Combinatorics of Borel ideals - repozitorium.omikk.bme.hu

Combinatorics of Borel idealsPh.D. thesis

Barnabás Farkas

BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS

DEPARTMENT OF ALGEBRA, INSTITUTE OF MATHEMATICS

Supervisor: Prof. Lajos Soukup

ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS

HUNGARIAN ACADEMY OF SCIENCES

2011

Preface

In this thesis we will discuss five loosely related topics about Borel ideals on count-able sets.

First of all, we would like to motivate the reader a little bit. How can we measurethe “smallness” of a subset of the natural numbers (which will be denoted by ω)?The mathematical approach to this problem gave us the notion of ideal: A family ofsets I ⊆ P(ω) = X : X ⊆ ω is an ideal on ω if it contains the finite sets, ω /∈ I,it is ⊆-descending (A ⊆ B ∈ I ⇒ A ∈ I), and it is closed for the union operation(A, B ∈ I ⇒ A∪ B ∈ I). The following families are well-known classical examples ofideals on ω: the Fréchet ideal of finite subsets of ω (Fin), the density zero ideal:

Z=

A⊆ω : limn→∞

|A∩ n|n= 0

,

and the summable ideal:

I1/n =

A⊆ω :∑

n∈A

1

n+ 1<∞

.

The study of these ideals and their generalizations has become a central topic ofinfinite combinatorics and forcing theory in the past few years.

An ideal I on ω is analytic if I ⊆ P(ω) ' 2ω is an analytic set in the usual product(Polish space) topology of the Cantor set. I is a P-ideal if for each countable C ⊆ I

there is an A ∈ I such that C ⊆∗ A for each C ∈ C, where ⊆∗ is the almost containingrelation, i.e. A ⊆∗ B iff A\B is finite. I is tall if each infinite subset of ω contains aninfinite element of I. Fin, Z, and I1/n are analytic P-ideals, moreover Z and I1/n aretall as well, and it is easy to see that I1/n ( Z.

Why analytic ideals? Because analyticity is one the most general but still usefulnotions of definability in descriptive set theory, and in forcing constructions it is muchsimpler to handle definable objects whose definitions are absolute.

Why P-ideals? Because P-ideals can be seen as an analogue of σ-ideals (for examplethe measure zero subsets of a measure space) for countable sets.

We will frequently use Solecki’s characterization theorem: Let I be an ideal on ω.Then I is an analytic P-ideal if and only if

I= Exh(ϑ) =n

A⊆ω : limn→∞

ϑ(A\n) = 0o

for some lower semicontinuous submeasure ϑ on ω.

Let me mention some natural questions concerning tall Borel (or analytic) ideals:

• What is the smallest (infinite or sometimes uncountable) cardinality of a familyA ⊆ P(ω)\I which is ⊆-maximal respect to the property: A∩ B ∈ I for eachdistinct A, B ∈A? (This property will be called I-almost disjointness.)

• What is the smallest cardinality of a family J⊆ I such that there is no A∈ I whichalmost contains all elements of J? (This cardinal will be denoted by add∗(I).)

iii

• What is the smallest cardinality of a family C ⊆ I such that for all X ∈ [ω]ω =X ⊆ω : |X |=ω there is an A∈ C such that |X ∩A|=ω? (This cardinal will bedenoted by cov∗(I).)

• Which forcing notions destroy I (i.e. P ∃ X ∈ [ω]ω ∀ A∈ I∩ V |X ∩ A|<ω)?

• What can we say about the cofinal type of the pre-ordered set (I,⊆∗)?

• Which ideals can be permuted into I?

Outline

In Chapter 1, we give a short summary of basic notions and results of methodof forcing, descriptive set theory, absoluteness, and cardinal invariants of the contin-uum, in particular we present the Cichon’s diagram and the method of Galois-Tukeyconnections.

In Chapter 2, we summarize basic notions and notations of the theory of idealson ω. We recall the useful characterizations of meager ideals, Fσ ideals, and analyticP-ideals. We define four basic pre-orders on the family of ideals and discuss someimportant results related to the Katetov-order, such as characterization of forcing in-destructibility of ideals. Finally, we give some examples of nice ideals on ω.

In Chapter 3, we investigate I-almost-disjoint families and the generalized almostdisjointness numbers a(I) and a(I). We prove some inequalities between these andclassical cardinal invariants, and discuss the method of proving inequalities of theform a(I) ≤ a. We investigate the existence of forcing-indestructible I-MAD familiesunder CH. At last, answering a question of L. Soukup, we prove a result about certainforcing-indestructible extensions of I-AD families in generic extensions.

In Chapter 4, we investigate general bounding and dominating properties of forcingnotions. We show some examples of Borel Galois-Tukey connections between relationsassociated to analytic P-ideals and classical ones. We show their consequences for in-equalities between cardinal invariants and for implications between certain propertiesof forcing notions. At last, we show that the natural generalization of the unboundingand dominating numbers for meager ideals do not give new invariants.

In Chapter 5, we investigate possible generalizations and applications of a measure-theoretic covering theorem due to M. Elekes. We present some counterexamples, andgive positive answer in the category case for a big class of ideals containing tall analyticP-ideals. Furthermore, we discuss the connection between covering properties andforcing indestructibility of ideals.

In Chapter 6, first we present Hechler’s famous theorem about cofinal subsets of(ωω,≤∗) and its known generalizations for the meager and the null ideal. Then weprove Hechler’s theorem for tall analytic P-ideals. At last, we show an application ofthese theorems noticed by L. Soukup.

In Chapter 7, we introduce a new way of associating cardinal invariants to ideals:we introduce the v-intersection number of an ideal where v is a pre-orders on ideals.

iv

These new invariants are natural generalizations of the pseudo-intersection number p.We give some analytic motivations for the considered problems connected to sequen-tial properties of spaces of measures with weak∗ topology. We show the consistency ofsome inequalities of these new invariants and classical ones, and other related com-binatorial questions. Furthermore, we discuss ideals generated by MAD families andtowers, and we analyze related maximality properties of these families.

Acknowledgement

First of all, I am very grateful to my supervisor Lajos Soukup for his support andkind help to find my way in set theory, and for his patience when I obstinately insistedon my favourite topic and I did not work on other ones.

Besides, I would like to thank my colleagues at the department Lajos Rónyai,György Serény, András Simon, and Ferenc Wettl their support and advices.

I also thank my mother, my brother, and my girlfriend their support and theirpatience when I was quite uptight during the work on this thesis.

Budapest, November 2011 Barnabás Farkas

v

vi

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Preliminaries 11.1 Method of forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Descriptive set theory and absoluteness . . . . . . . . . . . . . . . . . . . . 31.3 Cardinal invariants of the continuum . . . . . . . . . . . . . . . . . . . . . . 4

2 Ideals on countable sets 92.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Ideals associated to submeasures . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Orders on ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Idealized MADness 173.1 The almost-disjointness numbers of ideals . . . . . . . . . . . . . . . . . . . 173.2 Forcing-indestructible I-MAD families . . . . . . . . . . . . . . . . . . . . . 223.3 Forcing-indestructible extensions of I-AD families . . . . . . . . . . . . . . 25

4 The extended Cichon’s diagram 294.1 R-bounding and R-dominating . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Star-invariants and their underlying relations . . . . . . . . . . . . . . . . . 304.3 Idealized version of b and d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Covering properties of ideals 415.1 The J-covering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Around the category case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 When the J-covering property “strongly” fails . . . . . . . . . . . . . . . . . 465.4 J-covering properties of (P(ω), I) . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Generalizations of Hechler’s theorem 516.1 Hechler’s original theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Hechler’s theorem for M and N . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Hechler’s theorem for tall analytic P-ideals . . . . . . . . . . . . . . . . . . . 546.4 An application: coding with spectrum . . . . . . . . . . . . . . . . . . . . . 57

vii

7 Generalizations of the pseudo-intersection number 597.1 Intersection numbers of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 597.2 The convex Fréchet-Urysohn property . . . . . . . . . . . . . . . . . . . . . 617.3 Consistency results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647.4 Permuting MAD families into ideals . . . . . . . . . . . . . . . . . . . . . . . 68

viii

Chapter 1

Preliminaries

1.1 Method of forcing

Forcing is the most useful and powerful method of proving relative consistency results,that are metatheorems of the form “If ZFC is consistent then so is ZFC+Φ” or sim-ply “Con(ZFC) implies Con(ZFC+Φ)” where Φ is a sentence in the language of settheory. In this case we say that Φ is (relative) consistent with ZFC. The first and well-know example of using this method was Paul Cohen’s famous theorem from 1963: Thenegation of the Continuum Hypothesis (CH) is relative consistent with ZFC1.

Theory of forcing has become a central topic of set theory in the past 50 years notonly because of its applications in proving relative consistency result but also becauseit raised new types of problems.

We refer the reader to [35] and [33] for the basic theory of forcing and iteratedforcing (see also [5] and [28]). In this section we list our notations and some proper-ties of forcing notions that will be used later in the thesis.

Forcing notions will be denoted by P, Q, F, N etc. If p, q ∈ P then p ≤ q means “pextends q” or “p is stronger than q” or “p has more information than q.” We say that pand q are compatible (p‖q) if there is an r ≤ p, q, and they are incompatible (p ⊥ q) ifthey have no common extensions.

V always stands for a countable transitive model (c.t.m. in short) of (a large enoughfinite fragment of) ZFC. If P ∈ V is a forcing notion, then the class of P-names in V willbe denoted by VP, we will write a, x , Y etc. for elements of VP, and the extension ofV by a (V,P)-generic filter G ⊆ P will be denoted by V[G].

As it is usual, for simplifying our notations in the forcing language, if P ∈ V andx ∈ V , then we will write x for its canonical P-name as well. Furthermore, it is alsousual to write VP |= . . . and P . . . instead of 1P P . . . where 1P is the largest elementof P.

We recall the definition of nice names. If P is a forcing notion and S is an arbitraryset, then a nice P-name for a subset of S is a P-name X of the form X =

⋃

s × As :s ∈ S where As ⊆ P is an antichain for each s ∈ S. Clearly, in this case P X ⊆ S.

1Using other methods (namely the constructible universe), in 1940 Gödel proved the relative consis-tency of CH (with ZFC). This result can also be proved by forcing.

1

2 CHAPTER 1. PRELIMINARIES

Moreover, we know that if Y is a P-name, then there is a nice P-name X for a subset ofS such that P

Y ⊆ S→ Y = X

.

Let P be a forcing notion and H ⊆ P. H is n-linked if every ≤ n of its elements arecompatible. H is linked if it is 2-linked. H is centered if it is n-linked for each n.

P has the countable chain condition (ccc, in short) if all antichains A⊆ P are count-able. P is σ-n-linked (resp. σ-centered) if it is a countable union of n-linked (resp.centered) subsets.

Clearly, if P is σ-centered, then it is σ-n-linked for each n, and so it has the ccc.We know that if P has the ccc, then P does not collapse cardinals (i.e. if κ is a cardinal,then P“κ is a cardinal”).

Let K be a class of forcing notions. Martin’s Axiom for K, MA(K) is the followingstatement: If P ∈ K and D is a family of dense subsets of P with cardinality less thanc= 2ω, then there is a filter G ⊆ P such that G ∩ D 6= ; for each D ∈D.

Assume P= (P,≤P) is a subforcing ofQ= (Q,≤Q), i.e. P ⊆Q and≤P=≤Q P. Thenwe say that P is a complete subforcing of Q and write P ≤c Q if maximal antichains ofP are maximal antichains in Q as well. In this case, if G is a (V,Q)-generic filter, thenG ∩ P is a (V,P)-generic filter.

A forcing notion P is ωω-bounding if

P ∀ f ∈ωω ∩ V[G] ∃ g ∈ωω ∩ V f ≤∗ g

where G is the canonical P-name of the generic filter and f ≤∗ g iff n ∈ ω : f (n) >g(n) is finite. P adds a dominating real if

P ∃ d ∈ωω ∩ V[G] ∀ f ∈ωω ∩ V f ≤∗ d.

Let us denote Slm the set of slaloms, that are functions of the form S :ω→ [ω]<ω

with |S(n)| ≤ n for each n. A forcing notion P has the Sacks property if

P ∀ f ∈ωω ∩ V[G] ∃ S ∈ Slm∩ V ∀∞ n ∈ω f (n) ∈ S(n)

where “∀∞ n ∈ ω” stands for “for all but finitely many n ∈ ω” (and similarly, “∃∞

n ∈ ω” means “there are infinitely many n ∈ ω”). If P has the Sacks property, then Pis ωω-bounding because if f ∈ ωω ∩ V[G], S ∈ Slm ∩ V , and ∀∞ n ∈ ω f (n) ∈ S(n),then f ≤∗ g where g ∈ωω ∩ V , g(n) =max(S(n)).

The Cohen forcing. Let κ be a cardinal. The Cohen forcing Cκ if the com-plete Boolean algebra Borel(2κ)/Mκ (omitting its null element) where Borel(2κ) isthe Boolean algebra of Borel subsets of 2κ, i.e. the κth power of the discrete space2 = 0,1, and Mκ is the ideal of (Borel) meager (i.e. countable union of nowheredense sets) subsets of 2κ.

There are several forcing equivalent representations of the Cohen forcing: Cκ isforcing equivalent with Borel(2κ)\Mκ partially ordered by ⊆, with the poset of finitepartial functions from κ to 2 (or to ω) partially ordered by ⊇, and with the κ stagefinite support iteration of C= Cω as well. Cκ has the ccc, and C is σ-centered.

If V is a c.t.m. and G is a (V,C)-generic filter, then⋂

G (computed in the realworld with real Borel sets coded in V (see the next section)) contains a single real

1.2. DESCRIPTIVE SET THEORY AND ABSOLUTENESS 3

(∈ 2ω ∩ V[G]) called Cohen real over V . It is well-known that if V is a c.t.m., then areal c is Cohen over V (i.e. there is a (V,C)-generic filter G such that c =

⋂

G) iff cis not contained in any Borel meager sets coded in V .

Cohen reals in VC show that C is not ωω-bounding so it does not have the Sacksproperty. Furthermore, we know that C does not add dominating reals.

The random forcing. Let κ be a cardinal. The random forcing Bκ is the completeBoolean algebra Borel(2κ)/Nκ where Nκ is the ideal of Lebesgue null subsets of 2κ. Bκis forcing equivalnt with Borel(2κ)\Nκ. B is σ-n-linked for each n but not σ-centered.

If V is a c.t.m. and G is a (V,B)-generic filter, then⋂

G contains a single real calledrandom real over V . It is well-known that if V is a c.t.m., then a real r is random overV iff r is not contained in any Borel null sets coded in V .

Bκ is ωω-bounding but it does not have the Sacks property (witnessed by the ran-dom real). Note that Bκ is not forcing equivalent with the finite support iteration ofB = Bω because (for example) Bκ does not add Cohen reals (and conversely, Cκ doesnot add random reals) but the limit of any nontrivial finite support iterations does.

The Hechler forcing. The Hechler forcing D contains all pairs of the form p =(sp, f p) where sp ∈ ω<ω, f p ∈ ωω are strictly increasing, p ≤ q iff sp ⊇ sq, f p ≥ f q,and ∀ n ∈ |sp|\|sq| sp(n)≥ f q(n). D is σ-centered.

If V is a c.t.m. and G is a (V,D)-generic filter, then d =⋃

sp : p ∈ G ∈ωω∩V[G]is called a Hechler real over V . It is easy to see that a Hechler real over V is a dominatingreal over V , i.e. it shows that D adds dominating reals. Furthermore, D adds Cohenreals but does not add random reals.

Let us denote Dκ the κ stage finite support iteration of D.

The localization forcing. Let T =⋃

n∈ω∏

k<n[ω]≤k be the tree of initial slaloms.

p = (sp, F p) ∈ LOC iff sp ∈ T, F p ⊆ ωω, and |F p| ≤ |sp|. p ≤ q iff sp ⊇ sq, F p ⊇ Fq,and ∀ n ∈ |sp|\|sq| ∀ f ∈ Fq f (n) ∈ sp(n). LOC is σ-n-linked for each n but it is notσ-centered.

If V is a c.t.m. and G is a (V,LOC)-generic filter, then S =⋃

sp : p ∈ G ∈Slm ∩ V[G] is a slalom over V , that is ∀ f ∈ ωω ∩ V ∀∞ n ∈ ω f (n) ∈ S(n). Inparticular, d(n) = max(S(n)) is a dominating real over V . Furthermore, LOC addsboth Cohen and random reals.

1.2 Descriptive set theory and absoluteness

We refer the reader to [33], [42], and [34] for basics of descriptive set theory andrelated absoluteness results. In this section, we give a short list of some classicalresults important to us.

Analytic subsets of ωω (i.e. elements of Σ∼11(ω

ω)) can be coded by countable ob-jects, namely trees (or more generally, relations) onω×ω. The analytic set constructedfrom the tree T will be denoted by p[T]. Using this fact, we can talk about the (light-face) Σ1

1(a) sets where a ∈ ωω is a parameter: an analytic set A ∈ Σ∼11(ω

ω) is Σ11(a) if

A = p[T] where T is recursive in a, and A is Σ11 if T is recursive. Similarly, we have

Π11(a) and Π1

1 sets. Note that if k ∈ ω then the same procedure works for (ωω)k aswell.


From now on, transitive model means a transitive model of (a large enough finitefragment of) ZFC. The following theorem is usually called as “analytic absoluteness”.

Theorem 1.2.1. (Mostowski’s Absoluteness Theorem) Σ∼11(ω

ω) properties are absolutefor transitive models. That is, if M is a transitive model and T ∈ M is recursive ina ∈ωω ∩M, then the formula (with parameter a) x ∈ p[T] is absolute for M.

On the next level of the projective hierarchy we find the Σ∼12(ω

ω) sets. As in the case

of analytic sets, we can talk about Σ12(a) sets where a ∈ ωω. These sets can be coded

by trees on ω×ω1 and if A is Σ12(a) then there is a tree T ∈ L[a] such that A= p[T]

(where L[a] is the class of sets constructible form a).

Theorem 1.2.2. (Shoenfield’s Absoluteness Theorem) If M ⊆ N are transitive models,ωN

1 ⊆ M, and a ∈ M, then Σ12(a) properties are absolute between M and N.

The following theorem helps us to “identify” Borel sets in transitive models withBorel sets from the real world. If U ⊆ X × Y is a relation, then let Ux = y ∈ Y :(x , y) ∈ U.

Theorem 1.2.3. (Borel-codes) There are a Π11 set BC ⊆ ωω (the set of Borel-codes), a

Σ11 set U ⊆ ωω ×ωω, and a Π1

1 set V ⊆ ωω ×ωω (the coding “functions”) such thatUx = Vx for each x ∈ BC and Borel(ωω) = Ux : x ∈ BC.

If M is a transitive model and (B ⊆ ωω is Borel)M , then there is an x ∈ (BC)M =BC ∩ M (by analytic absoluteness) such that (B = Ux)M so B = (U M )x = (U ∩ M)x =Ux ∩M . And conversely, if x ∈ M then (Ux ∩M is a Borel set)M .

As an immediate consequence of this theorem, Ux ⊆ Uy , Ux = Uy , Ux = ;, Ux =Uy ∪ Uz , Ux = Uy ∩ Uz , Ux = ωω\Uy , and Ux =

⋃

Uyn: n ∈ ω are Π1

1 propertiesof Borel-codes, that is, for instance, (x , y) ∈ ωω ×ωω : x , y ∈ BC and Ux ⊆ Uy ⊆ωω ×ωω is Π1

1. In particular, they are absolute for transitive models. Besides, “Uxis meager” and “λ(Ux) = r” are also absolute for transitive models (where λ is theLebesgue measure on ωω and r ∈ R).

