ν) ∈ Fν , and(b) for every η ∈ limλ(p) the set {α < λ : succp(η�α) ∈ Fη�α} belongs to

F .The order of QH

F ,Fis the reverse inclusion.

(2) A condition p in PHF ,F

is a complete λ–tree p ⊆⋃

α<λ

∏

ξ<α

H(ξ) satisfying

(a) above and(b)+ for some set A ∈ F we have

(∀ν ∈ p

)(lh(ν) ∈ A ⇒ succp(ν) ∈ Fν

).

The order of PHF ,F

is the reverse inclusion.

Proposition 4.7. Assume that λ,H, F , F are as in 4.6. Let p = (P , S,D) bea D�–parameter such that λ \ S ∈ F . Then both QH

F ,Fand PH

F ,Fare strategically

(<λ)–complete forcing notions of size 2λ which are also B–noble over p.

313313


Proof. Like 1.12(2), 4.1, 4.3. �

The property of being B–noble seems to be a relative of properness for D–semidiamonds introduced in [RS01] and even more so of properness over D–diamondsstudied in Eisworth [Eis03]. However, technical differences make it difficult tosee what are possible dependencies between these notions (see Problem 7.2). Inthis context, let us note that there are forcing notions which are proper over semidiamonds, but are not B–noble over any D�–parameter p. Let us consider, forexample, a forcing notion P∗ defined as follows:

a condition in P∗ is a function p such that

(a) dom(p) ⊆ λ+, rng(p) ⊆ λ+, |dom(p)| < λ, and(b) if α1 < α2 are both from dom(p), then p(α1) < α2;

the order ≤ of P∗ is the inclusion ⊆.

Proposition 4.8 (See [RS01, Prop. 4.1, 4.2]). P∗ is (<λ)-complete forcingnotion which is proper over all semi diamonds.

Proposition 4.9. P∗ is not B–noble over any D�–parameter p on λ.

Proof. Let q ∈ P∗ be such that that λ ∈ dom(q) and let W˜

0 be a P∗–namesuch that �P∗“ W

˜0 =

⋃ΓP∗�λ ”. Clearly

q �P∗ “ W˜

0 is a function with dom(W˜

0) ⊆ λ and rng(W˜

0) ⊆ λ ”.

Let W˜

be a P∗–name for a member of λλ such that

q �P∗ “ W˜

0 ⊆ W˜

and(∀α ∈ λ \ dom(W

˜0))(W˜(α) = α

)”.

Now suppose that p = (P , S,D) is a D�–parameter and 〈Aα : α < λ〉 is a sequenceof subsets of λ such that |Aα| < λ for α < λ. The following claim implies that P∗

(above the condition q) is not B–noble over p (remember 3.3(b)).

Claim 4.9.1. �P∗ “{α < λ : W

˜(α) ∈ Aα

}/∈ D[P∗] ”.

Proof of the Claim. Suppose that p ≥ q and B˜

i, f˜i (for i < λ) are P∗–

names for members of D ∩V and members of λλ, respectively, such that

p �P∗ “ �i<λ

B˜

i ∩ �i<λ

setp(f˜i) ⊆ {α < λ : W

˜(α) ∈ Aα} ”.

Build inductively a sequence 〈pi, Bi, fi : i < λ〉 such that for each i < λ:

(i) pi ∈ P∗, p ≤ p0 ≤ pj ≤ pi for j < i,(ii) Bi ∈ D ∩V, fi ∈ λλ and(iii) pi �P∗“ B

˜i = Bi and f

˜j�i = fj�i for all j ≤ i”, and

(iv) i < sup(dom(pi) ∩ λ

).

Since B = �i<λ

Bi ∩ �i<λ

setp(fi) ∈ D, we may pick a limit ordinal δ ∈ B such

that(∀i < δ

)(sup

((dom(pi) ∪ rng(pi)) ∩ λ

)< δ

). Put α = sup(Aδ) + 888 and

p+ =⋃

i<δ

pi ∪ {(δ, α)}. Then p+ ∈ P∗ is a condition stronger than all pi for i < δ

and p+ �P∗ “ δ ∈ �i<λ

B˜

i ∩ �i<λ

setp(f˜i) and W

˜(δ) = α /∈ Aδ ”, a contradiction. �

A similar construction can be carried out above any condition q such that forsome α ∈ dom(q) we have cf(α) = λ (the set of such conditions is dense in P∗). �

314314


5. QEE vs PE

E and Cohen λ–reals

The forcing notions QEE and PE

E (introduced in 1.11 and 4.2, respectively) mayappear to be almost the same. However, at least under some reasonable assumptionson E, E they do have different properties.

Suppose that V ⊆ V∗ are transitive universes of ZFC (with the same ordinals)such that <λλ ∩V = <λλ ∩V∗. We say that a function c ∈ λ2 ∩V∗ is a λ–Cohenover V if for every open dense set U ⊆ <λ2 (where <λ2 is equipped with the partialorder of the extension of sequences), U ∈ V, there is α < λ such that c�α ∈ U .

Proposition 5.1. Assume that

(a) λ is a strongly inaccessible cardinal,(b) S is the set of all strong limit cardinals κ < λ of countable cofinality,(c) E is a normal filter on λ such that S ∈ E,(d) E = 〈Eν : ν ∈ <λλ〉 is a system of (<λ)–complete non-principal filters on

λ.

Then the forcing notion PEE adds a λ–Cohen over V.

Proof. Let κ ∈ S. We will say that a tree T ⊆ <κκ is κ–interesting if

(�)0 | limκ(T )| = 2κ and for some increasing cofinal in κ sequence 〈δn : n <ω〉 ⊆ κ we have

(∀n < ω

)(∀ν ∈ T

)(lh(ν) ≤ δn ⇒ sup

(rng(ν)

)< δn+1

).

Note that there are only 2κ many κ–interesting trees (for κ ∈ S). Therefore wemay fix a well ordering of the family of κ–interesting trees of length 2κ and chooseby induction a function fκ : κκ −→ <κ2 \ {〈〉} such that

(�)1 if T ⊆ <κκ is a κ–interesting tree, then(∀σ ∈ <κ2 \ {〈〉}

)(∃η ∈ limκ(T )

)(fκ(η) = σ

).

Let W˜

be a PEE–name such that �

PEEW˜

=⋃{

root(p) : p ∈ ΓPEE

}. Plainly, � W

˜∈

λλ. Next, let C˜

be a PEE–name such that �

PEEC˜

= {κ ∈ S : W˜�κ ∈ κκ} and let τ

˜be a PE

E–name such that

�PEE

“ τ˜is the concatenation of all elements of the sequence

〈fκ(W˜�κ) : κ ∈ C

˜〉, i.e., τ

˜= 〈. . .�fκ(W

˜�κ)�. . .〉κ∈C

˜”.

Plainly, � τ˜∈ λ2 (remember, 〈〉 /∈ rng(fκ) for κ ∈ S).

We are going to argue that

�PEE“ τ˜is λ-Cohen over V ”.

To this end suppose that U ⊆ <λ2 is an open dense set and p ∈ PEE . Let

B ={κ ∈ S :

(∀η ∈ p ∩ κλ

)(succp(η) ∈ Eη

)}

(so B ∈ E). By induction on n < ω choose δn, Tn so that

(�)2 δn ∈ B, δn < δn+1, Tn ⊆ ≤δnλ is a complete tree (thus every chain in Tn

has a �–bound in Tn),(�)3 Tn ⊆ Tn+1 ⊆ p, Tn+1 ∩ δnλ = Tn ∩ δnλ, T0 ⊆ ≤δ0δ0, |T0 ∩ δ0δ0| = 1,(�)4 if ν ∈ Tn ∩ <δnλ, then succTn

(ν) �= ∅ and

2 ≤ |succTn(ν)| ⇒ lh(ν) ∈ {δ0, . . . , δn−1},

315315


(�)5 if ν ∈ Tn+1 ∩ δnλ, then |succTn+1(ν)| = δn and

sup(rng(ν)

)< δn+1 < min

(succTn+1

(ν)).

Next put κ = sup(δn : n < ω) and T =⋃

n<ωTn. Clearly κ ∈ S and T ⊆ <κκ is a

κ–interesting tree such that(∀η ∈ limκ(T )

)(κ = min{δ ∈ S : δ0 < δ & η�δ ∈ δδ}

).

Let T0 ∩ δ0δ0 = {η0} and C(η0) = {δ ∈ S ∩ (δ0 + 1) : η0�δ ∈ δδ}, and let τ0be the concatenation of all elements of the sequence 〈fδ(η0�δ) : δ ∈ C(η0)〉, i.e.,τ0 = 〈. . .�fδ(η0�δ)�. . .〉δ∈C(η0) ∈ <λ2. Pick σ ∈ <λ2 such that τ0�σ ∈ U . Itfollows from (�)1 that we may find η ∈ limκ(T ) such that σ = τ0

�fκ(η). Now notethat (p)η �

PEEσ�τ

˜. �


(a) λ is a measurable cardinal,(b) E = 〈Eν : ν ∈ <λλ〉 is a system of normal ultrafilters on λ,(c) E is a normal filter on λ,

(d) p ∈ QEE and τ

˜is a QE

E–name such that p � τ˜∈ λ2,

(e) δ = 〈δα : α < λ〉 is an increasing continuous sequence of non-successor

ordinals below λ such that δ0 = 0 and 22|δα|

< |δα+1| for all α < λ.

Then there are a condition q ∈ QEE and a sequence 〈Aα : α < λ〉 such that

(i) q ≥ p and Aα ⊆ [δα,δα+1)2, |Aα| < |δα+1| for α < λ, and(ii) q �

QEE

(∀α < λ

)(τ˜�[δα, δα+1) ∈ Aα

).

In particular, the forcing notion QEE does not add any λ–Cohen over V.

Proof. Let 〈να : α < λ〉 be an enumeration of <λλ such that να�νβ impliesα < β. By induction on α < λ we will construct a sequence 〈Aα, pα, Xα : α < λ〉so that for each α < λ we have:

()1 Aα ⊆ [δα,δα+1)2, |Aα| < |δα+1|, pα ∈ QEE , Xα ⊆ pα, |Xα| ≤ δα,

()2 if α < β < λ, then pα ≤ pβ and Xα ⊆ Xβ,()3 p0 = p, X0 = {root(p0)}, and if α is limit then pα =

⋂

β<α

pβ and Xα =⋃

β<α

Xβ,

()4 if ν ∈ Xα+1, then succpα(ν) ∈ Eν ,

()5 if α is limit, ν ∈ Xα and α ∈⋂

β<α

succpβ(ν), then for some η ∈ Xα+1 we

have ν�〈α〉 � η,()6 if να ∈ pα, then there is η ∈ Xα+1 such that να � η and if (additionally)

succpα(να) ∈ Eνα

, then η = να,()7 if α is limit and ν ∈ αα ∩ pα and succpα

(ν) ∈ Eν , then ν ∈ Xα+1,()8 pα+1 � τ

˜�[δα, δα+1) ∈ Aα.

Suppose that we have determined pβ, Xβ for β < α and Aβ for β + 1 < α so thatthe relevant instances of ()1–()8 are satisfied. If α is limit or 0, then pα, Xα aredefined by ()3 (and Aα will be chosen at the next step). One easily verifies thatpα, Xα satisfy the requirements in ()1–()4.

So suppose now that α = γ+1 (and we have defined pγ , Xγ and Aβ for β < γ).We may easily choose a set Xα ⊆ pγ such that

316316


()9 Xγ ⊆ Xα, |Xα| < δα and Xα satisfies ()4–()7 (with α there corre-sponding to γ here), and

()10 if η0, η1 ∈ Xα, ν = η0 ∩ η1, then ν ∈ Xα, and()11 if 〈ηξ : ξ < ζ〉 ⊆ Xα is �–increasing, then there is η ∈ Xα such that

(∀ξ < ζ)(ηξ � η).

Next, for each η ∈ Xα choose a function ση : [δγ , δγ+1) −→ 2 and a condition

qη ∈ QEE so that

()12 root(qη) = η, (pγ)η ≤ qη,(∀ν ∈ Xα

)(η�ν ⇒ ν(lh(η)) /∈ succqη(η)

)and

qη � τ˜�[δγ , δγ+1) = ση.

(Possible by assumption (b) and 2.4.) Put pα =⋃

η∈Xα

qη and Aγ = {ση : η ∈ Xα}.

Plainly, |Aγ | ≤ |Xα| < δγ+1 and pα ∈ QEE (to verify that pα is a complete λ-tree use

()10 + ()11; the other requirements easily follow from the fact that |Xα| < λ).One also easily checks that pα � τ

˜�[δγ , δγ+1) ∈ Aγ .

After the inductive construction is carried out, we put q =⋂

α<λ

pα. It follows

from ()2+()5+()6 that q is a complete λ–tree, |succq(ν)| = 1 or succq(ν) ∈ Eν

for each ν ∈ q, and⋃

α<λ

Xα = {ν ∈ q : succq(ν) ∈ Eν}.

Suppose now that η ∈ limλ(q). Then for each α < λ we have η ∈ limλ(pα) and

hence Bαdef= {ξ < λ : succpα

(η�ξ) ∈ Eη�ξ} ∈ E. Let

C = {δ < λ : δ is limit and η�δ ∈ δδ}

(it is a club of λ). Since E is a normal filter, C ∩ �α<λ

Bα ∈ E. Suppose δ ∈

C ∩ �α<λ

Bα. Then by ()3 +()7 we have η�δ ∈ Xδ+1 and thus succq(η�δ) ∈ Eη�δ.

Consequently, q ∈ QEE .

Finally, it follows from ()8 that q �(∀α < λ

)(τ˜�[δα, δα+1) ∈ Aα

). �

Let us note that forcing notions of the form QEE may add λ–Cohens if the filters

Eν are far from being ultrafilters.


(a) E is a normal filter on λ = λ<λ,(b) E = 〈Eν : ν ∈ <λλ〉 is a system of (<λ)–complete filters on λ,(c) for every ν ∈ <λλ there is a family {Aν

α : α < λ} of pairwise disjoint setsfrom (Eν)

+.

Then both the forcing notions QEE and PE

E add λ–Cohens over V.

Proof. We will sketch the argument for QEE only (no changes are needed for

the case of PEE).

For each ν ∈ <λλ choose a function hν : λ −→ <λ2 \ {〈〉} such that

(∀σ ∈ <λ2

)(∃α < λ

)(∀ξ ∈ Aν

α

)(hν(ξ) = σ

).

317317


Let W˜

be a QEE–name such that �

QEEW˜

=⋃{

root(p) : p ∈ ΓQ

EE

}and let τ

˜be a

QEE–name such that

�Q

EE

“ τ˜is the concatenation of all elements of the sequence

⟨hW

˜�ξ(W˜(ξ)

): ξ < λ

⟩, i.e.,

τ˜=

⟨h〈〉

(W˜(0)

)�h〈W

˜(0)〉

(W˜(1)

)�. . .�hW

˜�ξ(W˜(ξ)

)�. . .〉ξ<λ ”.