1.3 Cardinal invariants of the continuum

Loosely speaking, a cardinal invariant is a first order definable (almost always) un-countable cardinal less or equal than the continuum c = 2ω. First of all, we presenta (definitely not complete) list of classical cardinal coefficients of the continuum. Formore details and for the proofs see [7].

If f , g ∈ ωω write f ≤∗ g if ∀∞ n ∈ ω f (n) ≤ g(n), i.e. the set n ∈ ω : f (n) >g(n) is finite. This is a pre-order on ωω. The unbounding and dominating numbers of(ωω,≤∗), denoted by b and d are defined in the natural way: b is the minimal size ofa ≤∗-unbounded family, i.e.

b=min

|U | : U ⊆ωω and ∀ f ∈ωω ∃ g ∈ U g ∗ f

,

and d is the minimal size of a ≤∗-dominating family, i.e.

d=min

|D| : D ⊆ωω and ∀ f ∈ωω ∃ g ∈ D f ≤∗ g

.

1.3. CARDINAL INVARIANTS OF THE CONTINUUM 5

A family A⊆ [ω]ω is an almost-disjoint (AD) family if |A∩ B|<ω for each distinctA, B ∈ A. An infinite(!) AD family A is maximal (MAD) if ∀ X ∈ [ω]ω ∃ A ∈ A

|X ∩ A| = ω, i.e. A is ⊆-maximal among the AD families. The almost-disjointnessnumber a is the smallest cardinality of a MAD family.

A family X ⊆ [ω]ω has the strong finite intersection property (SFIP, in short) if|⋂

X′| = ω for each finite (and nonempty) X′ ⊆ X. A set Y ∈ [ω]ω is a pseudo-intersection of X if Y ⊆∗ X for each X ∈ X where A⊆∗ B iff A\B is finite. The pseudo-intersection number p is the smallest cardinality of a family with the SFIP but withouta pseudo-intersection.

A sequence T = (Tα)α<γ in [ω]ω is a tower if γ is regular, it is ⊆∗-descending (i.e.Tβ ⊆∗ Tα if α < β < γ), and it has no pseudo-intersection. The tower number t is thesmallest cardinality of a tower.

A set D ⊆ [ω]ω is open if it is ⊆∗-downward closed. D is dense if ∀ X ∈ [ω]ω

∃ D ∈ D D ⊆ X . The distributivity number h is the smallest cardinality of a family ofdense and open sets with empty intersection.

We know that

ω< cf(p) = p≤ cf(t) = t≤ cf(h) = h≤ cf(b) = b≤ cf(d)≤ d≤ c

and b ≤ a hold in ZFC. In the most cases, there are no more connections betweenthese invariants because for instance, if V |= GCH then VMω2 |= t < h (where Mω2

is the ω2 stage countable support iteration of the Mathias-forcing M), VDω2 |= h < b,VCℵω2 |= b < cf(d) < d, VCω2 |= a < d; furthermore, d < a is also consistent with ZFC(see [10]). One of the most mysterious open problems of this topic is the equality of pand t.

Next, we recall the definition of ideals, their main properties, and their cardinalinvariants. Let X be a nonempty set. An I ⊆ P(X ) is an ideal on X if ; ∈ I , I is⊆-downward closed, and I is closed under finite unions. Let I be an ideal on X .

• I is principal if there is a Y ⊆ X such that I = P(Y ).

• I is proper if X /∈ I .

From now on, if we say that I is an ideal on X , then we always assume that I is properand [X ]<ω ⊆ I , in particular I is non-principal. We will use only one exception: theideal ; on any X .

• I is a prime ideal if for every A ⊆ X , either A ∈ I or X\A ∈ I ; equivalently, I is⊆-maximal among ideals on X .

• I is a σ-ideal if⋃

J ∈ I for each countable J ⊆ I .

Write I+ = P(X )\I for the family of I -positive sets and I∗ = X\A : A ∈ I for the dualfilter of I . Filters are the dual objects of ideals, that is, F ⊆ P(X ) is a filter if X ∈ F , F is⊆-upward closed, and F is closed for finite intersections. A filter is principal / proper /an ultrafilter if its dual ideal, F∗ = X\H : H ∈ F is principal / proper / a prime ideal.


The following four cardinals are the additivity, cofinality, uniformity, and coveringnumbers of an ideal I on X :

add(I) =min

|J | : J ⊆ I , ∪J /∈ I

cof(I) =min

|J | : J ⊆ I , ∀A∈ I ∃B ∈ J A⊆ B

non(I) =min

|Y | : Y ⊆ X , Y /∈ I

cov(I) =min

|J | : J ⊆ I , ∪J = X

It is easy to see that the following inequalities hold for each ideal I where an arrowfrom a to b is to read as a ≤ b.

cov(I)

add(I)

-

cof(I)

-

non(I)

-

-

The two most important examples of σ-ideals are the ideal M of meager subsets ofthe reals (or ωω or 2ω or in general, an uncountable Polish space), that is the ideal ofsets which can be covered by countable many nowhere dense sets; and the ideal N of(Lebesgue) measure zero subsets of the reals (or ωω or 2ω).

The following diagram called Cichon’s diagram shows the very deep connectionbetween M, N, and the unbounding and dominating numbers.

cov(N) - non(M) - cof(M) - cof(N)

b

6

- d

6

add(N)

6

- add(M)

6

- cov(M)

6

- non(N)

6

Of course, ω1 ≤ add(N) and cof(N) ≤ c. Furthermore, add(M) = minb,cov Mand cof(M) = maxnon(M),d. In this level, there are no more connections betweenthese invariants: If we cut the diagram into two parts, that is, we partition the invari-ants from it into to parts L (as “left”) and R (as “right”), such that

(1) there are no b ∈ L and a ∈ R with a ≤ b according to the diagram;

(2a) if add(M) ∈ L then b,cov(M) ∩ L 6= ;;

(2b) if cof(M) ∈ R then non(M),d ∩ R 6= ;;

then it is consistent with ZFC that all a ∈ L are equal to ω1 and all b ∈ R are equal toω2 (see [3]).

1.3. CARDINAL INVARIANTS OF THE CONTINUUM 7

There is an uniform way of defining cardinal invariants and proving inequalitiesbetween them. From now on, we will use the notions of [45] instead of [7]. A sup-ported relation is a triple R = (A, R, B) where R ⊆ A× B, dom(R) = A, ran(R) = B,and we always assume that for each b ∈ B there is an a ∈ A such that ⟨a, b⟩ /∈ R. LetR−1 = ⟨b, a⟩ : ⟨a, b⟩ ∈ R, and if R ⊆ A× B and X ⊆ A, then let R[X ] = b ∈ B : ∃a ∈ X ⟨a, b⟩ ∈ R.

The unbounding and dominating numbers of R are the following cardinals:

b(R) =min

|A′| : A′ ⊆ A and ∀ b ∈ B A′ * R−1[b]

d(R) =min

|B′| : B′ ⊆ B and A= R−1[B′]

Note that b(R) and d(R) are defined for each R, but in general b(R) ≤ d(R) doesnot hold. Of course, if R is a “pre-ordered set”, that is R = (Q,≤,Q) where (Q,≤)is a pre-ordered set in the usual sense, then b(R) ≤ d(R). Furthermore, if R is apre-ordered set, then it is easy to see that b(R) is regular.

Let’s see some examples: b= b(ωω,≤∗,ωω) and d= d(ωω,≤∗,ωω); if I is an idealon X , then add(I) = b(I ,⊆, I), cof(I) = d(I ,⊆, I), non(I) = b(X ,∈, I), and cov(I) =d(X ,∈, I).

We recall the definition of Galois-Tukey connections between supported relations.Let R1 = (A1, R1, B1) and R2 = (A2, R2, B2) be supported relations. A pair of functionsφ : A1 → A2, ψ : B2 → B1 is a Galois-Tukey (GT, in short) connection from R1 to R2, innotation (φ,ψ) : R1 GT R2 if a1R1ψ(b2) whenever φ(a1)R2 b2. In a diagram:

ψ(b2) ∈ B1ψ←− B2 3 b2

R1 ⇐= R2

a1 ∈ A1φ−→ A2 3 φ(a1)

We write R1 GT R2 if there is a GT-connection from R1 to R2. If R1 GT R2and R2 GT R1 also hold then we say R1 and R2 are Galois-Tukey (GT) equivalent, innotation R1 ≡GT R2. It is trivial from the definition that R1 GT R2 implies b(R1) ≥b(R2) and d(R1)≤ d(R2).

Notice that (A1, R1, B1) GT (A2, R2, B2) is equivalent to the existence of a functionφ : A1 → A2 with that φ[A′] is R2-unbounded (i.e. there is no b2 ∈ B2 such thatφ[A′] ⊆ R−1

2 [b2]) whenever A′ is R1-unbounded. If R1 and R2 are pre-ordered sets,then such a map is called Tukey map from R1 to R2 and we say that R1 is Tukey-reducibleto R2.

If R = (A, R, B) is a supported relation then let R⊥ = (B,¬R−1, A) its dual relationwhere ⟨b, a⟩ ∈ ¬R−1 iff ⟨a, b⟩ /∈ R. Clearly, (R⊥)⊥ = R and b(R) = d(R⊥). Further-more, it is easy to see that (φ,ψ) : R1 GT R2 iff (ψ,φ) : R⊥2 GT R

⊥1 .

In the most cases, if we prove an inequality between bounding / dominating likecardinal invariants, then actually we prove that there is a GT-connection between theunderlying relations. For instance, Cichon’s diagram comes from the following GT-


connections:

(2ω,∈,N) - (2ω,∈,M)⊥ - (M,⊆,M) - (N,⊆,N)

(ωω,≤∗,ωω)⊥

6

- (ωω,≤∗,ωω)

6

(N,⊆,N)⊥

6

- (M,⊆,M)⊥

6

- (2ω,∈,M)

6

- (2ω,∈,N)⊥

6

We will also use the following relations: (`1+,≤(∗),`1

+) and (ωω,v∗, Slm). `1+ is the

set of positive summable sequences and ≤(∗) is the (almost) everywhere dominatingrelation. Slm is the set of slaloms, that is Slm =

∏

n∈ω[ω]≤n so S ∈ Slm iff S : ω →

[ω]ω and |S(n)| ≤ n for each n, and f v∗ S iff ∀∞ n ∈ ω f (n) ∈ S(n), i.e. the setn ∈ω : f (n) /∈ S(n) is finite. The following theorem will be very useful when we willwork with (N,⊆,N).

Theorem 1.3.1. ([26, Corollary 524H])

(`1+,≤∗,`1

+)≡GT (ωω,v∗, Slm)≡GT (N,⊆,N).

Chapter 2

Ideals on countable sets

2.1 Basic properties

In this section, we list certain properties of ideals on ω and related notions that will beneeded later. Let us denote Fin the Fréchet ideal on ω, i.e. Fin= [ω]<ω.

If A⊆ P(ω) then let

⟨A⟩id =n

S ⊆ω : ∃A′ ∈ [A]<ω S ⊆∗⋃

A′o

.

If ω /∈ ⟨A⟩id then ⟨A⟩id is the ideal generated by A.The character of an ideal I, denoted by χ(I) is the minimal cardinality of a family

generating I, and the character of a filter is the character of the dual ideal. In particular,χ(Fin) = 0 but if χ(I)≥ω, then χ(I) = cof(I).

If I is an ideal on ω (or on any set) and X ∈ I+, then let I X be the restriction of Ito X , that is I X = A∈ I : A⊆ X = X ∩ B : B ∈ I.

If I is an ideal on ω (or on a countable set), then

• I is analytic (resp. Borel, Σ∼0α, meager, a null set etc.) if I ⊆ P(ω) ' 2ω is analytic

(Borel, Σ∼0α, meager, a null set etc.) in the usual product topology (and measure)

of the Cantor set;

• I is a P-ideal if for each countable C ⊆ I there is an A ∈ I such that C ⊆∗ A foreach C ∈ C, i.e. the pre-ordered set (I,⊆∗) is σ-directed;

• I is tall if each infinite subset of ω contains an infinite element of I, i.e. I∗ doesnot have a pseudo-intersection.

One can easily see that an ideal cannot be closed, open, or (using the Baire Cate-gory Theorem) Gδ so the minimal Borel-complexity of an ideal is Fσ, and for exampleI1/n is Fσ.

Informally, the next theorem says that ideals (and filters because of the measurepreserving homeomorphism P(ω)→ P(ω), X 7→ω\X ) are “small” if we can “measure”them.

Theorem 2.1.1. (Sierpinski, see [3, Theorem 4.1.1]) Let I be an ideal on ω.

9

10 CHAPTER 2. IDEALS ON COUNTABLE SETS

(1) If I has the Baire property (i.e. there is an open set U ⊆ P(ω)' 2ω such that U4Iis meager), then I is meager.

(2) If I is measurable, then it is a null set.

It is well-know that analytic sets are Lebesgue measurable and have the Baire prop-erty (see e.g. [33, Theorem 11.18]) so we obtain the following:

Corollary 2.1.2. Analytic ideals and filters are meager null sets.

There is a nice characterization of meager filters:

Theorem 2.1.3. ([44], in english [3, Theorem 4.1.2]) Let F be a filter on ω. Then thefollowing are equivalent:

(i) F is meager.

(ii) The set of increasing enumerations of elements of F is ≤∗-bounded.

(iii) There is a partition Pn : n ∈ω of ω into finite sets such that

∀ F ∈ F ∀∞ n ∈ω F ∩ Pn 6= ;.

Property (iii) is also called feebleness. This property can be reformulated by thefollowing way: A filter F ⊆ P(ω) is feeble iff there is a finite-to-one function f : ω→ωsuch that f ′′[F] is the Fréchet filter Fin∗.

The following theorem is about the characters of meager ideals and filters.

Theorem 2.1.4. (R. C. Solomon and P. Simon, [7, Theorem 9.10]) If an ideal (or filter)has character less than b then it is meager but there is a non-meager ideal (resp. filter)generated by b sets.

2.2 Ideals associated to submeasures

We present a natural way of defining nice ideals on ω.

Definition 2.2.1. A function ϑ : P(ω)→ [0,∞] is a submeasure on ω if

(0) ϑ(;) = 0;

(1) if X ⊆ Y ⊆ω then ϑ(X )≤ ϑ(Y );

(2) if X , Y ⊆ω then ϑ(X ∪ Y )≤ ϑ(X ) + ϑ(Y );

(4) ϑ(n)<∞ for n ∈ω.

A submeasure ϑ is lower semicontinuous (lsc in short) if

(5) ϑ(X ) = limn→∞ ϑ(X ∩ n) for each X ⊆ω.

A submeasure ϑ is finite if ϑ(ω)<∞.

2.2. IDEALS ASSOCIATED TO SUBMEASURES 11

Notice that if ϑ is an lsc submeasure on ω then it is σ-subadditive, i.e.

ϑ

⋃

n∈ωAn

≤∑

n∈ωϑ(An)

holds for An ⊆ω.If ϑ is an lsc submeasure on ω then for X ⊆ω let

‖X‖ϑ = limn→∞

ϑ(X\n).

It is easy to see that ‖A∪ B‖ϑ ≤ ‖A‖ϑ+ ‖B‖ϑ if A, B ⊆ω but ‖ · ‖ϑ is not necessarilyσ-subadditive. We assign two ideals to a submeasure ϑ as follows

Fin(ϑ) =

X ⊆ω : ϑ(X )<∞

,

Exh(ϑ) =

X ⊆ω : ‖X‖ϑ = 0

.

It is easy to see that if Fin(ϑ) 6= P(ω), then it is an Fσ ideal; and similarly if Exh(ϑ) 6=P(ω), then it is an Fσδ P-ideal. From now on, if we are dealing with Fin(ϑ) (resp.Exh(ϑ)) then we always assume that Fin(ϑ) 6= P(ω), i.e. ϑ(ω) =∞ (resp. Exh(ϑ) 6=P(ω), i.e. ‖ω‖ϑ > 0).

It is straightforward to see that if ϑ is an lsc submeasure on ω then

Exh(ϑ) is tall iff limn→∞ ϑ(n) = 0.

Furthermore, we can assume the following without changing Exh(ϑ):

(i) ϑ(k)> 0 for each k ∈ω;

(ii) ϑ is finite so ‖ω‖ϑ <∞ too.

Indeed, let ϑ1(A) = ϑ(A)+∑

k∈A 2−k, then ϑ1 is also an lsc submeasure onω, ϑ1(k)>0 for each k ∈ ω, and Exh(ϑ1) = Exh(ϑ). If ϑ2(A) = maxϑ1(A), 1 then ϑ2 is an lscsubmeasure, ϑ2(ω)≤ 1 so it satisfies (i) and (ii), and Exh(ϑ2) = Exh(ϑ1) = Exh(ϑ).

The following characterization theorem gives us the most useful method to workon combinatorics of Fσ ideals and analytic P-ideals.

Theorem 2.2.2. ([37] and [40]) Let I be an ideal on ω.

• I is an Fσ ideal iff I= Fin(ϑ) for some lsc submeasure ϑ.

• I is an analytic P-ideal iff I= Exh(ϑ) for some (finite) lsc submeasure ϑ.

• I is an Fσ P-ideal iff I= Fin(ϑ) = Exh(ϑ) for some lsc submeasure ϑ.

Therefore each analytic P-ideal is Fσδ, so a Borel subset of 2ω.Furthermore, Solecki proved (see [40]) that an ideal I on ω is an analytic P-ideal

(or I = P(ω)) iff it is Polishable, that is, there is Polish group topology on I withrespect to the symmetric difference as group operation such that the Borel structure ofthis topology coincides with the Borel structure inherited from 2ω.


Using the previous theorem, we do not need analytic absoluteness to see that if I isan analytic P-ideal coded in the transitive model M , then IM = I∩M . Lsc submeasuresare determined by their restrictions to [ω]<ω and there is an lsc submeasure ϑ suchthat ϑ [ω]<ω ∈ M and M |= I = Exh(ϑ PM (ω)). Furthermore, it is easy to seethat A ∈ Exh(ϑ) is absolute for transitive models. In particular, we can talk about the“same” ideal in forcing extensions of V by using ϑ.

2.3 Orders on ideals

Recall some classical pre-orders on the family of ideals on ω:

Katetov-order: I≤KB J iff ∃ f :ωfin-to-1−−→ω

A∈ I=⇒ f −1[A] ∈ J

.

Katetov-Blass-order: I≤KB J iff ∃ f :ωfin-to-1−−→ω

A∈ I=⇒ f −1[A] ∈ J

.

Rudin-Keisler-order: I≤RK J iff ∃ f :ωfin-to-1−−→ω

A∈ I⇐⇒ f −1[A] ∈ J

.

Rudin-Blass-order: I≤RB J iff ∃ f :ωfin-to-1−−→ω

A∈ I⇐⇒ f −1[A] ∈ J

.

We have the following trivial implications between these pre-orders:

I≤RK J I≤RB J

I≤K J? I≤KB J

?

Of course, we can use these pre-orders for filters as well, e.g. F ≤K G iff F∗ ≤K G∗.

We refer the reader to [19], [31], and [38] for algebraic and combinatorial aspectsof these pre-orders. In [38] using these pre-orders and some classical ideals (see e.g.the next section), some really nice characterization theorems were proved.

It is easy to see that a filter F is feeble (i.e. meager) iff Fin ≤RB F∗. According to

Corollary 2.1.2 and Theorem 2.1.3, analytic filters are feeble. This fact has an easyproof if we assume additionally that our ideal is a P-ideal:

Proposition 2.3.1. Fin≤RB I for any analytic P-ideal I.

Proof. Let I = Exh(ϑ) for some finite lsc submeasure ϑ on ω. Since ω /∈ I we have‖ω‖ϑ = limn→∞ ϑ(ω\n) = ε > 0. Hence by the lsc property of ϑ for each n > 0 thereis m> n such that ϑ([n, m))> ε/2.