One easily verifies that �“ τ˜∈ λ2 is a λ–Cohen over V ”. �

The result in 5.2 would be specially interesting if we only knew that it ispreserved in λ–support iterations. Unfortunately, at the moment we do not knowif this is true (see Problem 7.3(1)). However, we may consider properties strongerthan adding λ–Cohens and then our earlier results give some input.

Definition 5.4. Suppose that V ⊆ V∗ are transitive universes of ZFC (withthe same ordinals) such that <λλ∩V = <λλ∩V∗. We say that a function c ∈ λ2∩V∗

is a

(1) strongly λ–Cohen over V if it is a λ–Cohen (i.e., for every open denseset U ⊆ <λ2 from V there is α < λ such that c�α ∈ U) and(�) if 〈ηα, βα : α < λ〉 ∈ V is such that α < βα < λ and ηα ∈ [α,βα)2 for

α < λ, then

V∗ |= {α < λ : ηα � c} is stationary;

(2) strongly⊕ λ–Cohen over V if it is a λ–Cohen and(⊕) if 〈ηα, βα : α < λ〉 ∈ V is such that α < βα < λ and ηα ∈ [α,βα)2 for

α < λ, then

V∗ |= {α < λ : ηα ⊆ c} is stationary.

(3) More generally, if D is a normal filter on λ, D ∈ V then we say that c ∈λ2∩V∗ is D–strongly⊕ λ–Cohen over V if in (⊕) we replace “stationary”by “∈ (DV∗

)+” (where DV∗is the normal filter generated by D in V∗).

Similarly for strongly.

Remark 5.5. (1) To explain our motivation for 5.4, let us recall thatif c ∈ λ2 is λ–Cohen over V and 〈ηα, βα : α < λ〉 ∈ V is such thatα < βα < λ and ηα ∈ [α,βα)2 for α < λ, then

V∗ |= “ both {α < λ : ηα ⊆ c} and {α < λ : ηα � c} are unbounded in λ ”.

(2) Let 〈ηα, βα : α < λ〉 ∈ V be such that α < βα < λ and ηα ∈ [α,βα)2 forα < λ. Let C = (<λ2,�) (so this is the λ–Cohen forcing notion) and let c

˜be the canonical C–name for the generic λ–real (i.e., �C c˜=

⋃ΓC). Let Q

˜be a C–name for a forcing notion in which conditions are closed boundedsets d ⊆ λ such that (∀α ∈ d)(ηα ⊆ c

˜) ordered by end extension. Then

C ∗Q˜

is essentially the λ-Cohen forcing and

�C∗Q˜“ c˜is not strongly λ–Cohen over V ”.

Hence, if we add a λ–Cohen then we also add a non-strong λ–Cohen.(3) Note that strongly⊕ λ–Cohen implies strongly λ–Cohen. (Simply, for a

sequence 〈ηα, βα : α < λ〉 consider 〈1− ηα, βα : α < λ〉.)

318318


Proposition 5.6. Assume that λ is a strongly inaccessible cardinal and p =(P , λ,Dλ) is a D�–parameter such that Pδ = δδ and Dλ is the filter generated byclub subsets of λ.

(1) If a forcing notion Q is reasonably B–bounding over p, then

�Q “ there is no strongly⊕ λ–Cohen over V ”.

(2) If Q = 〈Pα,Q˜

α : α < γ〉 is a λ–support iteration such that for every α < λ,

�Pα“ Q˜

α is reasonably B–bounding over p[Pα] ”,

then

�Pγ“ there is no strongly⊕ λ–Cohen over V ”.

Proof. (1) Note that, in VQ, Dλ[Q] is the filter generated by clubs of λ.Let p ∈ Q and let η

˜be a Q–name such that p � η

˜∈ λ2. Let st be a winning

strategy of Generic in the game �B+p (p,Q).

Let us consider a play of �B+p (p,Q) in which Generic follows the instructions

of st while Antigeneric plays as follows. In the course of the play, in addition tohis innings qαt , Antigeneric constructs aside a sequence

⟨κα, 〈ηαt : t ∈ Iα〉 : α < λ

⟩

such that if⟨Iα, 〈pαt , qαt : t ∈ Iα〉 : α < λ

⟩is the sequence of the innings of the two

players then the following two demands are satisfied.

()1 κα is a cardinal such that 2|Iα|+α+ℵ0 < κα and ηαt ∈ κα2 (for t ∈ Iα),()2 qαt �Q η

˜�κα = ηαt for each t ∈ Iα.

Since the play is won by Generic, there is a condition q ≥ p such that

q �Q “ {α < λ : (∃t ∈ Iα)(qαt ∈ ΓQ)} contains a club of λ ”.

It follows from ()1 that for each α < λ we may choose ε0α < ε1α from the in-terval (α, κα) such that

(∀t ∈ Iα

)(ηαt (ε

0α) = ηαt (ε

1α)). For each α < λ choose

να : [α, κα) −→ 2 so that να(ε0α) = 0 and να(ε

1α) = 1. Then

q �Q “ {α < λ : να � η˜} contains a club of λ ”.

(2) Similar, but we have to work with trees of conditions as in the proof of 1.10. �

6. Marrying B–bounding with fuzzy proper

In this section we introduce a property of forcing notions which, in a sense,marries the B–bounding forcing notions of [RS05, Definition 3.1(5)] with the fuzzyproper forcings introduced in [RS07, §A.3]. This property, defined in the languageof games, is based on two games: the servant game �servant

S,D which is the part

coming from the fuzzy properness and the master game �masterS,D which is related to

the reasonable boundedness property. Later in this section we will even formulatea true preservation theorem for a slightly modified game.

In this section we assume the following:

Context 6.1. (1) λ is a strongly inaccessible cardinal,(2) D is a normal filter on λ,(3) S ∈ D, 0 /∈ S, all successor ordinals below λ belong to S, λ \ S is un-

bounded.

Definition 6.2. Let Q be a forcing notion.

319319


(1) A Q–servant over S is a sequence q = 〈qδt : δ ∈ S & t ∈ Iδ〉 such that|Iδ| < λ (for δ ∈ S) and qδt ∈ Q (for δ ∈ S, t ∈ Iδ).

(2) Let q be a Q–servant over S and q ∈ Q. We define a game �servantS,D (q, q,Q)

as follows. A play of �servantS,D (q, q,Q) lasts at most λ steps during which the

players, COM and INC, attempt to construct a sequence 〈rα, Aα : α < λ〉such that

• rα ∈ Q, q ≤ rα, Aα ∈ D and α < β < λ ⇒ rα ≤ rβ.The terms rα, Aα are chosen successively by the two players so that

• if α /∈ S, then INC picks rα, Aα, and• if α ∈ S, then COM chooses rα, Aα.

If at some moment of the play one of the players has no legal move,then INC wins; otherwise, if both players always had legal moves and thesequence 〈rα, Aα : α < λ〉 has been constructed, then COM wins if andonly if(�)

(∀δ ∈ S

)([δ ∈

⋂

α<δ

Aα & δ is limit ] ⇒ (∃t ∈ Iδ)(qδt ≤ rδ)

).

(3) If COM has a winning strategy in the game �servantS,D (q, q,Q), then we will

say that q is an (S,D)–knighting condition for the servant q.

Definition 6.3. Let Q be a strategically (<λ)–complete forcing notion.

(1) For a condition p ∈ Q we define a game �masterS,D (q,Q) between two players,

Generic and Antigeneric. The game is a small modification of �rbBp (p,Q)

(see 1.8) — the main difference is in the winning condition. A play of�masterS,D (p,Q) lasts λ steps and during a play a sequence

⟨Iα, 〈pαt , qαt : t ∈ Iα〉 : α < λ

⟩

is constructed. Suppose that the players have arrived to a stage α < λ ofthe game. Now,(ℵ)α first Generic chooses a non-empty set Iα of cardinality < λ and a

system 〈pαt : t ∈ Iα〉 of conditions from Q,(�)α then Antigeneric answers by picking a system 〈qαt : t ∈ Iα〉 of condi-

tions from Q such that (∀t ∈ Iα)(pαt ≤ qαt ).

At the end, Generic wins the play⟨Iα, 〈pαt , qαt : t ∈ Iα〉 : α < λ

⟩of

�masterS,D (p,Q) if and only if letting q = 〈qαt : α ∈ S & t ∈ Iα〉 (it is a

Q–servant over S) we have

(�)D,Smaster there exists an (S,D)–knighting condition q ≥ p for the servant q.

(2) A forcing notion Q is reasonably merry over (S,D) if (it is strategi-cally (<λ)–complete and) Generic has a winning strategy in the game�masterS,D (p,Q) for any p ∈ Q.

Theorem 6.4. Assume that λ, S,D are as in 6.1. Let Q = 〈Pα,Q˜

α : α < γ〉be a λ–support iteration such that for each α < γ:

�Pα“ Q˜

α is reasonably merry over (S,D)”.

Then

(a) Pγ = lim(Q) is λ–proper, and(b) for every Pγ–name τ

˜for a function from λ to V and a condition p ∈ Pγ ,

there are q ≥ p and 〈Aξ : ξ < λ〉 such that (∀ξ < λ)(|Aξ| < λ) and

q � “ {ξ < λ : τ˜(ξ) ∈ Aξ} ∈

(DPγ

)+”.

320320


Proof. (a) The proof starts with arguments very much like those in [RS05,Theorems 3.1, 3.2], so we will state only what should be done (without actuallydescribing how the construction can be carried out). The major difference comeslater, in arguments that the chosen condition is suitably generic.

Suppose that N ≺ (H(χ),∈, <∗χ) is such that

<λN ⊆ N, |N | = λ and Q, S,D, . . . ∈ N.

Let p ∈ N ∩ Pγ and 〈τ˜α : α < λ〉 list all Pγ–names for ordinals from N . For each

ξ ∈ N ∩γ fix a Pξ–name st˜

0ξ ∈ N for a winning strategy of Complete in �λ

0 (Q˜

ξ, ∅˜Q

˜ξ)

such that it instructs Complete to play ∅˜Q

˜ξas long as her opponent plays ∅

˜Q

˜ξ.

Let us pick an increasing continuous sequence 〈wδ : δ < λ〉 of subsets of γ suchthat

⋃

δ<λ

wδ = N ∩ γ, w0 = {0} and |wδ| < λ.

By induction on δ < λ choose

()δ Tδ, pδ, qδ, r−δ , rδ, 〈ε˜δ,ξ, p˜δ,ξ, q

˜δ,ξ : ξ ∈ wδ〉, and st

˜ξ for ξ ∈ wδ+1 \ wδ

so that if the following conditions (∗)0–(∗)11 are satisfied (for each δ < λ).

(∗)0 All objects listed in ()δ belong to N and they are known after stage δof the construction.

(∗)1 r−δ , rδ ∈ Pγ , r−0 (0) = r0(0) = p(0), and for each α < δ < λ we have

(∀ξ ∈ wα+1)(rα(ξ) = r−δ (ξ) = rδ(ξ)) and p ≤ r−α ≤ rα ≤ r−δ ≤ rδ.(∗)2 If ξ ∈ dom(rδ) \ wδ, then

rδ�ξ � “ the sequence 〈r−α (ξ), rα(ξ) : α ≤ δ〉 is a legal partial play of�λ0

(Q˜

ξ, ∅˜Q

˜ξ


˜0ξ ”


ξ is a Pξ–name for a winning strategy ofGeneric in �master

S,D (rδ(ξ),Q˜

ξ) such that if 〈pαt : t ∈ Iα〉 is given by thatstrategy to Generic at stage α, then Iα is an ordinal below λ. Also st0 isa suitable winning strategy of Generic in �master

S,D (p(0),Q0).

(∗)3 Tδ = (Tδ, rkδ) is a standard (wδ, 1)γ–tree, |Tδ| < λ.

(∗)4 pδ = 〈pδt : t ∈ Tδ〉 and qδ = 〈qδt : t ∈ Tδ〉 are standard trees of conditionsin Q, pδ ≤ qδ.

(∗)5 If t ∈ Tδ, rkδ(t) = γ, then the condition qδt decides the values of all names〈τ˜α : α ≤ δ〉.

(∗)6 For t ∈ Tδ we have(dom(p) ∪

⋃

α<δ

dom(rα) ∪ wδ

)∩ rkδ(t) ⊆ dom(pδt ) and

for each ξ ∈ dom(pδt ) \ wδ:

pδt �ξ �Pξ“ if the set {rα(ξ) : α < δ} ∪ {p(ξ)} has an upper bound in Q

˜ξ,

then pδt (ξ) is such an upper bound ”.

(∗)7 If ξ ∈ wδ, then ε˜δ,ξ is a Pξ–name for an ordinal below λ, p

˜δ,ξ, q

˜δ,ξ are

Pξ–names for ε˜δ,ξ–sequences of conditions in Q

˜ξ.

(∗)8 If ξ ∈ wδ+1 \ wδ, then

�Pξ“ 〈ε

˜α,ξ, p

˜α,ξ, q

˜α,ξ : α < λ〉 is a play of �master

S,D (rδ(ξ),Q˜

ξ)

in which Generic uses st˜

ξ ”.

(∗)9 If t ∈ Tδ, rkδ(t) = ξ < γ, then the condition pδt decides the value of ε˜δ,ξ,

say pδt �“ε˜δ,ξ = εtδ,ξ”, and {(s)ξ : t�s ∈ Tδ} = εtδ,ξ and

qδt �Pξ“ p˜δ,ξ(ε) ≤ pδt〈ε〉(ξ) and q

˜δ,ξ(ε) = qδt〈ε〉(ξ) for ε < εtδ,ξ ”.

321321



(t1)ξ, then

pδt0�ξ �Pξ“ the conditions pδt0(ξ), p

δt1(ξ) are incompatible ”.

(∗)11 dom(r−δ ) = dom(rδ) =⋃

t∈Tδ

dom(qδt )∪dom(p) and if t ∈ Tδ, ξ ∈ dom(rδ)∩

rkδ(t) \ wδ, and qδt �ξ ≤ q ∈ Pξ, rδ�ξ ≤ q, then

q �Pξ“ if the set {rα(ξ) : α < δ} ∪ {qδt (ξ), p(ξ)} has an upper bound in Q

˜ξ,


After the construction is carried out we define a condition r ∈ Pγ as follows. Welet dom(r) = N ∩ γ and for ξ ∈ dom(r) we let r(ξ) be a Pξ–name for a conditionin Q

˜ξ such that if ξ ∈ wα+1 \ wα, α < λ (or ξ = 0 = α), then

�Pξ“r(ξ) ≥ rα(ξ) is an (S,D)–knighting condition for the servant

q˜

ξ def= 〈q

˜δ,ξ(ε) : δ ∈ S & ε < ε

˜δ,ξ〉 ”.

Clearly r is well defined (remember (∗)8). Note also that rδ ≤ r for all δ < λ andp ≤ r. We will argue that r is an (N,Pγ)–generic condition. To this end supposetowards contradiction that r∗ ≥ r, α∗ < λ and r∗ � τ

˜α∗ /∈ N .