So there is a partition Pn : n < ω of ω into finite pieces such that ϑ(Pn) > ε/2for each n ∈ω. Define the function f :ω→ω by the stipulation f [Pn] = n. Then fwitnesses Fin≤RB I.

It is easy to see that if BI is the set of Borel-codes of Borel ideals, then BI is Π11.

Furthermore, if ≤ is one of the pre-orders defined above, then I ≤ J is a Σ12 property,

that is, the set (x , y) ∈ BI : Ux ≤ Ux is Σ12 (where U is the coding “function”, see

Theorem 1.2.3). In particular, using Shoenfield’s Absoluteness Theorem, this propertyis absolute between any pair of transitive models M ⊆ N with ωN

1 ⊆ M .

We prove some easy but useful facts about the Katetov-order.

2.3. ORDERS ON IDEALS 13

Proposition 2.3.2. ([38, Proposition 1.7.2] and A. Blass (private communication))The Katetov-order is c+-downward and c+-upward directed, that is, every family of idealson ω with cardinality at most c has both a ≤K-lower bound and a ≤K-upper bound.

Proof. Assume Iα : α < c is a family of ideals on ω. We will show that it has a ≤K-lower bound I and a ≤K-upper bound J. Let Aα : α < c ⊆ [ω]ω be an almost disjointfamily on ω (i.e. |Aα∩Aβ |<ω for each α < β < c) and fix bijections eα : ω→ Aα. Theideal I generated by

⋃

e′′α[Iα]: α < c is then a Katetov-lower bound of Iα : α < cbecause eα witnesses I≤K Iα for each α.

Now, let fα : α < c ⊆ ωω be an independent family of functions, that is, forevery α0, . . . ,αk−1 < c and n0, . . . , nk−1 ∈ ω there is some x ∈ ω such that fαi

(x) = nifor all i < k (see [18, Theorem 3] for the proof of the existence of such a family).Let J be the ideal generated by the family

⋃

( f −1α )

′′[Iα]: α < c. Notice that nofinite union of elements from the family covers ω, so ω /∈ J. Indeed, if Ai ∈ Iαi

andni ∈ω \Ai for i < k ∈ω, then there is an x ∈ω such that fαi

(x) = ni for each i < k sox /∈⋃

f −1αi[Ai]: i < k. Clearly, Iα ≤K J is witnessed by fα for each α.

Proposition 2.3.3. Meager filters are cofinal in the Katetov-order.

Proof. Let F be an arbitrary filter on ω. Fix a partition (Pn) of ω into finite sets suchthat |Pn| = n, Pn = pn

k : k < n. Let G be the filter generated by the sets eF = pnk : k ∈

F ∩ n for F ∈ F. Clearly, eF ∩ Pn 6= ; for each n > min(F) so G is meager by Theorem2.1.3. Now, the function g : ω→ω defined by g(pn

k) = k witnesses that F ≤K G.

At last in this section, we recall an important characterization theorem due to M.Hrušák and J. Zapletal.

Assume J is a tall ideal on ω and P is a forcing notion. We say that J is P-indestructible if P ∃ A ∈ J |X ∩ A| = ℵ0 for each P-name X for an infinite subsetofω, i.e. in VP the ideal generated by J is tall. If J is not P-indestructible, then we alsosay: P destroys J. Forcing-indestructibility of ideals has been widely studied for years.

The Gδ-closure of a set A⊆ 2<ω (or ω<ω) is

[A] =

f ∈ 2ω (orωω) : ∃∞ n f n ∈ A

.

The trace ideal of a σ-ideal I on 2ω (or on ωω) is

tr(I) =

A⊆ 2<ω (orω<ω) : [A] ∈ I

.

If I is a σ-ideal on a Polish space X , then PI denotes the forcing notion Borel(X )\ Ipartially ordered by the reverse inclusion. Properties of forcing notions of the form PIis a central topic of the theory of forcing methods. For more details and notions, forinstance the property continuous readings of names (CRN), see [46] or [32].

The following theorem shows the crucial role of the Katetov-order in forcing inde-structibility of ideals:

Theorem 2.3.4. (see [32]) Let I be a σ-ideal on 2ω or on ωω, and let J be a tall idealon ω. Assume furthermore that PI is proper and has the CRN. Then J is PI -indestructibleif, and only if JK tr(I) X for any X ∈ tr(I)+.


In some classical cases we know a little bit more: we do not need to investigateJK tr(I) X for any X ∈ tr(I)+ only JK tr(I) because of homogeneous properties oftr(I) (for more details see [38, Section 2.1.1.]).

For example:

• J is Cohen-indestructible iff JK tr(M).

• J is random-indestructible iff JK tr(N).

• J is Sacks-indestructible iff JK tr([2ω]≤ω).

For other types of general characterization theorems see [14].

2.4 Examples

We define some important (classes of) Borel ideals.

Summable ideals. Let h : ω → [0,∞) be a function such that∑

n∈ω h(n) = ∞.The summable ideal associated to h is

Ih =

A⊆ω :∑

n∈A

h(n)<∞

.

It is easy to see that a summable ideal Ih is tall iff limn→∞ h(n) = 0, and thatsummable ideals are Fσ P-ideals. The function ϑh : ω→ [0,∞], ϑh(A) =

∑

n∈A h(n) isan lsc (sub)measure on ω and Ih = Fin(ϑh) = Exh(ϑh).

The classical summable ideal I1/n = Ih where h(n) = 1/(n+ 1), or h(0) = 1 andh(n) = 1/n if n> 0.

We know that there are tall Fσ P-ideals which are not summable ideals (see [19,Example 1.11.1] for the details): Consider I = Fin(ϑ) where ϑ is the following sub-measure on ω:

ϑ(A) =∑

n<ω

min

n, |A∩ [2n, 2n+1)|

n2 .

Density ideals. Let (Pn)n∈ω be a sequence of pairwise disjoint finite sets of ωand let ~µ = (µn)n∈ω be a sequences of measures, µn is concentrated on Pn such thatlim supn→∞µn(Pn)> 0. The density ideal generated by ~µ is

Z~µ =

A⊆ω : limn→∞

µn(A∩ Pn) = 0

.

A density ideal

Z~µ is tall iff limn→∞

maxi∈Pn

µn(i) = 0

and density ideals are Fσδ P-ideals. The function ϑ~µ(A) = supn∈ωµn(A∩ Pn) is an lscsubmeasure on ω and Z~µ = Exh(ϑ~µ).

The density zero ideal Z =

A ⊆ ω : limn→∞ |A∩ n|/n = 0

is a tall density idealbecause let Pn = [2n, 2n+1) and µn(A) = |A|/2n for A ⊆ Pn. It is easy to see thatI1/n ( Z.

We will need the following:

2.4. EXAMPLES 15

Lemma 2.4.1. ([19, Lemma 1.13.4]) If I is a density ideal, then there is a sequence ofmeasures ~µ = (µn)n∈ω, µn is concentrated on Pn such that µn(Pn) ≥ 1 for each n andI= Z~µ.

Erdos-Ulam ideals. These ideals are special density ideals. Assume f :ω→ (0,∞)and

∑

n∈ω f (n) =∞. Then the Erdos-Ulam ideal associated to f is

EU f =

A⊆ω : limn→∞

∑

k∈A∩n f (k)∑

k∈n f (k)= 0

.

It is easy to see that EU f is a tall Fσδ P-ideal. Z is an Erdos-Ulam ideal because Z= EU fwhere f ≡ 1.

I. Farah proved (see [19, Theorem 1.13.3]) that each Erdos-Ulam ideal EU f isequal to a density ideal Z~µ where µn’s are probability measures. We do not have to

use this theorem to represent EU f as Exh(ϑ): if ϑ f (A) = supn∈ω

∑

k∈A∩n f (k)∑

k∈n f (k)

then

EU f = Exh(ϑ f ).

Some tall Fσ ideals:

(1) Tall summable ideals (see above).

(2) The eventually different ideals:

ED=

A⊆ω×ω : ∃ m ∈ω ∀∞ n ∈ω |(A)n| ≤ m

=n

A⊆ω×ω : lim supn→∞

|(A)n|<∞o

where (A)n = k ∈ ω : (n, k) ∈ A, and EDfin = ED ∆ where ∆ = (n, m) ∈ ω×ω :m≤ n. ED and EDfin are not P-ideals.

(3) The van der Waerden ideal:

W=

A⊆ω : A does not contain arbitrary long arithmetic progressions

.

Because of van der Waerden’s well-known theorem, W is a proper ideal. W is nota P-ideal. Szemerédi’s famous theorem says that W ⊆ Z (see [43]). The strongerstatement W⊆ I1/n is a still open Erdos prize problem ($3000).

For several interesting theorems about this ideal see J. Flašková’s papers.

(4) The random graph ideal:

Ran=

homogeneous subsets of the random graph

id.

Ran is not a P-ideal. For more details about this ideal see [38].

(5) Solecki’s ideal: Let CO(2ω) be the family of clopen subsets of 2ω (clearly|CO(2ω)| = ω), and let Ω = A ∈ CO(2ω) : λ(A) = 1/2 where λ is the Lebesguemeasure on 2ω. The ideal S is generated by Ix : x ∈ 2ω where Ix = A∈ Ω : x ∈ A.

S is not a P-ideal, for more details and characterizations see [38].

Some tall Fσδ ideals:

(1) Tall density ideals (see above).


(2) The ideal of nowhere dense subsets of Q: Nwd =

A⊆ Q : int(A) = ;

whereint stands for the interior operation on subsets of the reals, and A is the closure of A inR. Nwd is not a P-ideal.

Some tall Fσδσ (non P-)ideals:

(1) The ideal Conv is generated by the infinite subsets of Q ∩ [0,1] that are con-vergent in [0, 1], in other words

Conv=

A⊆Q∩ [0, 1] : |accumulation points of A (in R)|<ω

.

(2) The Fubini product of Fin by itself:

Fin⊗ Fin=

A⊆ω×ω : ∀∞ n ∈ω |(A)n|<ω

.

Some non tall ideals:

(1) An important Fσ ideal:

Fin⊗ ;=

A⊆ω×ω : ∀∞ n ∈ω (A)n = ;

,

(2) and its Fσδ brother:

;⊗ Fin=

A⊆ω×ω : ∀ n ∈ω |(A)n|<ω

.

Fubini products of ideals. We mentioned three important examples of Fubiniproducts of ideals. In general, if I and J are ideals on ω then their Fubini product isthe following ideal on ω×ω:

I⊗ J=

A⊆ω×ω : n : (A)n /∈ J ∈ I

.

Ideals generated by AD families. Let A ⊆ [ω]ω be an infinite AD family. Then⟨A⟩id is not a P-ideal, and it is tall iff A is a MAD family. If A is a MAD family, then⟨A⟩id is non analytic (see [36]) but it is meager by the following fact, by Theorem2.1.3, and by the characterization of feebleness using ≤RB.

Fact 2.4.2. If A is an almost disjoint family, then Fin≤RB ⟨A⟩id.

Proof. We can assume that there is a partition ofω in this AD family: An : n ∈ω ⊆A.Let An = an

k : k ∈ω. Define f : ω→ω by f (aij) = i+ j. f witnesses that Fin≤RB ⟨A⟩id

because if X ∈ [ω]ω, then | f −1[X ]∩ An|=ω for each n so f −1[X ] /∈ ⟨A⟩id.

Chapter 3

Idealized MADness

3.1 The almost-disjointness numbers of ideals

In this section we introduce a natural generalization of (M)AD families and the almost-disjointness number. All results without references in this section are from [23] and[22].

Let I be an ideal on ω. A family A ⊆ I+ is I-almost-disjoint (I-AD in short), ifA∩ B ∈ I for each A, B ∈ [A]2. An infinite I-AD family A is an I-MAD family if foreach X ∈ I+ there exists an A∈A such that X ∩A∈ I+, i.e. A is ⊆-maximal among theI-AD families.

Let a(I) denote the minimum of cardinalities of I-MAD families, a= a(Fin).

Example 3.1.1. We claim that a(Fin⊗ ;) = a. a(Fin⊗ ;) ≤ a because if A is aMAD family, then A×ω : A ∈ A is a Fin ⊗ ;-MAD family. Conversely, if B is aFin⊗;-MAD family, then there is an f ∈ωω such that B ∈B : B∩∆ f /∈ Fin⊗; isinfinite where∆ f =

⋃

n∈ωn×[0, f (n)]. It is easy to see that a(Fin⊗;)>ω (or wecan use the next proposition) so in particular B ∈ B : |B ∩∆ f | = ω is a MAD familyon ∆ f .

Remark 3.1.2. It is easy to see that a(I⊗ J)≤ a(I) for all ideals I and J.

Proposition 3.1.3. a(I)>ω for any Fσ ideal I.

Proof. Let I = Fin(ϑ) for some lsc submeasure ϑ, and assume that An : n < ω ⊆ I+

is an I-AD family. For each n ∈ ω let Bn ⊆ An \ ∪Am : m < n be finite (by the lscproperty of ϑ) such that ϑ(Bn) > n, and put B =

⋃

Bn : n ∈ ω ∈ I+. Then clearlyϑ(B ∩ An)<∞ i.e. B ∩ An ∈ I for each n ∈ω.

Theorem 3.1.4. a(Z~µ) =ω for any tall density ideal Z~µ.

Proof. Write ~µ= (µn)n∈ω where µn is concentrated on Pn. Using Lemma 2.4.1, we canassume that µn(Pn)≥ 1 for each n.

Since I is tall, limn→∞maxk∈Pnϑ(k) = 0 so limn→∞ |Pn|=∞. It is easy to see that

for all but finitely many n there is a P ′n ⊆ Pn such that 1 ≤ µn(P ′n) ≤ 2. Assume that ifn≥ N then there is such a P ′n. If A=

⋃

Pn\P ′n : n≥ N ∈ Z+~µ

then it will be an elementof the desired countable Z~µ-MAD family.

17

18 CHAPTER 3. IDEALIZED MADNESS

Now for n≥ N choose a kn ∈ω and a partition P ′n,k : k < kn of P ′n such that

limn→∞

kn =∞ and ∀ n≥ N ∀ k < kn µn(P′n,k)≥

µn(P ′n)

2k+1.

Put Ak =⋃

P ′n,k : n ≥ N ∧ k < kn. We show that Ak : k ∈ω is a Z~µ-MAD family onω\A (i.e. a Z~µ (ω\A)-MAD family) so if A∈ Z+

~µthen Ak : k ∈ω ∪ A is a Z~µ-MAD

family, if not then Ak : k ∈ω is a Z~µ-MAD family.

If n ≥ N and k < kn then µn(Ak ∩ P ′n) = µn(P ′n,k) ≥µn(P ′n)2k+1 . Since for an arbitrary k

for all but finitely many n we have k < kn it follows that

limsupn→∞

µn(Ak ∩ P ′n) = limsupn→∞

µn(P′n,k)≥ lim sup

n→∞

µn(P ′n)

2k+1≥

1

2k+1> 0,

thus Ak ∈ Z+~µ

.

Assume that X ⊆ ω\A and X ∈ Z+~µ

. There is a δ > 0 such that the set Y = n ∈ω : µn(X ∩ P ′n) > δ is infinite. For a large enough k we have 2

2k+1 <δ2

so if n ∈ Y andk < kn then

µn

P ′n \⋃

P ′n,i : i ≤ k

≤µn(P ′n)

2k+1≤

2

2k+1<δ

2.

So for each large enough n ∈ Y there is in ≤ k such that µn(X ∩ P ′n,in) > δ

2(k+1) .

Then in = i for infinitely many n ∈ Y so limsupn→∞µn(X ∩ Ai) ≥δ

2(k+1) , in particular

X ∩ Ai ∈ Z+~µ

.

By the following theorem of Farah we can characterize analytic P-ideals with count-able I-MAD families.

Theorem 3.1.5. (reformulation of [19, Theorem 5.11.1]) Let I be an analytic P-ideal.Then a(I)>ω if, and only if I is Fσ.

Problem 3.1.6. Is there any reasonable characterization of Borel (or even analytic)ideals with a(I) =ω?

At this point, it is not clear how large an I-AD family can be.

Observation 3.1.7. Assume that I and J are ideals on ω, I ≤RK J witnessed by afunction f : ω→ ω. If A is an I-AD family then, ( f −1)′′[A] = f −1[A] : A ∈ A is aJ-AD family.

Using that Fin≤RB I for each meager ideal I (see Theorem 2.1.3 and the note beforeProposition 2.3.1), we obtain that there are I-MAD families of cardinality c for eachmeager ideal I.

For any meager ideal I let a(I) denote the minimum of cardinalities of uncountableI-MAD families. Clearly a(I)>ω implies a(I) = a(I).

3.1. THE ALMOST-DISJOINTNESS NUMBERS OF IDEALS 19

Example 3.1.8. a(;⊗Fin) =ω because the columns in ω×ω form a ;⊗Fin-MADfamily. We claim that a(; ⊗ Fin) = a. If A is a MAD-family, then we can consider itscopies on all columns and we obtain an uncountable ;⊗Fin-MAD family. Conversely,if B is an uncountable ;⊗ Fin-MAD family, then there is a column n×ω such thatB ∈B : |(B)n|=ω is infinite and hence it is a MAD family on this column.

After clarifying the case of countable I-MAD families at least for analytic P-ideals,it is natural to ask: what can we say about lower bounds of a(I)?

Theorem 3.1.9. b≤ a(I) provided that I is an analytic P-ideal.

Proof. I = Exh(ϑ) for some finite lsc submeasure ϑ. Let A be an uncountable I-ADfamily of cardinality smaller than b. We show that A is not maximal.

There exists an ε > 0 such that the set Aε =

A∈A : ‖A‖ϑ > ε

is uncountable. LetA′ = An : n ∈ ω ⊆ Aε be a set of pairwise distinct elements of Aε. We can assumethat these sets are pairwise disjoint. For each A ∈ A\A′ choose a function fA ∈ ωω

such that

(∗A) ϑ

(A∩ An) \ fA(n)

< 2−n for each n ∈ω.

Using the assumption |A| < b there exists a strictly increasing function f ∈ ωω

such that fA ≤∗ f for each A ∈ A\A′. For each n pick g(n) > f (n) such that ϑ

An ∩[ f (n), g(n))

> ε, and let

X =⋃

n∈ω

An ∩ [ f (n), g(n))

.

Clearly, X ∈ Z+~µ

because for each n<ω there is m such that Am ∩ [ f (m), g(m))⊆ X\nand so ϑ(X \ n)≥ ϑ

Am ∩ [ f (m), g(m))

> ε, hence ‖X‖ϑ ≥ ε.We have to show that X ∩ A ∈ Z~µ for each A ∈ A. If A = An for some n then

X ∩ An = An ∩ [ f (n), g(n)), i.e. the intersection is finite.Assume now that A ∈ A\A′. Let δ > 0. We show that if k is large enough then

ϑ((A∩ X ) \ k)< δ.There is N ∈ω such that 2−N+1 < δ and fA(n) ≤ f (n) for each n ≥ N . Let k be so

large that k contains the finite set⋃

n<N[ f (n), g(n)).Now we have

(X ∩ A)\k =⋃

n∈ω

An ∩ A∩ [ f (n), g(n))

\k

and

An ∩ A∩ [ f (n), g(n))

\k = ; if n< N so

(X ∩ A)\k =⋃

n≥N

An ∩ A∩ [ f (n), g(n))

\k ⊆

⋃

n≥N

(An ∩ A)\ f (n)

⊆⋃

n≥N

(An ∩ A)\ fA(n)

.

Thus by (∗A) we have

ϑ

(X ∩ A) \ k

≤∑

n≥N

ϑ

An ∩ A\ fA(n)

≤∑

n≥N

1

2n = 2−N+1 < δ.