For each ξ ∈ N ∩ γ fix a Pξ–name st˜

∗ξ for a winning strategy of COM in

the game �servantS,D (q

˜

ξ, r(ξ),Q˜

ξ). Moreover, for each ξ < γ fix a Pξ–name st˜

0ξ for

a winning strategy of Complete in �λ0 (Q˜

ξ, ∅˜Q

˜ξ) such that it instructs Complete to

play ∅˜Q


˜Q

˜ξ.

By induction on δ < λ we will build a sequence⟨r∗δ , r

+δ , 〈A˜

ξδ,i, A

ξδ,i : i < λ & ξ ∈ N ∩ γ〉 : δ < λ

⟩

such that the following demands (∗)12–(∗)16 are satisfied:

(∗)12 r∗α ∈ Pγ , r∗ ≤ r∗α ≤ r+α ≤ r∗δ for α < δ < λ,

(∗)13 A˜

ξδ,i is a Pξ–name for an element of D ∩V (for ξ ∈ N ∩ γ, i < λ),

(∗)14 if δ ∈ λ \ S and ξ ∈ wδ, then r∗δ�ξ �Pξ(∀α < δ)(∀i < δ)(A

˜

ξα,i = Aξ

α,i),

(∗)15 if β < δ < λ and ξ ∈ wβ+1 \wβ , then for some Pξ–names 〈s˜ξα : α ≤ β〉 we

have

r∗δ�ξ � “ the sequence 〈s˜ξα, �

i<λA˜

ξα,i : α ≤ β〉�〈r∗α(ξ), �

i<λA˜

ξα,i : β < α ≤ δ〉

is a legal partial play of �servantS,D (q

˜

ξ, r(ξ),Q˜

ξ)

in which Generic follows st˜

∗ξ ”,

(∗)16 dom(r+δ ) = dom(r∗δ ), r+δ �wδ = r∗δ�wδ and for each ξ ∈ dom(r+δ ) \ wδ we

have

r+δ �ξ � “ the sequence 〈r∗α(ξ), r+α (ξ) : α ≤ δ〉 is a legal partial play of�λ0

(Q˜

ξ, ∅˜Q

˜ξ


˜0ξ ”.

So suppose that we have arrived to a stage δ < λ of the construction and

• r∗α, r+α for α < δ,

• A˜

ξα,i for i < λ, α < δ and ξ ∈

⋃

β<δ

wβ ,

• Aξα,i for α, i < sup(δ \ S) and ξ ∈

⋃

β<sup(δ\S)

wβ

322322


have been determined.Case 1: δ /∈ S.Note that by our assumption on S (in 6.1), δ is not a successor ordinal, so wδ =⋃

α<δ

wα (or δ = 0 and w0 = {0}). By (∗)16 + (∗)15 we may choose a condition r∗δ

stronger than all r+α (for α < δ) and stronger than r∗ and such that for each ξ ∈ wδ

• if α < δ and i < δ, then r∗δ�ξ forces a value to A˜

ξα,i, say

r∗δ�ξ �PξA˜

ξα,i = Aξ

α,i.

For ξ ∈ wδ and i < λ we also let A˜

ξδ,i be a Pξ–name for the interval (δ, λ). The

condition r+δ is fully determined by (∗)16.Case 2: δ ∈ S is a successor ordinal, say δ = β + 1.

First, for each ξ ∈ wδ \ wβ we pick Pξ–names s˜ξα and A

˜

ξα,i (for α < δ, i < λ) such

that r+β �ξ �Pξr+β (ξ) = s

˜

ξ0 and

r∗δ�ξ � “ the sequence 〈s˜ξα, �

i<λA˜

ξα,i : α ≤ β〉 is a legal partial play of

�servantS,D (q

˜

ξ, r(ξ),Q˜

ξ) in which Generic follows st˜

∗ξ ”.

Next, we let dom(r∗δ ) = dom(r+β ) and for each ξ ∈ wδ we choose Pξ–names r∗δ (ξ)

and A˜

ξδ,i (for i < λ) such that

r+β �ξ �Pξr∗δ (ξ), �

i<λA˜

ξδ,i is the answer to the partial game as in (∗)15 given by st∗ξ .

For ξ ∈ dom(r∗δ ) \wδ we put r∗δ (ξ) = r+β (ξ). Then we define condition r+δ by (∗)16.Case 3: δ ∈ S is a limit ordinal.We let dom(r∗δ ) =

⋃

α<δ

dom(r+α ) and by induction on ξ ∈ dom(r∗δ ) we define r∗δ (ξ)

so that

• if ξ /∈ wδ, then r∗δ�ξ � (∀α < δ)(r+α (ξ) ≤ r∗δ (ξ)) (exists by (∗)16),• if ξ ∈ wδ then for some Pξ–names A

˜

ξδ,i for members of D ∩V

r∗δ�ξ �Pξr∗δ (ξ), �

i<λA˜

ξδ,i is the answer to the partial game as in (∗)15 given by st∗ξ .

The condition r+δ is given by (∗)16.After the above construction is carried out we note that

{δ < λ : (∀ξ ∈ wδ)(∀α, i < δ)(δ ∈ Aξ

α,i)}∈ D,

so we may choose an ordinal δ ∈ S \ (α∗ + 1) which is a limit of points from λ \ Sand such that δ ∈

⋂

α<δ

�i<λ

Aξα,i for all ξ ∈ wδ. The following claim provides the

desired contradiction (remember (∗)0 + (∗)5).

Claim 6.4.1. For some t ∈ Tδ such that rkδ(t) = γ the conditions qδt and r∗δare compatible

Proof of the Claim. The proof is very much like that of Claim 2.7.1. Let〈εβ : β ≤ β∗〉 = wδ ∪ {γ} be the increasing enumeration. For each ξ < γ fix a Pξ–name st

˜∗ξ for a winning strategy of Complete in �λ

0 (Q˜

ξ, ∅˜Q

˜ξ) such that it instructs

Complete to play ∅˜Q


˜Q

˜ξ.

323323


By induction on β ≤ β∗ we will choose conditions sβ, s∗β ∈ Pεβ and t = 〈(t)εβ :

β < β∗〉 ∈ Tδ such that letting tβ◦ = 〈(t)ε′β : β′ < β〉 ∈ Tδ we have

()a qδtβ◦

≤ sβ and r∗δ�εβ ≤ sβ,

()b dom(sβ) = dom(s∗β) and for every ζ < εβ ,

s∗β�ζ �Pζ“ 〈sβ′(ζ), s∗β′(ζ) : β′ < β〉 is a partial legal play of �λ

0 (Q˜

ζ , ∅˜Q

˜ζ)


∗ζ ”.

Suppose that β ≤ β∗ is a limit ordinal and we have already defined tβ◦ = 〈(t)εβ′ :

β′ < β〉 and 〈sβ′ , s∗β′ : β′ < β〉. Let ξ = sup(εβ′ : β′ < β). It follows from ()bthat we may find a condition sβ ∈ Pεβ such that sβ�ξ is stronger than all s∗β′ (for

β′ < β) and sβ�[ξ, εβ) = r∗δ�[ξ, εβ). Clearly r∗δ�εβ ≤ sβ and also qδtβ◦�ξ ≤ sβ�ξ

(remember ()a). Now by induction on ζ ∈ [ξ, εβ] we argue that qδtβ◦�ζ ≤ sβ�ζ.