Notice that if A ⊆ [ω]ω is an infinite AD family, then there is a tall ideal I suchthat A is I-MAD. So Theorem 3.1.9 does not hold for an arbitrary (tall) ideal on ω.Moreover, according to the following theorem of J. Brendle, Theorem 3.1.9 does nothold even for all tall Fσ ideals:

Theorem 3.1.10. (see [12]) Assume GCH. Then VDω2 |= a(EDfin)< b.

We present two more lower bounds for a(I) in special cases.

Proposition 3.1.11. b≤ a(Fin⊗ Fin).

Proof. Let A ⊆ (Fin⊗ Fin)+ be a Fin⊗ Fin-AD family with |A| < b. We show that A isnot maximal. Fix a countable family A0 = An : n ∈ω ⊆A. We can assume that thesesets are pairwise disjoint. For each A∈A\A0 let gA ∈ωω be the following function:

gA(n) =min

k ∈ω : ∀ `≥ k |(A∩ An)`|<ω

.

Because of our assumption on the cardinality of A, there is a strictly increasing g ∈ωω

such that gA ≤∗ g for each A ∈ A\A0. Since An ∈ (Fin⊗ Fin)+ we can assume that(An)g(n) is infinite for each n ∈ω.

Now for each A∈A\A0 let fA ∈ωω be the following function:

fA(n) =

(

0 if (A∩ An)g(n) = ; or |(A∩ An)g(n)|=ω,

max

(A∩ An)g(n)

if 0< |(A∩ An)g(n)|<ω.

As in the case of gA’s, we can choose a strictly increasing f ∈ωω such that fA ≤∗ f foreach A∈A\A0. Finally, let

B =⋃

n∈ωg(n) ×

(An)g(n)\ f (n)

∈

Fin⊗ Fin+.

It is easy to see that B ∩ A∈ Fin⊗ ; ⊆ Fin⊗ Fin for each A∈A.

Before the next proposition we recall an important characterization of add(M):

Theorem 3.1.12. ([3, Theorem 2.2.6]) add(M) > κ if, and only if for every familyDα : α < κ of dense open subsets of R and for every countable dense X ⊆ R, there existsa dense Y ⊆ X such that Y ⊆∗ Dα for each α < κ.

Proposition 3.1.13. a(Nwd) =ω and add(M)≤ a(Nwd).

Proof. [n, n+ 1)∩Q : n ∈ Z is a countable Nwd-MAD family.Assume A ⊆ Nwd+ is an uncountable Nwd-AD family with cardinality less than

add(M). There is an interval (r, q) with rational endpoints such that A′ = A ∈ A :(r, q) ⊆ A is uncountable. As usual, first we fix a countable subfamily A0 = An :n ∈ ω ⊆ A′, and we assume that the elements of A0 are pairwise disjoint. For eachA∈A\A0 and n ∈ω there is a dense open DA

n ⊆ (r, q) such that A∩ An ∩ DAn = ;.

Using the characterization of add(M), for every n we can find a Bn ⊆ An such thatBn is dense in (r, q) and Bn ⊆∗ DA

n for each A∈A\A0. These Bn’s will be the “columns.”

3.1. THE ALMOST-DISJOINTNESS NUMBERS OF IDEALS 21

Let Bn = bnk : k ∈ ω be enumerations and for each A ∈ A\A0 let fA ∈ ωω be the

following function:fA(n) =min

k : ∀ `≥ K bn` /∈ A

.

Since add(M) ≤ b, there is an f ∈ ωω such that fA ≤∗ f for each A ∈ A\A0. Fix anenumeration (rn, qn) : n ∈ ω of open subintervals of (r, q) with rational endpoints.Finally, for each n choose a kn ≥ f (n) such that bn

kn∈ (rn, qn), then bn

kn: n ∈ ω ∈

Nwd+ witnesses that A is not maximal.

In the rest of this section we present a natural method of proving inequalities be-tween almost-disjointness numbers of ideals. Assume I and J are ideals onω and thereis a regular embedding e : P(ω)/I → P(ω)/J, that is, e is an embedding in the senseof Boolean algebras and the image of any maximal antichains of P(ω)/I (i.e. I-MADfamilies) is also maximal in P(ω)/J. In other words, e is a complete embedding in thesense of forcing notions but avoiding confusions when we are working with Booleanalgebras, we will talk about regular embeddings.

In this case clearly a(I)≥ a(J) and a(I)≥ a(J). How could we find regular embed-dings? The simplest way is to find a function f :ω→ω such that

f −1[ · ] : P(ω)/I→ P(ω)/J is a regular embedding (reg)

where we do not distinguish equivalence classes and representatives. We know that anf has property (reg) iff the following hold:

(a) A∈ I iff f −1[A] ∈ J, i.e. f witnesses I≤RK J;

(b) ∀ X ∈ J+ ∃ Y ∈ I+ ∀ Z ∈ (I Y )+ f −1[Z] ∩ X ∈ J+. (This Y is called thepseudo-projection of X .)

If there exists such an f , then we write I≤rRKJ.

Proposition 3.1.14. Fin≤rRKJ holds for J= Z~µ,ED,EDfin,W, Fin⊗Fin, Fin⊗;, ;⊗

Fin. In particular, a(J)≤ a for these ideals.

Proof. J = Z~µ: Let µn be concentrated on Pn. By Lemma 2.4.1 we can assume thatµn(Pn)≥ 1 for each n. Let f :ω→ω be the function defined by f −1[0] =ω\

⋃

Pn :n≥ 1 and f −1[n] = Pn for n> 0. Clearly, f witnesses Fin≤RK Z~µ.

Let A be a (Fin-)MAD family. We show that ( f −1)′′[A] = f −1[A] : A∈A is a Z~µ-MAD family. To show the maximality, let X ∈ Z+

~µbe arbitrary, lim supn→∞µn(X ∩ Pn) =

ε > 0. Thus J = n ∈ ω : µn(X ∩ Pn) > ε/2 is infinite. So there is an A ∈ A such thatA∩ J is infinite. Then f −1[A] ∈ ( f −1)′′[A] and X ∩ f −1[A] ∈ Z+

~µbecause there are

infinitely many n such that we have Pn ⊆ f −1[A] and µn(X ∩ Pn)> ε/2.

J= ED,EDfin, Fin⊗Fin, Fin⊗;: It is easy to see that the projection onto the firstcoordinate, i.e. f (n, m) = n generates regular embeddings.

J= ;⊗ Fin: In this case, the projection onto the second coordinate works.

J = W (Flašková): It is easy to see that the function f : ω → ω defined byf −1(n) = [2n, 2n+1) generates a regular embedding.


It is easy to see that if X ∈ I+\I∗, then a(I)≤ a(I X ) and a(I)≤ a(I X ) because ifA is an I X -MAD family, then A∪ω\X is an I-MAD. In particular, a(ED)≤ a(EDfin).

We finish this section with a list of related questions:

Question 3.1.15.

• Does a(I)≤ a hold for each analytic P-ideal I?

• Is there any reasonable lower bound of a(W)?

• Do b≤ a(Nwd) or a(Nwd)≤ a hold?

• What can we say about the almost-disjointness numbers of Ran,S, and Conv?

• Is there any nontrivial upper bound for a(I) when I is an Fσ ideal?

3.2 Forcing-indestructible I-MAD families

If P is a forcing notion and A is a MAD family, then we say that A is P-indestructibleif P“A is a MAD family”, in other words, the ideal ⟨A⟩id is P-indestructible. Kunen[35, Ch.VIII, Theorem 2.3] constructed a Cohen-indestructible MAD family assumingCH. His method was later extended to other forcing notions and there were provedmany similar indestructibility results assuming typically that a cardinal invariant of thecontinuum is equal to c.

In general, if P is a classical forcing notion (such as the Cohen, the random, theSacks, the Laver, or the Miller forcing), then the existence of a P-indestructible MADfamily (in ZFC) is still an open problem.

We can generalize this notion for I-MAD families: If P is a forcing notion, I is aBorel ideal, and A is an I-MAD family, then A is P-indestructible if P“A is I-MAD.”

Observation 3.2.1. The property “I is a Borel ideal, and A is a countable I-MADfamily” is Π1

1 so it is absolute for transitive models. In particular, countable I-MADfamilies are P-indestructible for each forcing notion P.

Unfortunately, the property I≤rRKJ on Borel ideals seems to be Σ1

4. We introduce astronger version of this pre-order: we want to guarantee that the pseudo-projection ofany set X is “computable” form X .

Definition 3.2.2. Assume I and J are Borel ideals. We say that a functions f :ω→ωgenerates a Borel-regular embedding if there is a Borel-measurable F : P(ω) → P(ω)such that the following hold

(a′) f witnesses I≤RK J, and ∀ X ∈ J+ F(X ) ∈ I+;

(b′) ∀ X ∈ J+ ∀ Z ∈ (I F(X ))+ f −1[Z]∩ X ∈ J+.

If there exists such an f , then we write I≤B-rRKJ.

3.2. FORCING-INDESTRUCTIBLE I-MAD FAMILIES 23

It is not hard to see that this pre-order on Borel ideals is Σ12 so it is absolute between

any pair M ⊆ N of transitive models with ωN1 ⊆ M .

In Proposition 3.1.14 we actually proved that Fin ≤B-rRK

J holds for all J in the list.For example, Fin≤B-r

RKFin⊗Fin because let f be the projection onto the first coordinate

and let F(X ) = n ∈ ω : |(X )n| = ω for each X ⊆ ω ×ω. Similarly, Fin ≤B-rRK

ED

because let f be the projection as in the previous case and let

F(X ) =

min

n≥ k : |(X )n| ≥ k

: k ∈ω

if X ∈ ED+ and let F(X ) = ; if X ∈ ED.At this moment, I am unable to show that this pre-order is really stronger than ≤r

RK

(maybe it is not):

Question 3.2.3. Do there exist Borel ideals I and J such that I≤rRKJ but IB-r

RKJ.

Using the absoluteness of this pre-order on Borel ideals, we have a method ofconstructing P-indestructible I-MAD families form P-indestructible MAD families:

Fact 3.2.4. Assume I and J are Borel ideal, f witnesses I ≤B-rRK

J, P is a forcing notion,and A is a P-indestructible I-MAD family. Then f −1[A] : A ∈ A is a P-indestructibleJ-MAD family.

In particular, under CH, there are Cohen-indestructible J-MAD families for J =Z~µ,ED,EDfin,W, Fin⊗ Fin, Fin⊗ ;, ;⊗ Fin.

Although, I do not no whether Fin ≤B-rRK

J holds for all Fσ ideals or for all ana-lytic P-ideals, we can construct Cohen-indestructible J-MAD families for these J’s bymodifying Kunen’s proof.

Theorem 3.2.5. Assume CH and let I be an Fσ ideal or an analytic P-ideal. Then thereis an uncountable Cohen-indestructible I-MAD family.

Proof. The construction for Fσ ideals is similar to the other case but a bit simpler sowe prove the theorem for analytic P-ideals only.

We will define the uncountable Cohen-indestructible I-MAD family Aα : α <ω1 ⊆ I+ by recursion on α ∈ ω1. The family Aα : α < ω1 will be Fin-AD aswell. Our main concern is that we do have that Aξ : ξ < α is not maximal for α <ω1because it is not automatic (e.g. a(Z) =ω).

Let I = Exh(ϑ) be an analytic P-ideal. We can assume that ‖ω‖ϑ = 1. Let(pα, Xα,δα) :ω ≤ α < ω1 be an enumeration of all triples (p, X ,δ) such that p ∈ C,X is a nice C-name for a subset of ω, and δ is a positive rational number.

Partition ω into infinite sets An : n<ω such that ‖An‖ϑ = 1 for each n<ω.Assume α ≥ ω and we have Aξ ∈ I+ for ξ < α such that Aξ : ξ < α is an AD so

especially an I-AD family.

Claim. There is X ∈ I+ such that |X ∩ Aξ|<ω for each ξ < α.

Proof of the Claim. Write α = ξi : i < ω. Recursion on m ∈ ω we can choose Fm ∈[Akm]<ω for some km ∈ω such that ϑ(Fm)≥ 1/2 and Fm ∩

Aξ0∪Aξ1

∪ · · · ∪Aξm

= ;.Why? Assume that (Fi)i<m are chosen. Pick a km ∈ω\ξi : i ≤ m and let D ∈ω such


that Akm∩⋃

Aξi: i ≤ m ⊆ D. Since ϑ(Akm

\ D) ≥ 1 there is Fm ∈ [Akm\D]<ω with

ϑ(Fm)≥ 1/2.Let X =

⋃

Fm : m<ω. Then |Aξ ∩ X |<ω for ξ < α and ‖X‖ϑ ≥ 1/2.

If pα does not force (a)α and (b)α below then let Aα be X from the Claim.

(a)α ‖Xα‖ϑ > δα, (b)α ∀ ξ < α Xα ∩ Aξ ∈ I.

Assume pα (a)α∧(b)α. Let Bαk : k ∈ ω = Aξ : ξ < α and pαk : k ∈ ω = p′ ∈C : p′ ≤ pα be enumerations. Clearly, for each k ∈ω we have

pαk

Xα\⋃

Bαl : l ≤ k

ϑ> δα,

so we can choose a qαk ≤ pαk and a finite aαk ⊆ ω such that ϑ(aαk ) > δα and qαk aαk ⊆

Xα\⋃

Bαl : l ≤ k

\k. Let Aα =⋃

aαk : k ∈ ω. Clearly Aα ∈ I+ and Aξ : ξ ≤ α isan AD family.

Thus we obtain the AD family A= Aα : α <ω1 ⊆ I+.We show that A is a Cohen-indestructible I-MAD. Assume otherwise that there is

an α such that pα ‖Xα‖ϑ > δα ∧∀ ξ < ω1 Xα ∩ Aξ ∈ I, in particular pα (a)α∧(b)α.There is a pαk ≤ pα and an N such that pαk ϑ

(Xα ∩ Aα)\N

< δα. We can assumethat k ≥ N so pαk ϑ

(Xα ∩ Aα)\k

< δα. By the choice of qαk and aαk we haveqαk aαk ⊆ (Xα ∩ Aα)\k, so qαk ϑ

(Xα ∩ Aα)\k

> δα, contradiction.

We can also generalize the construction of Sacks-indestructible MAD families (see[7, page 476]).

Theorem 3.2.6. Assume CH. Let I be an Fσ ideal or an analytic P-ideal and let P bean ωω-bounding proper forcing notion of cardinality ω1. Then there is an uncountableP-indestructible I-MAD family.

Proof. As in the previous proof we will construct the desired I-MAD family Aα : α <ω1 by recursion on α such that this family will be an AD family as well. AssumeI= Exh(ϑ) is an analytic P-ideal and ‖ω‖ϑ = 1.

It is well-known that under CH, a proper forcing of size c adds at most c many newreals. Let (pα, Xα) : ω ≤ α < ω1 be an enumeration of all pairs such that p ∈ P andX is a P-name for a subset of ω from our list of essentially different names of reals.

Let An : n ∈ ω be a partition of ω such that ‖An‖ϑ = 1 for each n ∈ ω, andassume Aξ : ξ < α is done.

If pα Xα ∈ I or pα 1 ∀ ξ < α Xα ∩ Aξ ∈ I, then let Aα = X form the Claimof previous proof. Assume that pα forces Xα ∈ I+ and that Xα ∩ Aξ ∈ I for eachξ < α. Then let Bαk : k ∈ ω = Aξ : ξ < α be an enumeration, and let Cα0 = Bα0 ,Cαk = Bαk \

Bα0 ∪ Bα1 ∪ · · · ∪ Bαk−1

for k > 0. Furthermore, fix enumerations Cαk =cαk (m) : m ∈ω for each k.

Since pα ∀ k ∈ ω Xα ∩ Cαk ∈ I, there is a P-name f for an element of ωω suchthat

pα ∀ k ∈ω ϑ

Xα ∩

cαk (m) : m≥ f (k)

< 2−k−2‖Xα‖ϑ.

3.3. FORCING-INDESTRUCTIBLE EXTENSIONS OF I-AD FAMILIES 25

By the ωω-bounding property of P there is a qα ≤ pα and a g ∈ωω (in the groundmodel) such that qα f ≤ g. Let Aα = cαk (m) : k ∈ ω, m ≤ g(k). Clearly, Aξ : ξ ≤α is an AD family.

Using the σ-subadditivity of ϑ, we show that qα ϑ(Xα\Aα)< ‖Xα‖ϑ/2:

qα ϑ

Xα\Aα) = ϑ

⋃

k∈ω

Xα ∩

cαk (m) : m> g(k)

≤

≤∑

k∈ωϑ

Xα ∩

cαk (m) : m> g(k)

<∑

k∈ω2−k−2‖Xα‖ϑ = ‖Xα‖ϑ/2.

It implies that qα ‖Xα\Aα‖ϑ < ‖Xα‖ϑ/2 so by the subadditivity of ‖ · ‖ϑ we obtainthat qα ‖Xα ∩ Aα‖ϑ > ‖Xα‖ϑ/2 > 0, in particular Aα ∈ I+. We are done by an easyargument similar to the end of the previous proof.

The proof for Fσ ideals is a natural modification of this proof.

Question 3.2.7. Assume that I is as above. Can we construct an uncountable Miller-indestructible I-MAD family under CH?

3.3 Forcing-indestructible extensions of I-AD families

The motivation of this section is based on the following theorem and the related ques-tion after that.

Theorem 3.3.1. [27, Theorem 11] In VCω1 there are AD families A and B such that,in any generic extension of VCω1 by a ccc forcing notion P such that P ∈ V , A cannot beextended to a Cohen-indestructible MAD family and B cannot be extended to a random-indestructible MAD family.

L. Soukup asked if any AD family could be extended to a Cohen-indestructible MADfamily in a ccc forcing extension. Using Kunen’s idea we give a general positive answerfor this question. (In a special case see [20].)

We will need the following easy lemma:

Lemma 3.3.2. The formula which says that F is a forcing notion, p ∈ F, X is a niceF-name for a subset of ω, ϑ is an lsc submeasure on ω, ε ∈ R, and p F ϑ(X ) > ε (orp F ‖X‖ϑ > ε) is absolute for transitive models.

Proof. (Sketch:) We show that this formula can be expressed with bounded quantifiersonly, i.e. it is equivalent to a ∆0 formula in ZFC. Let X =

⋃

k × Ak : k ∈ ω whereAk ⊆ F is an antichain for each k. The set Yε = Y ∈ [ω]<ω : ϑ(Y ) > ε is definableform ϑ and ε by a ∆0 formula. We know that

p F ϑ(X )> ε

⇐⇒

p F ∃ Y ∈ Yε Y ⊆ X

.

The negation of the second formula is equivalent to ∃ q ≤ p ∀ Y ∈ Yε q F Y * X so itis enough to show that q F Y * X is equivalent to a ∆0 formula. It is easy to see that

q F Y * X

⇐⇒ ∀ r ∈ F

(∀ y ∈ Y ∃ a ∈ Ay r ≤ a)→ r ⊥ q

.

A similar argument shows that p F ‖X‖ϑ > ε is also absolute for transitive models.


Theorem 3.3.3. Assume F is a forcing notion, I is an Fσ ideal or an analytic P-ideal,and A is an infinite I-AD family. Then in a ccc forcing extension A can be extended to anF-indestructible I-MAD family.

Proof. We prove the theorem for analytic P-ideals only because for Fσ ideals the proofis very similar. Let I= Exh(ϑ) for some lsc submeasure ϑ, and let κ= |F|.

By recursion on κ+ we will define a finite support iteration of ccc forcing notions(Pα, Qβ : α ≤ κ+,β < κ+) and a sequence (Aα)α<κ+ such that Aα is a Pα+1-name,α“A∪ Aβ : β < α is an I-AD family”, and |Pκ+ | ≤ 2κ. In VPκ+ the family A∪ Aβ :β < κ+ will be an F-indestructible I-MAD family.

At stage α we will work with a condition p from F such that each p ∈ F will beworked at cofinally many stages in κ+.

Assume Pα and Aβ : β < α are done and we have a p ∈ F. From now on weare working in VPα . Let Aα = A ∪ Aβ : β < α and let Xα be the set of all F-namesX such that X is a nice F-names for a subsets of ω, there is a real ε(X ) > 0 withp F ‖X‖ϑ > ε(X ), and p F“Aα ∪ X is an I-AD family”. Let Qα be the followingforcing notion:(n, s, F,B,Y) ∈Qα iff

(1) n ∈ω and s ⊆ n;

(2) F is a finite subset of q ∈ F : q ≤F p ×ω;

(3) B is a finite subset of Aα;

(4) Y is a finite subset of Xα.

(n1, s1, F1,B1,Y1)< (n0, s0, F0,B0,Y0) iff

(a) n1 ≥ n0 and s1 ∩ n0 = s0,

(b) F1 ⊇ F0, B1 ⊇B0, and Y1 ⊇ Y0;

(c) (s1\s0)∩⋃

B0 = ;;

(d) ∀ (q, k) ∈ F0 ∀ X ∈ Y0 ∃ r ≤F q r F ϑ

(s1\k)∩ X

> ε(X ).

Notation: c = (nc , sc , F c ,Bc ,Yc) ∈Qα.

Claim. Qα is σ-centered, |Qα| ≤ 2κ, and the following sets are dense in Qα:

(i) c ∈Qα : (q, k) ∈ F c for each (q, k) ∈ q ∈ P : q ≤ p ×ω;

(ii) c ∈Qα : B ∈Bc for each B ∈Aα;

(iii) c ∈Qα : X ∈ Yc for each X ∈ Xα.

Proof. σ-centeredness: We show that conditions with the same first and second coor-dinates are compatible. Let (n, s, F0,B0,Y0), (n, s, F1,B1,Y1) ∈Qα and let

(F0×Y0)∪ (F1×Y1) =

⟨⟨q`, k`⟩, X`⟩ : ` < L

3.3. FORCING-INDESTRUCTIBLE EXTENSIONS OF I-AD FAMILIES 27

be an enumeration. For each ` < L we know that q` F X` ∩⋃

(B0 ∪ B1) ∈ I soq` F ‖X`\

⋃

(B0 ∪B1)‖ϑ > ε(X`). Therefore, we can choose an r` ≤F q` and a finiteK` ⊆ ω\maxn, k` such that ϑ(K`) > ε(X`) and r` F K` ⊆ X`\

⋃

(B0 ∪ B1). Lets′ = s ∪

⋃

K` : ` < L, n′ =max(s′) + 1, F ′ = F0 ∪ F1, B′ =B0 ∪B1, and Y′ = Y0 ∪Y1.Then (s′, n′, F ′,B′,Y′) is a common extension of our two conditions.|Qα| ≤ 2κ is trivial. (i), (ii), and (iii) are implied by the the following observation:

If (n, s, F,B,Y) ∈Qα, then this condition is compatible with (n, s, F∪(q, k),B∪B,Y∪X ) where (q, k) ∈ q ∈ P : q ≤ p×ω, B ∈Aα, and X ∈ Xα are arbitrary by the proofof σ-centeredness.

Let Qα be a Pα-name for Qα, and Aα be a Pα+1-name for the set⋃

sc : c ∈ Hαwhere Hα is a Pα+1-name for the Qα-generic filter. Then

• VPα+1 |= p F ‖Aα ∩ X‖ϑ ≥ ε(X ) holds for each X ∈ Xα by (i), (iii), (d), and byLemma 3.3.2 (in particular VPα+1 |= p F Aα ⊇ Aα ∩ X ∈ I+);

• VPα+1 |=“A∪ Aβ : β ≤ α is an I-AD family” because of the previous point, (ii),and (c).

At last we prove that Aκ+ =A∪ Aα : α < κ+ is an F-indestructible I-MAD familyin VPκ+ .

Assume on the contrary that there is a Pκ+-generic filter Gκ+ , a p ∈ F, a nice F-nameX ∈ V[Gκ+] for an infinite subset of ω, and a real ε > 0 such that

V[Gκ+] |= p F ‖X‖ϑ > ε ∧ p F “Aκ+ ∪ X is an I-AD family”.

Then there is an α < κ+ such that at the stage α we worked with p and X ∈ Xα(because |X | ≤ κ). Of course, ε(X ) is not necessarily equal to ε but it does not matter.Then in particular V[Gκ+ ∩ Pα+1] |= p F Aα ∩ X ∈ I+ so by Lemma 3.3.2 this alsoholds in V[Gκ+], a contradiction.

Corollary 3.3.4. Assume I is an Fσ ideal or an analytic P-ideal, and A is an infiniteI-AD family. Then in a ccc forcing extension A can be extended to a Cohen-indestructibleI-MAD family.

Unfortunately, our theorem does not say anything about those forcing notionswhich are not absolute for transitive models (e.g. the random forcing) so the followingquestion is still open.

Problem 3.3.5. Assume that I is as above and A is an infinite I-AD family. Does thereexist a ccc forcing extension in which A can be extended to a random-indestructible I-MAD family? What can we say about other classical forcing notions, such as the Sacksor the Miller forcing?

Remark 3.3.6. Using Theorem 2.3.4 we can prove that that any AD family can beextended to a random-indestructible MAD family in a ccc forcing extension (see [22]).


Chapter 4

The extended Cichon’s diagram

4.1 R-bounding and R-dominating

In this section we generalize some classical bounding and dominating properties offorcing notions. Then we discuss special cases related to analytic P-ideals. The resultswithout references in this section are mainly reformulated and improved versions oftheorems from [23]. We have to note that there are other known ways of proving theseresults but the ideas are the same and we prefer the language of (Borel) GT-connectionsbecause this is an uniform method for proving all theorems in this section.

A supported relation R = (A, R, B) is called Borel relation iff A, B ⊆ ωω and R ⊆ωω ×ωω are Borel sets. Similarly, a GT-connection (φ,ψ) : R1 GT R2 between Borelrelations is called Borel GT-connection iff φ and ψ are Borel functions (or equivalently,φ,ψ ⊆ ωω ×ωω are Borel sets). In this case, we write (φ,ψ) : R1 B

GTR2, and we

write R1 BGTR2 if there is a Borel GT-connection from R1 to R2.

For instance, all GT-connections mentioned in Section 1.3 are Borel. Notice that Mand N can be seen as Borel sets because for example, in all proofs it is enough to workwith Gδ elements of N (they are cofinal in (N,⊆)), and we can consider the (Borel)set of Borel-codes of Gδ null sets.

A pair of transitive models (V, W ) is an extension if V ⊆W .

Definition 4.1.1. Let R = (A, R, B) be a Borel supported relation. An extension (V, W )is

R-bounding if∀ a ∈ A∩W ∃ b ∈ B ∩ V ⟨a, b⟩ ∈ R;

R-dominating if∃ b ∈ B ∩W ∀ a ∈ A∩ V ⟨a, b⟩ ∈ R.

A forcing notion P is R-bounding (resp. R-dominating) if

P “(V, V[G]) is R-bounding (R-dominating).”

The following is trivial from the definitions.

Fact 4.1.2. Let (V, W ) be an extension and assume V |= R1 BGT

R2. Then if (V, W ) isR2-bounding / dominating then (V, W ) is R1-bounding / dominating as well.

29

30 CHAPTER 4. THE EXTENDED CICHON’S DIAGRAM

Remark 4.1.3. We should be careful when saying “P is not R-bounding” (resp. “Pis not R-dominating”) because in general, it does not imply the stronger propertyP“(V, V[G]) is not R-bounding” which is equivalent that P is R⊥-dominating. Al-though, in all cases if we prove that a forcing notion is not R-bounding it means thatthis stronger property holds.

Several properties of forcing notions can be defined as R-bounding / -dominatingwith a suitable R. For instance,

• P is ωω-bounding iff P is (ωω,≤∗,ωω)-bounding;

• P adds a dominating real iff P is (ωω,≤∗,ωω)-dominating;

• P has the Sacks property iff P is (ωω,v∗, Slm)-bounding;

• P adds a slalom over the ground model iff P is (ωω,v∗, Slm)-dominating;

• P adds a Cohen real iff P is (2ω,∈,M)⊥-dominating;

• P 2ω ∩ V ∈M iff P is (2ω,∈,M)-dominating.

4.2 Star-invariants and their underlying relations

First of all, the ∗-invariants of a tall ideal I on ω are the following cardinals:

add∗(I) =min

|J| : J⊆ I, ∀ A∈ I ∃ B ∈ J B *∗ A

;

cof∗(I) =min

|C| : C⊆ I, ∀ A∈ I ∃ C ∈ C A⊆∗ C

;

non∗(I) =min

|X| : X⊆ [ω]ω, ∀ A∈ I ∃ X ∈ X |A∩ X |<ω

;

cov∗(I) =min

|J| : J⊆ I, ∀ X ∈ [ω]ω ∃ B ∈ J |X ∩ B|=ω

.

Clearly, I is a P-ideal iff add∗(I)>ω.Using the unbounding and dominating numbers of supported relations we have

that add∗(I) = b(I,⊆∗, I), cof∗(I) = d(I,⊆∗, I), non∗(I) = b([ω]ω, R∞, I), and cov∗(I) =d([ω]ω, R∞, I) where ⟨A, B⟩ ∈ R∞ iff |A∩ B|=ω.

Example 4.2.1. (see [38])

• add∗(ED) = non∗(ED) =ω, cov∗(ED) = non(M), and cof∗(ED) = c;

• add∗(EDfin) =ω and cof∗(EDfin) = c;

• non(M) =maxcov∗(EDfin),b and cov(M) =mind,non∗(EDfin);

• add∗(Ran) = non∗(Ran) =ω and cov∗(Ran) = cof∗(Ran) = c;

• add∗(S) = non∗(S) =ω, cov∗(S) = non(N), and cof∗(S) = c;

• add∗(Nwd) = non∗(Nwd) =ω, cov∗(Nwd) = cov(M), and cof∗(Nwd) = cof(M);

• add∗(Conv) = non∗(Conv) =ω and cov∗(Conv) = cof∗(Conv) = c;

4.2. STAR-INVARIANTS AND THEIR UNDERLYING RELATIONS 31

• add∗(Fin⊗Fin) = non∗(Fin⊗Fin) =ω, cov∗(Fin⊗Fin) = b, and cof∗(Fin⊗Fin) =d.

In this section, our goal is to understand deeper these cardinal invariants and re-lated bounding / dominating properties of forcing notions if I is a tall analytic P-ideal.

If I is a tall ideal on ω, then the associated σ-ideal on [ω]ω is the following ideal:

I =

X ∈ [ω]ω : |X ∩ A|=ω : A∈ I

id.

One can easily prove that add∗(I) = add(I), cof∗(I) = cof(I), non∗(I) = non(I),and cov∗(I) = cov(I). In particular, we have:

cov∗(I)

add∗(I)

-

cof∗(I)

-

non∗(I)

-

-

Of course, we do not need I to see these inequalities between the ∗-invariants,they come from trivial GT-connections between their defining relations:

([ω]ω, R∞, I)

(I,⊆∗, I)⊥

-

(I,⊆∗, I)-

([ω]ω, R∞, I)⊥

-

-

We say that a forcing notion (or extension) is I-bounding / dominating if it is (I,⊆∗

, I)-bounding / dominating. In other words, a forcing notion P is I-bounding iff allnew elements of I in VP can be covered by an old element of I from the ground model,i.e. I∩ V is cofinal in (I∩ VP,⊆∗); and P is I-dominating iff it adds a new element of Iwhich almost contains all the old elements of I, i.e. I∩ V is bounded in (I∩ VP,⊆∗).

Forcing-indestructibility of Borel ideals can also be seen as a bounding property:

I is P-indestructible iff P is ([ω]ω, R∞, I)-bounding.

Similarly, the corresponding dominating property means the following:

P is ([ω]ω, R∞, I)-dominating iff P [ω]ω ∩ V ∈ I.

One can easily show the following fact.

Fact 4.2.2. The following implications hold for tall (in (1) and (3)) Borel (in (2) and(4)) ideals:


• If I≤RB J then (I,⊆∗, I)(B)GT(J,⊆∗,J), in particular:

(1) add∗(I)≥ add∗(J) and cof∗(I)≤ cof∗(J).

(2) If P is J-bounding / dominating, then P is I-bounding / dominating.

• If I≤KB J then ([ω]ω, R∞,J)(B)GT([ω]ω, R∞, I), in particular

(3) non∗(J)≥ non∗(I) and cov∗(J)≤ cov∗(I).

(4a) If I is P-indestructible, then J is also P-indestructible.

(4b) If P [ω]ω ∩ V ∈ I, then P [ω]ω ∩ V ∈ J.

It easy to see that I≤K J also implies cov∗(J)≤ cov∗(I).Before our first theorem, we prove the well-know fact that our bound of the size

of slaloms is not essential. If a = (an) ∈ ωω is a strictly increasing sequence, then letSlm(a) be the set of a-slaloms, that is, S ∈ Slm(a) iff S : ω→ [ω]<ω with |S(n)| ≤ anfor each n.

Lemma 4.2.3. If a = (an) ∈ωω is strictly increasing, then

(ωω,v∗, Slm(a))≡BGT(ωω,v∗, Slm).

Proof. (ωω,v∗, Slm(a))BGT(ωω,v∗, Slm) is trivial because of the identities.

Conversely, we show that (ωω,v∗, Slm) BGT(ωω,v∗, Slm(a)). Fix a bijection j :

ω → ω<ω. If f ∈ ωω then let φ( f )(n) = j−1( f an+1). If S ∈ Slm(a) and n ∈[ak, ak+1), then let

ψ(S)(n) =

j(`)(n) : ` ∈ S(k) and | j(`)|= ak+1

and if n ∈ [0, a0) then let S(n) = ;. If n ∈ [ak, ak+1) then |S(k)| ≤ ak so |ψ(S)(n)| ≤ak ≤ n for each n so ψ(S) ∈ Slm.

We show that (φ,ψ) : (ωω,v∗, Slm) BGT(ωω,v∗, Slm(a)). Assume φ( f ) v∗ S,

φ( f )(k) ∈ S(k) if k ≥ K . Then j−1( f ak+1) ∈ S(k) for each k ≥ K so if n ∈ [ak, ak+1)for some k ≥ K then f (n) = ( f ak+1)(n) ∈ψ(S)(n). In other words, f v∗ ψ(S), andwe are done.

Theorem 4.2.4. Let I be an analytic P-ideal. Then

(I,⊆∗, I)BGT(ωω,v∗, Slm).

Moreover, if Ih is a tall summbale ideal, then

(I,⊆∗, I)≡BGT(ωω,v∗, Slm).

Proof. Let I= Exh(ϑ) and fix a bijection e :ω→ [ω]<ω.If A ∈ I and n ∈ ω, then let dA ∈ ωω, dA(n) = mink ∈ ω : ϑ(A\k) < 2−n, and let

φ(A) ∈ωω be defined by

φ(A)(n) = e−1A∩ [dA(n), dA(n+ 1))

.


If S ∈ Slm is a slalom, then let

ψ(S) =⋃

n∈ω

⋃

e(k) : k ∈ S(n) and ϑ(e(k))< 2−n.

For each n the set⋃

e(k) : k ∈ S(n)∧ϑ(e(k))< 2−n is finite and has submeasureless then n/2n so ψ(S) ∈ I.

We show that (φ,ψ) : (I,⊆∗, I) BGT(ωω,v∗, Slm). Assume A ∈ I, S ∈ Slm, and

φ(A) v∗ S, φ(A)(n) ∈ S(n) for n ≥ N . Clearly, ϑ

e

φ(A)(n)

< 2−n so if n ≥ N thene

φ(A)(n)

⊆ψ(S). Therefore A\dA(N)⊆ψ(S) so A⊆∗ ψ(S).Let Ih be a tall summable ideal. Using Lemma 4.2.3, it is enough to show that

(ωω,v∗, Slm(a))BGT(Ih,⊆∗, Ih) where a = (2n).

Since Ih is tall, we can fix a family Ank : n, k ∈ω of pairwise disjoint subsets of ω

such that∑

`∈Ankh(`) = 2−n for all n, k ∈ω.

If f ∈ ωω then let φ( f ) =⋃

Anf (n) : n ∈ ω ∈ Ih. If A ∈ Ih then let ψ0(A)(n) =

k ∈ω : Ank ⊆ A. If there would be infinitely many n such that |ψ0(A)(n)| ≥ 2n, then A

could not be in Ih so we can modify ψ0(A) at finitely many coordinates and we obtainψ(A) such that ψ(A) ∈ Slm(a).

We claim that (φ,ψ) : (ωω,v∗, Slm(a)) BGT(Ih,⊆∗, Ih). Assume φ( f ) ⊆∗ A,

φ( f )\N ⊆ A. Fix an M such that if m ≥ M then N ∩ Amk = ; for each k ∈ ω, and

ψ0(A)(m) = ψ(A)(m). If m ≥ M then Amf (m) ⊆ φ( f )\N ⊆ A so f (m) ∈ ψ0(A)(m) =

ψ(A)(m). It means that f v∗ ψ(A), we are done.

Using Theorem 1.3.1 (that is (N,⊆,N)≡BGT(ωω,v∗, Slm)), we have the following:

Corollary 4.2.5. Let I be a tall analytic P-ideal. Then the following hold:

(1) add(N)≤ add∗(I) and cof∗(I)≤ cof(N).

(2a) If P has the Sacks property (i.e. P is (N,⊆,N)-bounding), then P is I-bounding.

(2b) If P adds a slalom over the ground model (i.e. P is (N,⊆,N)-dominating, in otherwords, P

⋃

(N∩ V ) ∈N), then P is I-dominating.

More precisely in (2b), if e :ω→ [ω]<ω is a bijection in V and S ∈ Slm∩ VP is a slalomover V , then

⋃

n∈ω

⋃

e(k) : k ∈ S(n)∧ ϑ(e(k))< 2−n ∈ I∩ VP

almost contains all elements of I∩ V .Moreover, if Ih is a tall summable ideal, then

(3) add(N) = add∗(Ih) and cof(N) = cof∗(Ih);

(4a) P has the Sacks property iff P is Ih-bounding;

(4b) P adds a slalom over the ground model iff P is Ih-dominating.


Remark 4.2.6. In [30] using Fremlin’s and Farah’s results, the authors proved thatadd(N) = add∗(Z~µ) and cof(N) = cof∗(Z~µ) also hold for each tall density ideal Z~µ.Unfortunately, at this moment we are unable to show a (Borel) GT-connection from(N,⊆,N) to (Z,⊆∗,Z). Although, using a result of Fremlin, we will prove that theZ-bounding property is equivalent with the Sacks property.

One of the most important open problems of this topic is the following:

Problem 4.2.7. Is (I,⊆∗, I) ≡(B)GT(N,⊆,N) for each tall analytic P-ideal I? Is at least

add∗(I) = add(N) (and cof∗(I) = cof(N)) for each tall analytic P-ideal I?

Theorem 4.2.8. Let I= Exh(ϑ) be a tall analytic P-ideal. Then

(ωω,≤∗,ωω)BGT(I,⊆∗, I).

Proof. We can assume that ‖ω‖ϑ = 2.For each f ∈ ωω we can choose a sequence (Af

n)n∈ω of pairwise disjoint finite setssuch that Af

n ⊆ω\ f (n) and 2−n ≤ ϑ(Afn) < 2−n+1 for each n. Why? Since I is tall, the

setX =

n

k ∈ω\

f (n)∪⋃

Afk : k < n

: ϑ(k)< 2−no

.

is co-finite, and so 2= ‖ω‖ϑ ≤ ϑ(X ). Thus there is a finite Afn ⊆ X with 2−n ≤ ϑ(Af

n)<2−n+1. Let φ( f ) =

⋃

Afn : n ∈ω ∈ I. Clearly, φ could be chosen as Borel.