Suppose that ξ ≤ ζ < εβ and we know qδtβ◦�ζ ≤ sβ�ζ. By (∗)2+(∗)4+(∗)6 we know

that sβ�ζ � (∀i < δ)(ri(ζ) ≤ pδtβ◦(ζ) ≤ qδ

tβ◦(ζ)) and therefore we may use (∗)11 to

conclude that

sβ�ζ �Pζqδtβ◦(ζ) ≤ rδ(ζ) ≤ r(ζ) ≤ r∗δ (ζ) = sβ(ζ).

Then the condition s∗β ∈ Pεβ is determined ()b.

Now suppose that β = β′ + 1 ≤ β∗ and we have already defined sβ′ , s∗β′ ∈ Pεβ′

and tβ′

◦ ∈ Tδ. It follows from the choice of δ and (∗)14 that r∗δ�εβ′ � δ ∈⋂

α<δ

�i<λ

A˜

ξα,i

and hence, by the choice of r and (∗)15 we have (remember (�) of 6.2(2))

s∗β′ �Pεβ′ “

(∃ε < ε

˜δ,εβ′

)(q˜δ,εβ′ (ε) ≤ r∗δ (εβ′)

)”.

Therefore we may use (∗)9 to choose ε = (t)εβ′ and a condition sβ ∈ Pεβ such that

• tβ◦def= tβ

′◦ ∪ {(εβ′ , ε)} ∈ Tδ, s

∗β′ ≤ sβ�εβ′ and

sβ�εβ′ �Pεβ′ “ qδtβ◦(εβ′) ≤ r∗δ (εβ′) = sβ(εβ′) ”,

• r∗δ�(εβ′ , εβ) = sβ�(εβ′ , εβ).

We finish exactly like in the limit case.

After the inductive construction is completed, look at t = tβ∗

◦ and sβ∗ . �

(b) Should be clear at the moment. �

Definition 6.5 (See [RS05, Def. 3.1]). LetQ be a strategically (<λ)–completeforcing notion.

(1) Let p ∈ Q. A game �rcBD (p,Q) is defined similarly to �rbB

p (p,Q) (see 1.8)

except that the winning criterion (�)prbB is weakened to(�)rcB there is a condition p∗ ∈ Q stronger than p and such that

p∗ �Q “{α < λ :

(∃t ∈ Iα

)(qαt ∈ ΓQ

)}∈ DQ ”.

(2) A forcing notion Q is reasonably B–bounding over D if for any p ∈ Q,Generic has a winning strategy in the game �rcB

D (p,Q).

Observation 6.6. If Q is reasonably B–bounding over D, then it is reasonablymerry over (S,D).

324324


It is not clear though, if forcing notions which are reasonably B–bounding over aD�–parameter p are also reasonably merry (see Problem 7.4). Also, we do not knowif fuzzy properness introduced in [RS07, §A.3] implies that the considered forcingnotion is reasonably merry (see Problem 7.5), even though the former propertyseems to be almost built into the latter one.

One may ask if being reasonably merry implies being B–bounding. There areexamples that this is not the case. The forcing notion Q2

D (see 6.8 below) wasintroduced in [RS05, Section 6] and by [RS05, Proposition 6.4] we know that it isnot reasonably B–bounding overD. However we will see in 6.12 that it is reasonablymerry over (S,D).

Definition 6.7 (See [RS05, Def. 5.1]). (1) Let α < β < λ. An (α, β)–extending function is a mapping c : P(α) −→ P(β) \ P(α) such thatc(u) ∩ α = u for all u ∈ P(α).

(2) Let C be an unbounded subset of λ. A C–extending sequence is a sequencec = 〈cα : α ∈ C〉 such that each cα is an (α,min(C \ (α + 1)))–extendingfunction.

(3) Let C ⊆ λ, |C| = λ, β ∈ C, w ⊆ β and let c = 〈cα : α ∈ C〉 be a C–extending sequence. We define pos+(w, c, β) as the family of all subsets uof β such that(i) if α0 = min

({α ∈ C : (∀ξ ∈ w)(ξ < α)}

), then u ∩ α0 = w (so if

α0 = β, then u = w), and(ii) if α0, α1 ∈ C, w ⊆ α0 < α1 = min(C \ (α0 + 1)) ≤ β, then either

cα0(u ∩ α0) = u ∩ α1 or u ∩ α0 = u ∩ α1,

(iii) if sup(w) < α0 = sup(C ∩ α0) /∈ C, α1 = min(C \ (α0 + 1)

)≤ β,

then u ∩ α1 = u ∩ α0.For α0 ∈ β ∩ C such that w ⊆ α0, the family pos(w, c, α0, β) consists ofall elements u of pos+(w, c, β) which satisfy also the following condition:(iv) if α1 = min

(C \ (α0 + 1)

)≤ β, then u ∩ α1 = cα0

(w).

Definition 6.8 (See [RS05, Def. 6.2]). We define a forcing notion Q2D as

follows.A condition in Q2

D is a triple p = (wp, Cp, cp) such that

(i) Cp ∈ D, wp ⊆ min(Cp),(ii) cp = 〈cpα : α ∈ Cp〉 is a Cp–extending sequence.

The order ≤Q2D=≤ of Q2

D is given byp ≤Q

2Dq if and only if

(a) Cq ⊆ Cp and wq ∈ pos+(wp, cp,min(Cq)) and(b) if α0, α1 ∈ Cq, α0 < α1 = min(Cq \ (α0 + 1)) and u ∈ pos+(wq, cq, α0),

then cqα0(u) ∈ pos(u, cp, α0, α1).

For p ∈ Q2D, α ∈ Cp and u ∈ pos+(wp, cp, α) we let p�αu

def= (u,Cp \α, cp�(Cp \α)).

In [RS05, Problem 6.1] we asked if λ–support iterations of forcing notions Q2D

are λ–proper. Now we may answer this question positively (assuming that λ isstrongly inaccessible). First, let us state some auxiliary definitions and facts.

Proposition 6.9. (1) Q2D is a (<λ)–complete forcing notion of cardinal-

ity 2λ.(2) If p ∈ Q2

D and α ∈ Cp, then

325325


• for each u ∈ pos+(wp, cp, α), p�αu ∈ Q2D is a condition stronger than

p, and• the family {p�αu : u ∈ pos+(wp, cp, α)} is pre-dense above p.

(3) Let p ∈ Q2D and α < β be two successive members of Cp. Suppose that

for each u ∈ pos+(wp, cp, α) we are given a condition qu ∈ Q2D such

that p�βcpα(u) ≤ qu. Then there is a condition q ∈ Q2D such that letting

α′ = min(Cq \ β) we have(a) p ≤ q, wq = wp, Cq ∩ β = Cp ∩ β and cqδ = cpδ for δ ∈ Cq ∩ α, and

(b)⋃{

wqu : u ∈ pos+(wp, cp, α)}⊆ α′, and

(c) qu ≤ q�α′cqα(u) for every u ∈ pos+(wp, cp, α).(4) Assume that p ∈ Q2

D, α ∈ Cp and τ˜is a Q2

D–name such that p �“τ˜∈ V”.

Then there is a condition q ∈ Q2D stronger than p and such that

(a) wq = wp, α ∈ Cq and Cq ∩ α = Cp ∩ α, and(b) if u ∈ pos+(wq, cq, α) and γ = min(Cq \ (α+ 1)), then the condition

q�γcq(u) forces a value to τ˜.

Proof. Fully parallel to [RS05, Proposition 5.1]. �

Definition 6.10. The natural limit of an ≤Q2D–increasing sequence p = 〈pξ :

ξ < γ〉 ⊆ Q2D (where γ < λ is a limit ordinal) is the condition q = (wq, Cq, cq)

defined as follows:

• wq =⋃

ξ<γ

wpξ , Cq =⋂

ξ<γ

Cpξ and

• cq = 〈cqδ : δ ∈ Cq〉 is such that for δ ∈ Cq and u ⊆ δ we have cqδ(u) =⋃

ξ<γ

cpξ

δ (u).