For a B ∈ I let ψ(B) ∈ωω be defined by

ψ(B)(n) =min

k ∈ω : ϑ(B \ k)< 2−n.

We show that (φ,ψ) : (ωω,≤∗,ωω)BGT(I,⊆∗, I).

Assume φ( f ) ⊆∗ B ∈ I. Then there is an N such that⋃

Afn : n ≥ N ⊆ B. If n ≥ N

then ϑ(B\ f (n))≥ ϑ(Afn)≥ 2−n so ψ(B)(n)≥ f (n). Therefore f ≤∗ ψ(B).


(1) add∗(I)≤ b and d≤ cof∗(I).

(2a) If P is I-bounding, then P is ωω-bounding.

(2b) If P is I-dominating, then P adds dominating reals.

We show that (ωω,≤∗,ωω) and (I,⊆∗, I) are not GT-equivalent for any tall analyticP-ideals.

Notice that for tall summable and tall density ideals it follows from the cardinalcharacteristics: from Corollary 4.2.5 (3), Remark 4.2.6, and from the consistent cutsof Cichon’s diagram. Or simply consider the random forcing: it is ωω-bounding but itdoes not have the Sacks property, i.e. it is not (Ih,⊆∗, Ih)-bounding by Corollary 4.2.5(4a) (in the strong sense: it is (Ih,⊆∗, Ih)⊥-dominating) for all tall summbale ideals.

We show that the converse of the implication of Corollary 4.2.9 (2b) does not holdfor any tall analytic P-ideals. The Hechler forcing is a counterexample according to thefollowing theorem.


Theorem 4.2.10. If P is σ-centered then P is not I-dominating (in the strong sense: P is(I,⊆∗, I)⊥-bounding) for any tall analytic P-ideal I.

Proof. Assume that I = Exh(ϑ) for some finite lsc submeasure ϑ. W.l.o.g. we canassume that ‖ω‖ϑ = 1.

Let P =⋃

Cn : n ∈ ω where Cn is centered for each n. Assume on the contrarythat P X ∈ I and P ∀ A∈ I∩ V A⊆∗ X for some P-name X .

For each A∈ I choose a pA ∈ P and a kA ∈ω such that

pA A\kA ⊆ X ∧ ϑ(X \ kA)< 1/2. ()

For each n, k ∈ ω let Cn,k = A ∈ I : pA ∈ Cn ∧ kA = k, and let Bn,k =⋃

Cn,k. Weshow that for each n and k

ϑ

Bn,k \ k

≤ 1/2.

Assume on the contrary that ϑ(Bn,k\k) > 1/2 for some n and k. There is a k′ suchthat ϑ

Bn,k ∩ [k, k′)

> 1/2 and there is a finite D ⊆ Cn,k such that Bn,k ∩ [k, k′) =(⋃

D) ∩ [k, k′). Choose a common extension q of pA : A ∈ D. Now we have q ⋃

A\k : A∈D ⊆ X and so

q 1/2< ϑ

Bn,k ∩ [k, k′)

= ϑ⋃

D

∩ [k, k′)

≤ ϑ

X ∩ [k, k′)

≤ ϑ

X\k

,

which contradicts ().So for each n and k the set ω \ Bn,k is infinite, so ω \ Bn,k contains an infinite

Dn,k ∈ I. Let D ∈ I such that Dn,k ⊆∗ D for each n, k ∈ ω. Then there is no n, k suchthat D ∈ Cn,k, contradiction.

We will use the following deep result of Fremlin.

Theorem 4.2.11. ([26, Theorem 526F]) There is a family Pf : f ∈ ωω of Borelsubsets of `+1 such that the following hold:

(i) `+1 = ∪Pf : f ∈ωω;

(ii) if f ≤ g then Pf ⊆ Pg ;

(iii) (Pf ,≤,`+1 )BGT(Z,⊆,Z) for each f .

Corollary 4.2.12. P is Z-bounding iff P has the Sacks property.

Proof. Let Pf : f ∈ ωω be a family satisfying (i), (ii), and (iii) in Theorem 4.2.11,and fix Borel GT-connections (φ f ,ψ f ) : (Pf ,≤,`+1 )

BGT(Z,⊆,Z) for each f ∈ωω.

Assume P is Z-bounding and P x ∈ `+1 . P is ωω-bounding by Corollary 4.2.9 sousing (ii) we have P `+1 = ∪Pf : f ∈ ωω ∩ V. We can choose a P-name f for anelement of ωω ∩ V such that P x ∈ P f . By the Z-bounding property of P there is aP-name A for an element of Z∩ V such that P φ f ( x) ⊆ A, so P x ≤ψ f (A) ∈ `

+1 ∩ V .

It means that P is (`+1 ,≤(∗),`+1 )-bounding so by Theorem 1.3.1 we obtain that P hasthe Sacks property.

The converse implication was proved in Corollary 4.2.5 (2a).


Problem 4.2.13. Does the I-bounding property imply the Sacks property for each tallanalytic P-ideal I? Does Z-dominating (or I-dominating) imply adding slaloms overthe ground model?

By Theorem 4.2.10 the Cohen forcing is not I-dominating, and it will be followedfrom the following theorem that C is not I-bounding for any tall analytic P-ideal I.

Theorem 4.2.14. Let I= Exh(ϑ) be a tall analytic P-ideal. Then

([ω]ω, R∞, I)BGT(ωω,∈,M)⊥.

Proof. Since I is tall, we can fix a partition Pn : n ∈ ω of ω into finite intervals suchthat ϑ(x) < 2−n for x ∈ Pn+1 (we cannot say anything about ϑ(x) for x ∈ P0).Then A ∈ I whenever |A ∩ Pn| ≤ 1 for each n. Let in

k : k < kn be the increasingenumeration of Pn.

In this proof, first we will define ψ :ωω→ I. If f ∈ωω then let

ψ( f ) = ink : f (n)≡ k mod kn.

Then ψ( f ) ∈ I because |ψ( f )∩ Pn|= 1 for each n.If X ∈ [ω]ω then let φ(X ) = g ∈ωω : |ψ(g)∩ X |<ω. It is enough to show that

φ(X ) ∈M because then clearly (φ,ψ) : ([ω]ω, R∞, I)BGT(ωω,∈,M)⊥. We have that

φ(X ) =⋃

N∈ω

⋂

n≥N

g ∈ωω : g(n)≡ k (kn) =⇒ ink /∈ X

and it is easy to see that the counatble intersection after the union is nowhere denseso φ(X ) ∈M.


(1) non∗(I)≥ cov(M) and cov∗(I)≤ non(M).

(2a) If P adds Cohen reals, then P [ω]ω ∩ V ∈ I.

(2b) If P destroys I, then P ωω ∩ V ∈M.

This corollary has an interesting application. Let I be an ideal on ω. We say that atower (Tα)α<γ is a tower in I∗ if Tα ∈ I∗ for α < γ. Under CH it is straightforward toconstruct towers in I∗ for each tall analytic P-ideal I. The existence of such towers isconsistent with arbitrary large continuum as well by the following theorem.

Theorem 4.2.16. In VCω1 there exist towers in I∗ for each tall analytic P-ideal I.

Proof. Let G be a (V,Cω1)-generic filter and I = Exh(ϑ) be a tall analytic P-ideal in

V[G] with some lsc submeasure ϑ on ω. There is a δ < ω1 such that ϑ [ω]<ω ∈V[Gδ] where Gδ = G ∩Cδ, so we can assume that ϑ [ω]<ω ∈ V .

Working in V[G] by recursion on ω1 we construct the tower T = (Tα)α<ω1in I∗

such that T α ∈ V[Gα].


Because I contains infinite elements we can construct in V a sequence (Tn)n∈ω inI∗ which is strictly ⊆∗-descending, i.e. |Tn\Tn+1| = ω for n ∈ ω. Assume (Tξ)ξ<α aredone.

Since I is a P-ideal there is T ′α ∈ I∗ ∩ V[Gα] with T ′α ⊆∗ Tβ for β < α.

By Corollary 4.2.15 (2a) there is a set Aα ∈ V[Gα+1] ∩ I such that |Aα ∩ X | = ωfor each X ∈ [ω]ω ∩ V[Gα]. Let Tα = T ′α\Aα ∈ V[Gα+1] ∩ I∗ so X *∗ Tα for anyX ∈ V[Gα]∩ [ω]ω. Hence V[G] |=“(Tα)α<ω1

is a tower in I∗”.

Remark 4.2.17. Recently, J. Brendle proved (unpublished) that consistently there areno towers in the dual filters of any tall analytic P-ideals. He used an ω2 stage finitesupport iteration of filter based Mathias forcings.

Finally, we show one more property of tall summable ideals.

Theorem 4.2.18. If Ih is a tall summable ideal, then

(2ω,∈,N)BGT([ω]ω, R∞, Ih).

Proof. Let λ be the Lebesgue-measure on 2ω. We can choose pairwise disjoint setsH(n) ∈ [ω]ω such that

∑

`∈H(n) h(`) = 1 and suph(`) : ` ∈ H(n) < 2−n for eachn ∈ ω. Let H(n) = `n

k : k ∈ ω be an enumeration. For each n fix a partitionBn

k : k ∈ω of 2ω into Borel-sets such that λ(Bnk) = h(`n

k) for each k ∈ω.If f ∈ 2ω then let φ( f ) = `n

k : f ∈ Bnk ∈ [ω]

ω.If A∈ Ih then let

ψ(A) =⋂

N∈ω

⋃

n≥N

⋃

`nk∈A

Bnk .

ψ(A) ∈N because for each fixed N ≤ n ∈ω we have

λ

⋃

`nk∈A

Bnk

=∑

`nk∈A

λ(Bnk) =

∑

`nk∈A

h(`nk) =

∑

`∈A∩H(n)

h(`)

and for each D ∈ω if N is large enough, then D ∩⋃

n≥N H(n) = ; so

λ

⋃

n≥N

⋃

`nk∈A

Bnk

≤∑

n≥N

∑

`∈A∩H(n)

h(`)≤∑

`∈A\D

h(`)

which tends to 0 if D→∞ (because A∈ Ih). This implies that ψ(A) ∈N.Finally, we show that (φ,ψ) : (2ω,∈,N)B

GT([ω]ω, R∞, I). Assume |φ( f )∩A|=ω

for f ∈ 2ω and A ∈ Ih. Using that |φ( f )∩ H(n)| = 1 for each n, we obtain that for allN ∈ ω there are n ≥ N and k ∈ ω such that `n

k ∈ φ( f )∩ A so f ∈⋃

n≥N

⋃

`nk∈A Bn

k andtherefore f ∈ψ(A).

Corollary 4.2.19. Let Ih be a tall summable ideal. Then the following hold:

(1) non(N)≥ non∗(Ih) and cov(N)≤ cov∗(Ih).

(2a) If P [ω]ω ∩ V ∈ Ih, then P 2ω ∩ V ∈N.


(2b) If P adds random reals, then P destroys Ih.

Remark 4.2.20. Corollary 4.2.19 (2b) does not hold even for tall density ideals be-cause it is known that the random forcing does not destroy Z. We refer the readerto [17] for a nice proof of this result, and to [14] and [32] for the theory of forcingindestructibility of ideals.

We summarize the results of this section in the following diagrams where I is a tallanalytic P-ideal and the arrows mean Borel GT-connections:

(2ω,∈,N) - (2ω,∈,M)⊥ - (M,⊆,M) - (N,⊆,N)

([ω]ω, R∞, I) -

-(t.s.)-

6 6

(I,⊆∗, I)

(∗) -

(ωω,≤∗,ωω)⊥ - (ωω,≤∗,ωω)

-

(I,⊆∗, I)⊥

6

-

- 6 6

([ω]ω, R∞, I)⊥

6

(N,⊆,N)⊥

6

-(∗)

-

(M,⊆,M)⊥ - (2ω,∈,M) -

-

(2ω,∈,N)⊥

6

(t.s.)-

An arrow with (t.s.) means that this Borel GT-connection holds for tall summableideals, and with (∗) means that this Borel GT-connection holds for each tall analyticP-ideal and it is a Borel GT-equivalence for tall summable ideals.

cov(N) - non(M) - cof(M) - cof(N)

cov∗(I) -

-(t.s.)-

6 6

cof∗(I)

(]) -

b - d

-

add∗(I)

6

-

- 6 6

non∗(I)

6

add(N)

6

-

(])-

add(M) - cov(M) -

-

non(N)

6

(t.s.) -

4.3. IDEALIZED VERSION OF b AND d 39

Arrows with (t.s.) mean that the inequalities hold for tall summable ideals, andwith (]) hold for all tall analytic P-ideals and they are equalities for tall summable andtall density ideals.

We give some known supplements for the second diagram (see [30]).Cardinal characteristics in ZFC:

• For every tall density and tall summable ideal I, cov∗(Z)≤ cov∗(I) and non∗(Z)≥non∗(I);

• mincov(N),b ≤ cov∗(Z)≤maxb,non(N);

• mind,cov(N) ≤ non∗(Z)≤maxd,non(N).

Problem 4.2.21. Is cov∗(Z) (resp. non∗(Z)) minimal (maximal) among covering∗

(uniformity∗) numbers of tall analytic P-ideals?

The following are consistent with ZFC:

• cov(N)< cov∗(I) for all tall analytic P-ideals

(so there is no GT-connection from ([ω]ω, R∞, I) to (2ω,∈,N));

• cov∗(I)< add(M) for all tall analytic P-ideals

(so there is no GT-connection from ([ω]ω, R∞, I)⊥ to (M,⊆,M));

• cov∗(Z)< cov(N)

(so there is no GT-connection from (2ω,∈,N) to ([ω]ω, R∞,Z));

• d < cov∗(I) for all totally bounded tall analytic P-ideals (that is, if I = Exh(ϑ),then ϑ(ω)<∞).

The following problem is still open and seems to be very difficult.

Problem 4.2.22. Is it consistent with ZFC that b or even d is smaller than cov∗(Z)?

4.3 Idealized version of b and d

One can naturally generalize the classical unbounding and dominating number forideals: Let I be an ideal on ω. If f , g ∈ ωω then we write f ≤I g if n ∈ ω : f (n) >g(n) ∈ I, in particular ≤∗=≤Fin. Let b(I) be the minimal size of a ≤I-unboundedfamily in the the pre-ordered set (ωω,≤I), i.e.

b(I) = b(ωω,≤I,ωω) =min

|U | : U ⊆ωω and ∀ f ∈ωω ∃ g ∈ U g I f

;

and let d(I) be the minimal size of a ≤I-dominating family in (ωω,≤I), i.e.

d(I) = d(ωω,≤I,ωω) =min

|D| : D ⊆ωω and ∀ f ∈ωω ∃ g ∈ D f ≤I g

.

Theorem 4.3.1. If I≤RB J then (ωω,≤I,ωω)≡B

GT(ωω,≤J,ω

ω).


Proof. Fix a finite-to-one function f :ω→ω witnessing I≤RB J.Define φ,ψ :ωω→ωω as follows:

φ(x)(i) =max

x

f −1[i]

,

ψ(y)( j) = y( f ( j)).

We prove two claims.

Claim. (φ,ψ) : (ωω,≤J,ωω)B

GT(ωω,≤I,ω

ω).

Proof of the claim. We show that if φ(x) ≤I y then x ≤J ψ(y). Indeed, A = i :φ(x)(i)> y(i) ∈ I. Assume that f ( j) = i /∈ A. Then φ(x)(i) =max

x

f −1[i]

≤y(i). Since y(i) =ψ(y)( j), so

x( j)≤max

x

f −1[ f ( j)]

≤ y( f ( j)) =ψ(y)( j)

Since f −1[A] ∈ J this yields x ≤J ψ(y).

Claim. (ψ,φ) : (ωω,≤I,ωω)B

GT(ωω,≤J,ω

ω).

Proof of the claim. We show that ifψ(y)≤J x then y ≤I φ(x). Assume on the contrarythat y 6≤I φ(x). Then X = i ∈ ω : y(i) > φ(x)(i) ∈ I+. By definition of φ, we haveX = i : y(i)>max

x

f −1[i]

.Let Y = f −1[X ] ∈ J+. For j ∈ Y we have f ( j) ∈ X and so

ψ(y)( j) = y( f ( j))> φ(x)( f ( j)) =max

x

f −1[ f ( j)]

≥ x( j).

Hence ψ(y) 6≤I x , contradiction.

These claims prove the statement of the theorem, so we are done.

Corollary 4.3.2. If I≤RB J holds then b(I) = b(J) and d(I) = d(J).

By Theorem 2.1.3 we know that Fin≤RB I holds iff I is meager (see the note beforeProposition 2.3.1) so we obtain the following:

Corollary 4.3.3. If I is a meager ideal, then b(I) = b and d(I) = d.

Chapter 5

Covering properties of ideals

5.1 The J-covering property

In this chapter we will discuss the generalizations of the following result due to M.Elekes (see [17]). All result of this chapter can be found in [1].

Theorem 5.1.1. Let (X ,A,µ) be a σ-finite measure space and let (An)n∈ω be a sequenceof sets from A that covers µ-almost every x ∈ X infinitely many times. Then there exists aset M ⊆ω of asymptotic density zero such that (An)n∈M also covers µ-almost every x ∈ Xinfinitely mamy times.

Using this result Elekes gave a nice new proof for the fact that the density zeroideal is random-indestructible. He asked about other variants of this theorem.

We can consider the following abstract setting.

Definition 5.1.2. Let X be an arbitrary set and I ⊆ P(X ) be an ideal of subsets of X .We say that a sequence (An)n∈ω of subsets of X is an I -a.e. infinite-fold cover of X if

x ∈ X : n ∈ω : x ∈ An is finite

∈ I , i.e. limsupn∈ω

An ∈ I∗.

Of course, the sequence (An) above can be indexed by any countable infinite set. As-sume furthermore that given a σ-algebra A of subsets of X and an ideal J on ω. Wesay that the pair (A, I) has the J-covering property if for every I -a.e. infinite-fold cover(An)n∈ω of X by sets from A, there is a set S ∈ J such that (An)n∈S is also an I -a.e.infinite-fold cover of X .

In this context, Elekes’ theorem says that if (X ,A,µ) is a σ-finite measure space,then (A,Nµ) has the Z-covering property where Nµ = H ∈ A : µ(H) = 0 (moreprecisely, Nµ is the ideal generated by null sets because we assume that I is an idealon the underlying set).

Clearly, in the previous definition it is enough to check infinite-fold covers insteadof I -a.e. infinite-fold covers.

Observe that, if I1 ⊆ I2 and (A, I1) possesses the J-covering property, then also(A, I2) possesses it.

41

42 CHAPTER 5. COVERING PROPERTIES OF IDEALS

Notice that if (A, I) has the J-covering property, then (A[I], I) also has this prop-erty where A[I] is the “I -completion of A”, that is

A[I] =

B ⊆ X : ∃ A∈A A4B ∈ I

.

For instance, if (Borel(R),N) has the J-covering property, then (LM(R),N) also hasthis property where LM(R) is the σ-algebra of Lebesgue measurable subsets of R.Similarly in the category case, it is enough to prove that (Borel(R),M) has the J-covering property, then it also holds for (BP(R),M) where BP(R) is the σ-algebra ofsets with the Baire property.

Furthermore, if (A, I) has the J-covering property then for all Y ∈ A \ I the pair(A Y, I Y ) also has this property where of course, A Y = Y ∩ A : A ∈ A is therestricted σ-algebra.

Clearly, if J is not tall, then there is no (A, I) with the J-covering property.We give another motivation to this notion. First we reformulate the J-covering

property:

Fact 5.1.3. (A, I) has the J-covering property (X =⋃

A) if, and only if for every(A, Borel([ω]ω))-measurable function F : X → [ω]ω, there is an S ∈ J such that x ∈X : |F(x)∩ S|<ω ∈ I .

Because of this fact and the observation: (P(X ), ;) has the J-covering propertyiff |X |< non∗(J), the J-covering property can be seen as “analytic uniformity.”

We present some easy negative results. First we show that, in Elekes’ theoremapplied to the Lebesgue measure, one cannot use I1/n instead of Z.

Example 5.1.4. We will show that (Borel(R),N) does not have the I1/n-covering prop-erty in a strong sense. First consider the interval (0,1) and a fixed infinite-fold cover(An)n∈ω of (0,1) of the form An = (an, bn), bn − an =

1n+1

. Then for each S ∈ I1/n

we have∑

n∈S λ(An) <∞ where λ stands for Lebesgue measure on R. Hence by theBorel-Cantelli lemma, λ(lim supn∈S An) = 0. Fix a homeomorphism h from (0, 1) ontoR of class C1. Then (h[An])n∈ω is an infinite-fold open cover of R. Since h is absolutelycontinuous, we have λ(limsupn∈S h[An]) = 0, which gives the desired claim.

This example motivates the following question which will be discussed in Section5.3 (see Example 5.3.1 and Theorem 5.3.2 below).

Question 5.1.5. Assume X is a Polish space and (Borel(X ), I) does not have the J-covering property. Does there exist an infinite-fold Borel cover (An)n∈ω of X such thatlimsupn∈S An ∈ I for all S ∈ J?

Let us denote Kσ be the σ-ideal on ωω generated by compact sets. We will use thefact that an H ⊆ωω is in Kσ iff there is an h ∈ωω such that H ⊆ x ∈ωω : x ≤∗ h.

Example 5.1.6. Consider the following infinite-fold cover (An)n∈ω of ωω by Fσ sets:Let B = x ∈ ωω : ∀∞ n x(n) = 0 and An = y ∈ ωω : y(n) 6= 0 ∪ B for n ∈ ω. Itis easy to see that if X ∈ [ω]ω with ω \ X ∈ [ω]ω, then ωω \ lim supn∈X An is dense,uncountable, and does not belong to Kσ.

5.1. THE J-COVERING PROPERTY 43

In particular, there is no ideal J on ω such that (Borel(ωω), I) has the J-coveringproperty if I = [ωω]≤ω, NWD (the ideal of nowhere dense sets), or Kσ. Conse-quently (by the Borel isomorphism theorem), given an uncountable Polish space X ,(Borel(X ), [X ]≤ω) does not have the J-covering property for any ideal J on ω.

How could we conclude a J1-covering property from a J0-covering property? Inspecial cases we can do it by the following easy observation:

Fact 5.1.7. Assume J0 ≤KB J1 and that (A, I) has the J0-covering property. Then (A, I)has the J1-covering property as well.

The next observation shows a connection between trace ideals and covering prop-erties.

Proposition 5.1.8. Assume I is a σ-ideal on 2ω (or on ωω) and (Borel(2ω), I) has theJ-covering property. Then JK tr(I) X for any X ∈ tr(I)+.

Proof. Assume on the contrary that f ∈ ωω shows that J ≤K tr(I) X for some X ∈tr(I)+. Let (An)n∈ω be the following infinite-fold cover of [X ] ∈ I+:

An =

x ∈ [X ] : ∃ k ∈ω (x k ∈ X and f (x k) = n)

.

If S ∈ J then lim supn∈S An = [ f −1[S]] ∈ I , a contradiction.

Using Theorem 2.3.4 we obtain the following:

Corollary 5.1.9. Let I is a σ-ideal on 2ω (or on ωω), and J is a tall ideal on ω. As-sume furthermore that PI is proper and has the CRN. If (Borel(2ω), I) has the J-coveringproperty, then J is PI -indestructible.

We do not need to use Theorem 2.3.4, the trace ideal or the CRN property inthis result. The following theorem is a natural generalization of Elekes’ result aboutrandom-indestructibility of Z.

Theorem 5.1.10. Let X be a Polish space, I be a σ-ideal on X , and assume that PI isproper. If (Borel(X ), I) has the J-covering property, then J is PI -indestructible.

Proof. Assume on the contrary that Y is a PI -name for an infinite subset of ω, i.e.PI

Y ∈ [ω]ω and B PI∀ A∈ J |Y ∩A|<ω for some B ∈ PI . Then there are a C ∈ PI ,

C ⊆ B, and a Borel function f : C → [ω]ω (coded in the ground model) such thatC PI

f (rgen) = Y where rgen is a name for the generic real (see [46, Prop. 2.3.1]).For each n ∈ω let

Yn = f −1[S ∈ [ω]ω : n ∈ S] ∈ Borel(X ).

Then (Yn)n∈ω is an infinite-fold cover of C (by Borel sets) because x ∈ Yn ⇐⇒ n ∈f (x) and | f (x)| = ω. Using the J-covering property of (Borel(X ) C , I C) we canchoose an A∈ J such that (Yn)n∈A is an I -a.e. infinite-fold cover of C , that is | f (x)∩A|=ω for I -a.e. x ∈ C , i.e. x ∈ C : | f (x) ∩ A| < ω ∈ I , so C PI

| f (rgen) ∩ A| = ω, andconsequently, C PI

|Y ∩ A|=ω, a contradiction.


5.2 Around the category case

If X is a Polish space, then let M(X ) be the σ-ideal of meager subsets of X .

Theorem 5.2.1. The pair (Borel(X ),M(X )) has the EDfin-covering property for eachPolish space X .

Proof. Let (A(n,m))(n,m)∈∆ be an infinite-fold cover of X by Borel sets. Without loss ofgenerality, we can assume that all A(n,m)’s are open and nonempty.

Enumerate Uk : k ∈ ω a base of X . We will define by recursion a sequence(nk, mk)k∈ω of elements of ∆. First, pick (n0, m0) ∈ ∆ such that A(n0,m0) ∩ U0 6= ;.Assume (ni , mi) are done for i < k. Then we can choose an (nk, mk) ∈ ∆ such thatnk 6= ni for i < k and A(nk ,mk) ∩ Uk 6= ;. We obtain the desired set S = (nk, mk) : k ∈ω ∈ EDfin: For every k ∈ω, the set

⋃

i≥k A(ni ,mi) is dense and open. Consequently,

limsup(n,m)∈S

A(n,m) =⋂

k∈ω

⋃

i≥k

A(ni ,mi)

is a dense Gδ set, hence it is residual.

Using the Fact 5.1.7 and Theorem 5.1.10 we obtain the following:

Corollary 5.2.2. If EDfin ≤KB J, then (Borel(X ),M(X )) has the J-covering property foreach Polish space X , and hence J is Cohen-indestructible.

Note that EDfin ≤KB J holds for a quite big class of interesting ideals:

Proposition 5.2.3. EDfin ≤KB J holds for each tall analytic P-ideal J.

Proof. Let J= Exh(ϕ) for some lsc submeasure ϕ. For k ∈ω let

d(k) =min

`0 ∈ω : ∀ `≥ `0 ϕ(`)< 2−k.

We can choose a strictly increasing sequence (nk)k∈ω ∈ωω such that d(k+1)−d(k)≤nk. Let f : ω→ ω be any one-to-one function with the property f [d(k+ 1) \ d(k)] ⊆(nk, m) : m≤ nk. Then f shows that EDfin ≤KB J.

Now, by Corollary 5.2.2 and Proposition 5.2.3 we obtain

Corollary 5.2.4. (Borel(X ),M(X )) has the J-covering property for each Polish space Xand for each tall analytic P-ideal J.

One can ask if the implications in Corollary 5.2.2 could be equivalences. The an-swer is no by the following examples and results after them.

Example 5.2.5. Fin⊗ Fin is Cohen-indestructible but (Borel(ωω),M) does not havethe Fin⊗ Fin-covering property.

Cohen-indestructibility of Fin⊗ Fin: Is easy to see that a forcing notion P destroysFin⊗ Fin iff P adds dominating reals.(Borel(ωω),M) does not have the Fin ⊗ Fin-covering property: Enumerate sn

m :m ∈ω the elements of ω<ω with first element n. Consider the following infinite-foldcover of ωω: A(n,m) = x ∈ ωω : sn

m ⊆ x for (n, m) ∈ ω×ω. It is trivial to see thatthere is no S ∈ Fin⊗Fin such that (A(n,m))(n,m)∈S is an M-a.e. infinite-fold cover of ωω.

5.2. AROUND THE CATEGORY CASE 45

Example 5.2.6. ED is also Cohen-indestructible: Let C = (2<ω,⊇) be the Cohenforcing and assume on the contrary that X is a C-name for an infinite subset of ω×ωsuch that C ∀ A∈ ED∩V |X∩A|<ω. Enumerate C= pn : n ∈ω and let f ∈ωω∩Vbe the following function: if pn C (X )n = ; then let f (n) = 0, if not then let

f (n) =min

k ∈ω : ∃ q ≤ pn q C k =min

(X )n

.

Clearly f ∈ ED and it is easy to see that C |X ∩ f |=ω.Of course, (Borel(ωω),M) does not have the ED-covering property because ED⊆

Fin⊗ Fin (and we can use Example 5.2.5).

It would be nice to know a characterization of forcing notions which destroy ED

(similar to the characterization in the case of Fin⊗ Fin):

Question 5.2.7. Is it true that a forcing notion P destroys ED iff P adds an eventuallydifferent real, i.e. a real r ∈ωω ∩ VP such that | f ∩ r|<ω for each f ∈ωω ∩ V? (The“if” part trivially holds.)

In the case of the first implication of Corollary 5.2.2 we have only consistent coun-terexamples.

Theorem 5.2.8. Assume t= c and |A| ≤ c, then there is no ≤KB-smallest element of

J : (A, I) has the J-covering property

.

Proof. If J : (A, I) has the J-covering property = ; then we are done. If (A, I) hasthe J0-covering property then we will construct a J such that J0 KB J but (A, I) hasthe J-covering property.

Enumerate ( fα)α<c all finite-to-one functions fromω toω, and enumerate ((Aαn)n∈ω :α < c) the infinite-fold covers of X =

⋃

A by sets from A. By recursion on c we willdefine a ⊆∗-increasing sequence (Sξ)α<c of infinite and co-infinite subsets of ω and theideal J generated by this sequence will be as required.

Assume (Sξ)ξ<α is done for some α < c. Because of our assumption on t we canchoose an infinite and co-infinite S′α such that Sξ ⊆∗ S′α for each ξ < α. The setfα[ω\S′α] contains an infinite element E of J0. We want to guarantee that f −1

α [E] /∈ J

because then fα can not witness J0 ≤KB J. Let H = f −1α [E] \ S′α ∈ [ω]

ω.Consider the αth cover (Aαn)n∈ω. If (An)n∈ω\H is an I -a.e. infinite-fold cover of X ,

then let Sα = S′α ∪ (ω \H).If not, then

C =

x ∈ X : n ∈ω \H : x ∈ Aαn is finite

/∈ I .

Using our assumption on J0 for (A[I] C , I C) and its infinite-fold cover (Aαn∩C)n∈H(with a copy of J0 on H), we can choose an infinite H ′ ⊆ H such that H \ H ′ isalso infinite and (Aαn ∩ C)n∈H ′ is an I C-a.e. infinite-fold cover of C . Finally, letSα = S′α ∪ (ω \H)∪H ′.

It is easy to see from the construction that J is as required.

Corollary 5.2.9. If t = c then there are ideals J0 and J1 such that Z KB J0 andEDfin KB J1 but (Borel(2ω),N) has the J0-covering property and (Borel(2ω),M) hasthe J1-covering property.


Without t= c we can use a simple forcing construction.

Theorem 5.2.10. After addingω1 Cohen reals by finite support iteration there is an idealJ such that EDfin KB J (in particular, Z KB J) but (Borel(2ω),N) and (Borel(2ω),M)have the J-covering property.

Proof. Let (cα)α<ω1be the sequence of generic Cohen-reals in 2ω, Cα = c−1

α [1]⊆ω,and let J be the ideal generated by these sets. A trivial density argument shows that Jis a proper ideal.

To show that (Borel(2ω),N) has the J-covering property in the extension, it isenough to see that if (An)n∈ω is an infinite-fold cover of 2ω by Borel sets in a groundmodel V , then (An)n∈C is an N-a.e. infinite-fold cover of 2ω in V[C] where C ⊆ω is aCohen real over V . By a simple density argument V[C] |= λ

⋃

n∈C\k An

= 1 for eachk ∈ω, so V[C] |= λ

lim supn∈C An

= 1.To show that (Borel(2ω),M) has the J-covering property in the extension, it is

enough to prove that if (An)n∈ω is an infinite-fold cover of 2ω by open sets in V ,then (An)n∈C is an M-a.e. infinite-fold cover of 2ω in V[C]. By a simple densityargument V[C] |=“

⋃

n∈C\k An is dense open” for each k ∈ω, so V[C] |=“limsupn∈C Anis residual.”

To show that EDfin KB J, it is enough to see that if f ∈ ∆ω ∩ V[(cξ)ξ<α] is afinite-to-one function for some α < ω1, then there is an A ∈ EDfin ∩ V[(cξ)ξ≤α] suchthat f −1[A] cannot be covered by finitely many of Cξ’s (ξ < ω1). Simply let A be aCohen function in

∏

n∈ω(n+1), i.e. the graph of a Cohen function in∆, for example ifc′α ∈ω

ω is a Cohen real over V[(cξ)ξ<α] then A=

(n, k) ∈∆ : c′α(n)≡ k mod (n+1)

is suitable. Using the presentation of this iteration by finite partial functions fromω1×ω to 2, we are done by a simple density argument.

Question 5.2.11. Does there exist an (analytic) ideal J in ZFC such that Z KB J but(Borel(2ω),N) has the J-covering property? Does there exist Katetov-Blass-smallestideal in the family of all analytic (or Borel) ideals J such that (Borel(2ω),N) has theJ-covering property?

Similarly, one can ask the analogous question for the meager ideal with EDfin in-stead of Z; and about the existence of Katetov-Blass-smallest ideal in the correspondingfamily as well.

5.3 When the J-covering property “strongly” fails

In this section we give a positive answer for Question 5.1.5 in a special case but first ofall, we present a counterexample:

Example 5.3.1. Consider X = (−1, 1) and let an ideal I on X consist of sets A ⊆ Xsuch that A∩(−1, 0] is meager and A∩(0,1) is of Lebesgue measure zero. Using Exam-ple 5.1.4 and Corollary 5.2.4 observe that (Borel(X ), I) yields the negative answer toQuestion 5.1.5 with J = I1/n. However, this question remains interesting if we restrictit to pairs (Borel(R), I) where I is a translation invariant ideal on R.

5.4. J-COVERING PROPERTIES OF (P(ω), I) 47

In the translation invariant case, we describe a class of ideas I which yields apositive answer to Question 5.1.5 provided J is a P-ideal.

Let Q stand for the set of rational numbers. For A, B ⊆ R and x ∈ R we writeA+ x = a+ x : a ∈ A and A+ B = a+ b : A ∈ A and b ∈ B. We say that an ideal Ion a Polish space X is a ccc ideal if every disjoint subfamily of Borel(X )\ I is countable.

Theorem 5.3.2. Assume that I is a translation invariant ccc σ-ideal on R fulfilling thecondition

Q+ A∈ I∗ for each A∈ Borel(R) \ I . (])

Fix a P-ideal J on ω. If (Borel(R), I) does not have the J-covering property, then thereexists an infinite-fold Borel cover (A′n)n∈ω of R with limsupn∈S A′n ∈ I for all S ∈ J.

Proof. Fix an infinite-fold Borel cover (An)n∈ω of R such that lim supn∈S An /∈ I∗ for allS ∈ J. We will show that there is a Borel set B ⊆ Rwith B /∈ I and (lim supn∈S An)∩B ∈ Ifor all S ∈ J. Suppose it is not the case. So, in particular (when B = R), we find S0 ∈ J

with X0 := limsupn∈S0An /∈ I . Then by transfinite recursion we define sequences

(Sα)α<γ and (Xα)α<γ with Sα ∈ J and Xα := (lim supn∈Sα An) \⋃

β<α Xβ /∈ I (whenB = R \

⋃

β<α Xβ /∈ I). Since I is ccc, this construction stops at a stage γ < ω1 with⋃

α<γ lim supn∈Sα An =⋃

α<γ Xα ∈ I∗. Since I is a P-ideal, there is S ∈ J which almostcontains each Sα for α < γ. Then limsupn∈S An ∈ I∗ which contradics our supposition.

So, fix a Borel set B /∈ I such that (limsupn∈S An) ∩ B ∈ I for all S ∈ J. Let Q =qk : k ∈ ω. Define B0 := B and Bk := (qk + B) \

⋃

i<k Bi for k ∈ ω. Then putA′n :=

⋃

k∈ω

(qk + An) ∩ Bk

for n ∈ ω. Since (An)n∈ω is an infinite-fold cover of R,we have lim supn∈ω

(qk + An)∩ Bk

= Bk for all k ∈ω. Note that (A′n)n∈ω is an I -a.e.infinite-fold cover of R since

limsupn∈ω

A′n ⊇⋃

k∈ω

limsupn∈ω

(qk + An)∩ Bk

=Q+ B

andQ+B ∈ I∗ by (]). Fix an S ∈ J. Since I is translation invariant and (limsupn∈S An)∩B ∈ I , we have limsupn∈S

(qk + An) ∩ Bk

∈ I for all k ∈ ω. Since I is a σ-ideal andBk ’s are pairwise disjoint, it follows that

lim supn∈S

A′n =⋃

k∈ω

limsupn∈S

(qk + An)∩ Bk

∈ I .

Of course, we can modify (A′n) to be an infinite-fold cover of R.

Theorem 5.3.2 can be generalized to any Polish group G with Q replaced by acountable dense subset of G. Condition (]) is related to the Steinhaus property, fordetails see [2]. Note that M, N, M⊗N and N⊗M satisfy (]) with Q replaced by anydense subset of R (resp. R2).

5.4 J-covering properties of (P(ω), I)

Let I, J be ideals on ω where J is tall. We will say that I has the J-covering propertywhenever the pair (P(ω), I) has the J-covering property.


It is trivial that non∗(J) > ω iff J is ω-hitting, that is, for every sequence (Xn)n∈ωin [ω]ω, there is an A∈ J such that |Xn ∩A|=ω for each n ∈ω. We will use a weakerversion of this property: An ideal J on ω is weakly ω-hitting if for each sequence(Xn)n∈ω in [ω]ω there is an A∈ J such that n ∈ω : |Xn ∩ A|=ω is infinite.

This property is really weaker than non∗(J) > ω by Lemma 5.4.2(4). Moreover, itis easy to see the following characterization:

Proposition 5.4.1. J is weakly ω-hitting if, and only if JKB Fin⊗ Fin.

By the following easy lemma, in the characterization of “I has the J-covering prop-erty” the interesting case is when J is weakly ω-hitting but not ω-hitting.

Lemma 5.4.2. Assume J is a tall ideal on ω. Then

(1) J is ω-hitting iff all ideals have the J-covering property.

(2) If J is not weakly ω-hitting, then there is no ideal with the J-covering property.

(3) If J is weakly hitting and all tall ideals have the J-covering property, then J isω-hitting.

(4) If J is weakly ω-hitting but not ω-hitting, then there is a tall ideal J0 such that (upto isomorphism) J is contained in J0 ⊗ Fin. And all ideals of this form are weaklyω-hitting but not ω-hitting.