Proposition 6.11. (1) Suppose p = 〈pξ : ξ < λ〉 is a ≤Q2D–increasing

sequence of conditions from Q2D such that

(a) wpξ = wp0 for all ξ < λ, and(b) if γ < λ is limit, then pγ is the natural limit of p�γ, and(c) for each ξ < λ, if δ ∈ Cpξ , otp(Cpξ ∩ δ) = ξ, then Cpξ+1 ∩ (δ + 1) =

Cpξ ∩ (δ + 1) and for every α ∈ Cpξ+1 ∩ δ we have cpξ+1α = c

pξα .

Then the sequence p has an upper bound in Q2D.

(2) Suppose that p ∈ Q2D and h

˜is a Q2

D–name such that p �“h˜: λ −→ V”.

Then there is a condition q ∈ Q2D stronger than p and such that

(⊗) if δ < δ′ are two successive points of Cq, u ∈ pos(wq, cq, δ), then thecondition q�δ′cqδ(u) decides the value of h

˜�(δ + 1).

Proof. Fully parallel to [RS05, Proposition 5.2]. �

Proposition 6.12. Assume that λ, S,D are as in 6.1. The forcing notion Q2D

is reasonably merry over (S,D).

Proof. Let p ∈ Q2D. We will describe a strategy st for Generic in the game

�masterS,D (p,Q2

D) - this strategy is essentially the same as the one in the proof of

[RS05, Proposition 5.4], only the argument that it is a winning strategy is different.In the course of a play the strategy st instructs Generic to build aside an

increasing sequence of conditions p∗ = 〈p∗α : α < λ〉 ⊆ Q2D such that for each α < λ:

(a) p∗0 = p and wp∗α = wp, and

(b) if α < λ is limit, then p∗α is the natural limit of p∗�α, and

326326


(c) if δ ∈ Cp∗α , otp(Cp∗

α ∩ δ) = α, then Cp∗α+1 ∩ (δ + 1) = Cp∗

α ∩ (δ + 1) and

for every ξ ∈ Cp∗α+1 ∩ δ we have c

p∗α+1

ξ = cp∗α

ξ , and

(d) after stage α of the play of �masterS,D (p,Q2

D), the condition p∗α+1 is deter-mined.

After arriving to the stage α, Generic is instructed to pick δ ∈ Cp∗α such that

otp(Cp∗α ∩ δ) = α, put γ = min(Cp∗

α \ (δ+1)) and play as her innings of this stage:

Iα = pos+(wp∗α , cp

∗α , δ) and pαu = p∗α�γc

p∗α

δ (u) for u ∈ Iα.

Then Antigeneric answers with 〈qαu : u ∈ Iα〉 ⊆ Q2D. Since p∗α�γc

p∗α

δ (u) ≤ qαu for

each u ∈ pos+(wp∗α , cp

∗α , δ), Generic may use 6.9(3) (with δ, γ, p∗α, q

αu here standing

for α, β, p, qu there) to pick a condition p∗α+1 such that, letting α′ = min(Cp∗α+1 \γ),

we have

(e) p∗α ≤ p∗α+1, wp∗α+1 = wp, Cp∗

α+1 ∩ γ = Cp∗α ∩ γ and c

p∗α+1

ξ = cp∗α

ξ for

ξ ∈ Cp∗α+1 ∩ δ, and

(f)⋃{

wqαu : u ∈ Iα}⊆ α′, and

(g) qαu ≤ p∗α+1�α′cp∗α+1

δ (u) for every u ∈ Iα.

This completes the description of st. Suppose that⟨Iα, 〈pαu , qαu : u ∈ Iα〉 : α < λ

⟩is

the result of a play of �masterS,D (p,Q2

D) in which Generic followed st and constructed

aside p∗ = 〈p∗α : α < λ〉. By 6.11, there is a condition p∗ ∈ Q2D stronger than all

p∗α (for α < λ). We claim that p∗ is an (S,D)–knighting condition for the servantq = 〈qαu : α ∈ S & u ∈ Iα〉. To this end consider the following strategy st∗ of COMin �servant

S,D (q, p∗,Q2D). After arriving to a stage α ∈ S of a play of �servant

S,D (q, p∗,Q2D),

when 〈rβ, Aβ : β < α〉 has been already constructed, COM plays as follows.If α is a successor or α /∈

⋂

β<α

Aβ, then she just puts rα, Aα such that:

(h) rβ ≤ rα for all β < α and if δ ∈ Cp∗α is such that otp(δ ∩ Cp∗

α) = α, thenwrα \ (δ + 1) �= ∅, and

(i) Aα =⋂

β<α

Aβ ∩⋂

β<α

Crβ \ (sup(wrα) + 1).

If α ∈⋂

β<α

Aβ is a limit ordinal, then COM first lets u =⋃

β<α

wrβ . It follows from

(h)+(i) from earlier stages that u ⊆ α, α ∈ Cp∗α and otp(α ∩ Cp∗

α) = α. Note thatα ∈

⋂

β<α

Crβ , u ∈⋂

β<α

pos+(wrβ , crβ , α) and u ∈ Iα. Let α′ = min(Cp∗ \ (α + 1)

)

and α′′ ∈⋂

β<α

Crβ \ (α+ 1). It follows from (c)+(g) that for each β < α we have

qαu ≤ p∗α+1�α′cp∗α+1

α (u) ≤ p∗�α′cp∗

α (u) ≤ rβ�α′′crβα (u).

Hence COM may choose a condition rα ≥ qαu stronger than all rβ (for β < α) andsatisfying (h). Then Aα is given by (i).

It follows directly from the description of st∗ that it is a winning strategy ofCOM in �servant

S,D (q, p∗,Q2D). �

The master game �masterS,D used to define the property of being reasonably merry

is essentially a variant of the A–reasonable boundedness game �rcA of [RS05, Def.3.1]. The related bounding property was weakened in [RS, Def. 2.9] by introducingdouble a–reasonably completeness game �rc2a. We may use these ideas to introduce

327327


a property much weaker than “reasonably merry”, though the description of theresulting notions becomes somewhat more complicated. As an award for additionalcomplication we get a true preservation theorem, however. In the rest of thissection, in addition to 6.1 we assume also

Context 6.13. μ = 〈μα : α < λ〉 is a sequence of cardinals below λ such that

(∀α < λ)(ℵ0 ≤ μα = μ|α|α ).

Definition 6.14. Let Q be a forcing notion.

(1) A double Q–servant over S, μ is a sequence

q = 〈ξδ, qδγ : δ ∈ S & γ < μδ · ξδ〉such that for δ ∈ S,

• 0 < ξδ < λ and qδγ ∈ Q (for γ < μδ · ξδ),•(∀i, i′ < ξδ

)(∀j < μδ

)(i′ < i ⇒ qδμδ·i′+j ≤ qδμδ·i+j

).

(Here μδ is treated as an ordinal and μδ · ξδ is the ordinal product of μδ

and ξδ.)(2) Let q be a double Q–servant over S, μ and let q ∈ Q. We define a game

�2serS,D,μ(q, q,Q) as follows. A play of �2ser

S,D,μ(q, q,Q) lasts at most λ stepsduring which the players, COM and INC, attempt to construct a sequence〈rα, Aα : α < λ〉 such that

• rα ∈ Q, q ≤ rα, Aα ∈ D and α < β < λ ⇒ rα ≤ rβ.The terms rα, Aα are chosen successively by the two players so that

• if α /∈ S, then INC picks rα, Aα, and• if α ∈ S, then COM chooses rα, Aα.

If at some moment of the play one of the players has no legal move,then INC wins; otherwise, if both players always had legal moves and thesequence 〈rα, Aα : α < λ〉 has been constructed, then COM wins if andonly if

(♥)(∀δ ∈ S

)([δ ∈

⋂

α<δ

Aα & δ is limit ] ⇒ (∃j < μδ)(∀i < ξδ)(qδμδ·i+j ≤ rδ)

).

(3) If COM has a winning strategy in the game �2serS,D,μ(q, q,Q), then we will

say that q is an knighting condition for the double servant q.

Definition 6.15. Let Q be a strategically (<λ)–complete forcing notion.

(1) For a condition p ∈ Q we define a game �2masS,D,μ(p,Q) between two players,

Generic and Antigeneric. A play of �2masS,D,μ(p,Q) lasts λ steps and during

a play a sequence⟨ξα, 〈pαγ , qαγ : γ < μα · ξα〉 : α < λ

⟩.

is constructed. (Again, μα · ξα is the ordinal product of μα and ξα.)Suppose that the players have arrived to a stage α < λ of the game. First,Antigeneric picks a non-zero ordinal ξα < λ. Then the two players start asubgame of length μα · ξα alternately choosing the terms of the sequence〈pαγ , qαγ : γ < μα · ξα〉. At a stage γ = μα · i+ j (where i < ξα, j < μα) ofthe subgame, first Generic picks a condition pαγ ∈ Q stronger than p andstronger than all conditions qαδ for δ < γ of the form δ = μα · i′+ j (wherei′ < i), and then Antigeneric answers with a condition qαγ stronger thanpαγ .

328328


At the end, Generic wins the play if(a) there were always legal moves for both players (so a sequence

⟨ξα, 〈pαγ , qαγ : γ < μα · ξα〉 : α < λ

⟩

has been constructed) and(b) for each α ∈ S, the conditions in {pαj : j < μα} are pairwise incom-

patible, and(c) letting

q = 〈ξδ, qδγ : δ ∈ S & γ < μδ · ξδ〉(it is a double Q–servant over S) we may find a knighting conditionq ≥ p for the double servant q.

(2) A forcing notion Q is reasonably double merry over (S,D, μ) if (it is strate-gically (<)–complete and) Generic has a winning strategy in the game�2masS,D,μ(p,Q) for any p ∈ Q.

Theorem 6.16. Assume that λ, S,D, μ are as in 6.1+6.13. Let Q = 〈Pα,Q˜

α :α < γ〉 be a λ–support iteration such that for each α < γ:

�Pα“ Q˜

α is reasonably double merry over (S,D, μ)”.

Then Pγ = lim(Q) is reasonably double merry over (S,D, μ) (so also λ–proper).

Proof. Combine the proof of [RS, Thm 2.12] (the description of the strategyhere is the same as the one there) with the end of the proof of 6.4(a). �

7. Open problems

Problem 7.1. Let p = (P , S,D) be a D�–parameter. Does “reasonably B-bounding over p” (see 1.8) imply “reasonably B-bounding over D” (of [RS05, Def.3.1])? Does “reasonably B-bounding over p” imply “B–noble over p” ? (Note 6.6.)

Problem 7.2. Are there any relations among the notions of properness overD–semi diamonds (of [RS01]), properness over D–diamonds (of [Eis03]) and B–nobleness (of 3.1)?

Problem 7.3. Does λ–support iterations of forcing notions of the form QEE

add λ–Cohens? Here we may look at iterations as in 3.3 or 2.7.

Problem 7.4. Does “reasonably B–bounding over p” (for a D�–parameter p)imply “reasonably merry over (S,D)” (for some S,D as in 6.1)?

Problem 7.5. Does “fuzzy proper over quasi D–diamonds for W” (see [RS07,Def. A.3.6]) imply “reasonably merry”? (Any result in this direction may requireadditional assumptions on W, Y in [RS07, A.3.1, A.3.3].)

References

[CS95] James Cummings and Saharon Shelah, Cardinal invariants above the continuum, Annalsof Pure and Applied Logic 75 (1995), 251–268, math.LO/9509228.

[Eis03] Todd Eisworth, On iterated forcing for successors of regular cardinals, FundamentaMathematicae 179 (2003), 249–266.

[HR01] Tapani Hyttinen and Mika Rautila, The canary tree revisited, The Journal of SymbolicLogic 66 (2001), 1677–1694.

[Jec03] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin,2003, The third millennium edition, revised and expanded.

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[MS93] Alan H. Mekler and Saharon Shelah, The canary tree, Canadian Journal of Mathematics.Journal Canadien de Mathematiques 36 (1993), 209–215, math.LO/9308210.

[RS] Andrzej Roslanowski and Saharon Shelah, Reasonable ultrafilters, again, Notre DameJournal of Formal Logic submitted, math.LO/0605067.

[RS99] , Norms on possibilities I: forcing with trees and creatures, Memoirs of the Amer-ican Mathematical Society 141 (1999), no. 671, xii + 167, math.LO/9807172.

[RS01] , Iteration of λ-complete forcing notions not collapsing λ+., International Journal

of Mathematics and Mathematical Sciences 28 (2001), 63–82, math.LO/9906024.[RS05] , Reasonably complete forcing notions, Quaderni di Matematica 17 (2005),

math.LO/0508272.[RS07] , Sheva-Sheva-Sheva: Large Creatures, Israel Journal of Mathematics 159 (2007),

109–174, math.LO/0210205.[She82] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-

Verlag, Berlin-New York, xxix+496 pp, 1982.[She98] , Proper and improper forcing, Perspectives in Mathematical Logic, Springer,

1998.[She00] , The Generalized Continuum Hypothesis revisited, Israel Journal of Mathematics

116 (2000), 285–321, math.LO/9809200.[She03a] , Not collapsing cardinals ≤ κ in (< κ)–support iterations, Israel Journal of

Mathematics 136 (2003), 29–115, math.LO/9707225.[She03b] , Successor of singulars: combinatorics and not collapsing cardinals ≤ κ

in (< κ)-support iterations, Israel Journal of Mathematics 134 (2003), 127–155,math.LO/9808140.

[SS02] Saharon Shelah and Zoran Spasojevic, Cardinal invariants bκ and tκ, Publications deL’Institute Mathematique - Beograd, Nouvelle Serie 72 (2002), 1–9, math.LO/0003141.

[Zap97] Jindrich Zapletal, Splitting number at uncountable cardinals, Journal of Symbolic Logic62 (1997), 35–42.

Department of Mathematics, University of Nebraska at Omaha, Omaha, NE 68182-

0243, USA

E-mail address: [email protected]: http://www.unomaha.edu/logic

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The He-

brew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathemat-

ics, Rutgers University, New Brunswick, NJ 08854, USA

E-mail address: [email protected]: http://shelah.logic.at

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CONM/533www.ams.orgAMS on the Webwww.ams.org

This book consists of several survey and research papers covering a wide range of topics in active areas of set theory and set theoretic topology. Some of the articles present, for the first time in print, knowledge that has been around for several years and known intimately to only a few experts. The surveys bring the reader up to date on the latest information in several areas that have been surveyed a decade or more ago. Topics covered in the volume include combinatorial and descriptive set theory, determinacy, iterated forcing, Ramsey theory, selection principles, set-theoretic topology, and universality, among others. Graduate students and researchers in logic, especially set theory, descriptive set theory, and set-theoretic topology, will find this book to be a very valuable reference.

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