Proof. (1): It is trivial by Fact 5.1.3 that if J is ω-hitting, then all ideals have the J-covering property. Conversely, clearly Fin has the J-covering property iff J is ω-hitting.

(2): Let (Xn)n∈ω witness that J is not weakly ω-hitting and let F(n) = Xn. Thenn ∈ ω : |F(n)∩ A| = ω is finite for each A ∈ J, so by Fact 5.1.3, F shows that I doesnot have the J-covering property for all I (because finite sets cannot be in I∗).

(3): Assume on the contrary that J is not ω-hitting witnessed by the sequence(Xn)n∈ω. For each A ∈ J let EA = n ∈ ω : |Xn ∩ A| = ω. Clearly, EA is co-infinite andEA∪B = EA ∪ EB for A, B ∈ J, so these sets generate an ideal I on ω. Moreover, it istrivial to see that I = EA : A ∈ J. I is tall because of the weak ω-hitting property ofJ. If F(n) = Xn then n ∈ω : |F(n)∩A|=ω ∈ I for each A∈ J, so I does not have theJ-covering property, a contradiction.

(4): Let J0 = I from the proof of (3). (Of course, we can assume that Xn’s arepairwise disjoint and

⋃

n∈ω Xn =ω.)Assume J0 is a tall ideal. The columns in ω×ω show that J0⊗Fin is not ω-hitting.

To prove the weak ω-hitting property, fix a sequence Xn ∈ [ω×ω]ω (n ∈ω). We canassume the followings:

(i) if there is a k such that |(Xn)k|=ω, then Xn ⊆ k ×ω;

(ii) if n 6= m and (Xn)k and (Xm)k are finite for all k ∈ω then

k ∈ω : (Xn)k 6= ; ∩ k ∈ω : (Xm)k 6= ;= ;.

5.4. J-COVERING PROPERTIES OF (P(ω), I) 49

If B = n ∈ω : ∃ k ∈ω Xn ⊆ k ×ω is finite, then let A=⋃

n∈ω\B Xn ∈ J0 ⊗ Fin.If |B|=ω then let B′ ⊆ B, |B′|=ω, B′ ∈ J0, and let A= B′×ω ∈ J0⊗ Fin. Clearly, theset n ∈ω : |Xn ∩ A|=ω is infinite in both of these cases.

In particular, if J is weakly ω-hitting but not ω-hitting, then there is a tall idealI which does not have the J-covering property, so the next natural question is thefollowing: Does there exist an ideal I with the J-covering property in this case?

In next theorem we characterize ideals with the J0 ⊗ Fin-covering property. Firstwe recall an important notion: Assume J is an ideal on ω. Then a filter F is a J-(ultra)filter if for each function f : ω → ω there is an X ∈ F such that f [X ] ∈ J (orequivalently, there is an A ∈ J such that f −1[A] ∈ F). For combinatorial propertiesof J-filters and investigation of their existence see for example [24] and J. Flašková’sother publications.

Theorem 5.4.3. Let J0 be a tall ideal. Then an ideal I has the J0⊗Fin-covering propertyiff I∗ is a J0-filter.

Proof. Assume that I has the J0⊗Fin-covering property and let f :ω→ω be arbitrary.Let F(n) = f (n)×ω. Then there is an A∈ J0⊗Fin such that X = n ∈ω : |F(n)∩A|=ω ∈ I∗. We can assume that A is of the form A0 × ω for some A0 ∈ J0. Clearlyf −1[A0] = X ∈ I∗, so f [X ] ∈ J0.

Conversely, assume that I∗ is a J0-filter and let F :ω→ [ω×ω]ω be arbitrary. Wecan assume the followings:

(i) if there is a k such that |(F(n))k|=ω, then (F(n))k =ω;

(ii) if n 6= m and (F(n))k and (F(m))k are finite for all k ∈ω then

k ∈ω : (F(n))k 6= ; ∩ k ∈ω : (F(m))k 6= ;= ;.

Let X = n ∈ω : ∃ kn ∈ω F(n) = kn ×ω. If X ∈ I then we do not have to dealwith it. If X ∈ I+ then let f : X → ω, f (n) = kn. Clearly (I X )∗ is also a J0-filter, sothere is an A0 ∈ J0 such that X \ f −1[A0] ∈ I.

It is easy to see that

A=

A0×ω

∪⋃

n∈ω\X

F(n) ∈ J0⊗ Fin

and n ∈ω : |F(n)∩ A|=ω ∈ I∗.

Question 5.4.4. Is there any similarly easy and reasonable characterization of idealswith the J-covering property for an arbitrary (weakly ω-hitting but not ω-hitting) J?


Chapter 6

Generalizations of Hechler’stheorem

6.1 Hechler’s original theorem

A partially ordered set (Q,≤) is σ-directed if each countable subset of Q has a strictupper bound in Q. Hechler’s original theorem is the following statement:

Theorem 6.1.1. ([29],[15]) Let (Q,≤) be a σ-directed partially ordered set. Then thereis a ccc forcing notion P such that in VP a cofinal subset of (ωω,≤∗) is order isomorphicto (Q,≤).

This theorem has a very important application. It says that we really cannot proveany more connections between b and d than ω< cf(b) = b≤ cf(d)≤ d≤ c.

Theorem 6.1.2. ([7, Theorem 2.5]) Assume GCH and let b′, d′, and c′ three cardinalssatisfying

ω< cf(b′) = b′ ≤ cf(d′)≤ d′ ≤ c′

and cf(c′)>ω. Then there is a ccc forcing extension in which b= b′, d= d′, and c= c′.

Proof. (Sketch.) Apply Hechler’s theorem to Q = [d′]<b′partially ordered by inclusion.

(Q,⊆) is σ-directed because any < b′ elements of Q has a strict upper bound by regu-larity of b′. There is an unbounded family of size b′. Furthermore, b′ ≤ cf(d′) impliesthat a subset of Q with cardinality smaller than d′ cannot be cofinal. Using GCH weobtain that |Q| = d′. These observations imply that VP |= b = b′,d = d′ where P is theccc forcing notion given by Hechler’s theorem.

Finally, P ∗Bc′ has the ccc, Bc′ does not change b and d, and VP∗Bc′ |= c= c′.

In Section 6.4, we will present another interesting application of this theorem.

6.2 Hechler’s theorem for M and N

In [41] L. Soukup asked if Hechler’s theorem hold for classical σ-ideals as partiallyordered sets with the inclusion. T. Bartoszynski, M.R. Burke, and M. Kada gave thefollowing positive answers.

51

52 CHAPTER 6. GENERALIZATIONS OF HECHLER’S THEOREM

Theorem 6.2.1. ([4]) Let (Q,≤) be a σ-directed partially ordered set. Then there is a cccforcing notion P such that in VP a cofinal subset of (M,⊆) is order isomorphic to (Q,≤).

Theorem 6.2.2. ([16]) Let (Q,≤) be a σ-directed partially ordered set. Then there isa ccc forcing notion P such that in VP a cofinal subset of (N,⊆) is order isomorphic to(Q,≤).

In this section, we recall the construction of forcing notion from 6.2.2 and its mainproperties.

We will use a special version of the localization forcing (see [16, Definition 3.1]):Let T =

⋃

n∈ω∏

k<n[ω]≤k be the tree of initial slaloms. p ∈ LOC∗ iff p = (sp, wp, F p)

where

(1) sp ∈ T, wp ∈ω, F p ⊆ωω,

(2) |F p| ≤ wp ≤ |sp|,

q ≤ p iff

(a) sq ⊇ sp, wq ≥ wp, and Fq ⊇ F p,

(b) ∀ n ∈ |sq|\|sp| ∀ f ∈ F p f (n) ∈ sq(n),

(c) wq ≤ wp + |sq| − |sp|,

(d) ∀ n ∈ |sq|\|sp| |sq(n)| ≤ wp + n− |sp|.

Lemma 6.2.3. ([16, Lemma 3.2, 3.3, and 3.4]) LOC∗ is σ-linked (so ccc) and adds aslalom over the ground model.

Let (Q,≤) be a partially ordered set such that each countable subset of Q has astrict upper bound in Q. Let Q∗ =Q ∪ Q and extend the partial order to this set withx <Q for each x ∈Q.

Fix a well-founded cofinal R⊆Q and a rank function on R∗ = R∪Q, % : R∗→ On.Extend % to Q∗ by letting %(x) =min%(y) : y ∈ R∗, x < y for x ∈Q\R. For x , y ∈Q∗

define x y iff x < y and %(x)< %(y). Further notations:

• Q x = y ∈Q : y x for x ∈Q∗,

• Dξ = x ∈ D : %(x) = ξ for D ⊆Q and ξ ∈ On,

• D<ξ = x ∈ D : %(x)< ξ for D ⊆Q and ξ ∈ On,

• D≤x = y ∈ D : %(y) = %(x), y ≤ x for D ⊆Q and x ∈Q.

If E ⊆ D ⊆ Q, we say that E is downward closed in D, E ⊆d.c. D in short, if y ∈ Ewhenever y ∈ D and y ≤ x ∈ E for some x .

Definition 6.2.4. ([16, Definition 3.1]) The forcing notions Na for a ∈Q∗ are definedby recursion on %(a).

p =

(spx , wp

x , F px ) : x ∈ Dp ∈ Na where Dp ∈ [Qa]<ω if the following hold:

6.2. HECHLER’S THEOREM FOR M AND N 53

(I) for x ∈ Dp, spx ∈ T, wp

x ∈ω, and F px is a set of nice Nx -names for elements of ωω

with |F px | ≤ wp

x ;

(II) for x ∈ Dp,∑

wpz : z ∈ Dp

≤x

≤ |spx |;

(III) for each ξ ∈ %[Dp] there is an `pξ∈ω such that |sp

x |= `pξ

for each x ∈ Dpξ.

If p ∈ Na and b ∈Qa, define p b ∈ Nb by letting

p b =

(spx , wp

x , F px ) : x ∈ Dp ∩Qb

.

p ≤Naq iff

(A) Dp ⊇ Dq;

(B) ∀ x ∈ Dq spx ⊇ sq

x ∧wpx ≥ wq

x ∧ F px ⊇ Fq

x

;

(C) ∀ x ∈ Dq ∀ n ∈ |spx |\|s

qx | ∀ f ∈ Fq

x

p x Nxf (n) ∈ sp

x(n)

;

(D) ∀ ξ ∈ %[Dq] ∀ x , y ∈ Dqξ

x < y ⇒∀ n ∈ `pξ\`qξ

spx(n)⊆ sp

y(n)

;

(E) ∀ ξ ∈ %[Dq]∑

wpx : x ∈ Dp

ξ

≤∑

wqx : x ∈ Dq

ξ

+ (`pξ− `q

ξ);

(F) ∀ ξ ∈ %[Dq] ∀ E ⊆d.c. Dqξ∀ n ∈ `p

ξ\`qξ

⋃

spx(n) : x ∈ E

≤∑

wqx : x ∈ E

+ (n− `qξ).

Proposition 6.2.5. ([16, Proposition 4.3])

(a) If p, q ∈ Na, p ≤Naq, and b ∈Qa, then p b ≤Nb

q b.

(b) ≤Nais a partial order.

(c) If a, b ∈Q∗ and p, q ∈ Na ∩Nb, then p ≤Naq ⇐⇒ p ≤Nb

q.

From now on we write ≤ (=≤NQ) instead of ≤Na

.

Definition 6.2.6. ([16, Definition 4.4]) For an A⊆d.c. Q, let NA = p ∈ NQ : Dp ⊆ A,and for p ∈ NQ, let p A=

(spx , wp

x , F px ) : x ∈ Dp∩A

∈ NA. Furthermore, if ξ ∈ On thenlet Nξ = NQ<ξ , p ξ= p Q<ξ ∈ Nξ, and p [ξ,∞) =

(spx , wp

x , F px ) : x ∈ Dp\Q<ξ

.

So we have Na = NQafor each a ∈Q∗, and NQ has the same meaning if we consider

Q either as an element of Q∗ or as a subset of Q.

Lemma 6.2.7. ([16, Lemma 4.6]) If A, B ⊆d.c. Q and A⊆ B, then NA ≤c NB.

Remark 6.2.8. In [16] Lemma 4.6, exactly the following stronger result was proved:If p ∈ NB, r ∈ NA, and r ≤ p A then there is a q ∈ NB satisfying q ≤ p, r.


Lemma 6.2.9. ([16, Lemma 4.10]) NQ has ccc.

We will use the following density arguments.

Lemma 6.2.10. ([16, Lemma 5.1, 5.2, 5.3, and 5.4]) If a ∈ A⊆d.c. Q, ξ ∈ On, N ∈ ω,and f is a nice Na-name for an element of ωω, then the following sets are dense in NA:

(i)

p ∈ NA : a ∈ Dp;

(ii)

p ∈ NA : ξ ∈ %[Dp]∧ `pξ≥ N

;

(iii)

p ∈ NA : a ∈ Dp ∧wpa ≥ |F

pa |+ 1

;

(iv)

p ∈ NA : a ∈ Dp ∧ f ∈ F pa

.

6.3 Hechler’s theorem for tall analytic P-ideals

Using the model of [16] we prove the following theorem (see also in [21]).

Theorem 6.3.1. Let (Q,≤) be a σ-directed partially ordered set. Then there is a cccforcing notion P such that in VP for each tall analytic P-ideal I coded in V a cofinal subsetof (I,⊆∗) is order isomorphic to (Q,≤).

Remark 6.3.2. Tallness is not really necessary in Theorem 6.3.1. It is enough to as-sume that I can be represented by Exh(ϑ) such that n ∈ ω : ϑ(n) < ε /∈ I for eachε > 0. This property of Exh(ϑ) is really weaker than tallness.

For an a ∈Q, let Sa be an NQ-name such that

NQSa =

⋃

spa : p ∈ G

.

Using (i) and (ii) from Lemma 6.2.10, NQSa ∈ Slm for each a ∈ Q. Furthermore

using (iv) and the definition of NQ we know that Sa is a slalom over V[G ∩Na], i.e.

NQ∀ f ∈ωω ∩ V[G ∩Na] f v∗ Sa. (]1)

At last, using the definition of NQ it is clear that if %(a) = %(b) and a < b then

NQ∀∞ n ∈ω Sa(n)⊆ Sb(n). (†)

Let I = Exh(ϑ) be a tall analytic P-ideal. We will use Corollary 4.2.5 (2b): Fix abijection e :ω→ [ω]<ω and for a slalom S ∈ Slm, let

I(S) =⋃

n∈ω

⋃

e(k) : k ∈ S(n)∧ ϑ(e(k))< 2−n ∈ I.

We prove that in VNQ the set I(Sa) : a ∈Q ⊆ I is

(i) cofinal, i.e. ∀ I ∈ I∩ VNQ ∃ a ∈Q I ⊆∗ I(Sa);

6.3. HECHLER’S THEOREM FOR TALL ANALYTIC P-IDEALS 55

(ii) order isomorphic to (Q,≤), i.e. I(Sa)⊆∗ I(Sb) iff a ≤ b.

The only difficult step is to show that a b implies I(Sa)*∗ I(Sb).It is clear from (]1) and from Corollary 4.2.5 (2b) that for each a ∈Q

NQ∀ I ∈ I∩ V[G ∩Na] I ⊆∗ I(Sa). (]2)

Lemma 6.3.3. NQ“I(Sa) : a ∈Q is cofinal in (I,⊆∗)”.

Proof. Let I be a nice NQ-name for an element of I. Using that NQ is ccc and that eachcountable subset of Q is (strictly) bounded in Q, there is an a ∈ Q such that I is anNa-name. Then NQ

I ⊆∗ I(Sa) by (]2).

Lemma 6.3.4. Assume a, b ∈Q and a ≤ b. Then NQI(Sa)⊆∗ I(Sb).

Proof. If a b then NQI(Sa) ∈ I∩ V[G ∩Nb] so we are done by (]2). If %(a) = %(b)

then we are done by (†).

We will need the following version of Lemma 6.2.10 (ii) which says that we canextend conditions in a natural way.

Lemma 6.3.5. Assume p ∈ NQ, ξ ∈ %[Dp], and m≥ `pξ. Then there is a q ≤ p such that

Dqξ= Dp

ξand q ξ forces that ∀ b ∈ Dq

ξ∀ n ∈ [`p

ξ, m]

sqb(n) =

n

f (n) : f ∈⋃

F pb′ : b′ ∈ Dp

≤b

o

.

Proof. First we choose an r ∈ Nξ, r ≤ p ξ which decides f [`pξ, m] for each f ∈

⋃


≤b

: r Nξ f [`pξ, m] = g f for some g f ∈ω

[`pξ,m].

Now let q be the following condition:

(i) q ξ= r, q [ξ+ 1,∞) = p [ξ+ 1,∞), and Dqξ= Dp

ξ;

(ii) if b ∈ Dqξ

then let |sqb|= m+ 1, sq

b `pξ= sp

b, wqb = wp

b, and Fqb = F p

b ;

(iii) if b ∈ Dqξ

and n ∈ [`pξ, m] then let

sqb(n) =

n

g f (n) : f ∈⋃


≤b

o

.

Clearly q ∈ NQ. We have to show that q ≤ p. (A), (B), (C), (D), and (E) holdtrivially.

To see (F) assume E ⊆d.c. Dpξ

and n ∈ [`pξ, m] (m+ 1= `q

ξ). Then

⋃

sqx(n) : x ∈ E

=

n

g f (n) : f ∈⋃

F px : x ∈ E

o

≤∑

|F px | : x ∈ E

≤

∑

wpx : x ∈ E

≤∑

wpx : x ∈ E

+ (n− `pξ).


In Lemma 6.3.6 we will use the following notation: if s ∈ T is an initial slalom thenlet

I(s) =⋃

n<|s|

⋃

e(k) : k ∈ s(n)∧ ϑ(e(k))< 2−n ∈ [ω]<ω.

Clearly, if p ∈ NQ and a ∈ Dp, then p NQI(sp

a)⊆ I(Sa).

Lemma 6.3.6. Assume a, b ∈Q and a b. Then NQI(Sa)*∗ I(Sb).

Proof. Let p ∈ NQ and N ∈ω. We have to find a q ≤ p such that q NQI(Sa)\N * I(Sb).

Using Lemma 6.2.10 (i) and (iii) we can assume that a, b ∈ Dp and |wpa | ≥ |F

pa |+ 1.

Let M =max|spa |, |s

pb|. Using Lemma 6.2.10 we can assume that M is large enough

such that ϑ(k)≥ 2−M for each k < N . For each m ∈ω let

Xm =

k ∈ω : 2−m−1 ≤ ϑ(k)< 2−m.

Let ξ = %(b). Using that Nb′ ≤c Nb if b′ ∈ Dp≤b by Lemma 6.2.7, we can define a

descending sequence in Nb: p b ≥ rM ≥ rM+1 ≥ . . . such that rm decides f [`pξ, m]

for each f ∈⋃


≤b

. Let Im : [`pξ, m]→ [ω]<ω be defined by

rm NbIm(n) =

⋃n

e( f (n)) : f ∈⋃


≤b

∧ ϑ

e( f (n))

< 2−no

.

Claim. There is an m≥ M such that Xm * I(spb)∪

⋃

Im(n) : n ∈ [`pξ, m].

Proof of the Claim. Assume on the contrary that there is no such an m. Then

Xm ⊆ I(spb)∪

⋃

Im(n) : n ∈ [`pξ, m]

for each m ≥ M . Clearly ω ⊆∗⋃

m≥M Xm by tallness1, the sets I(spb) and Im(n) are

finite, and if n≤ m1 ≤ m2 then Im1(n) = Im2

(n) so we have

ω⊆∗ I(spb)∪

⋃

m≥M

m⋃

n=`pξ

Im(n)⊆∗⋃

m≥M

Im(m).

Using that ϑ(Im(n))≤ |Dp≤b|

n2n we obtain that ω ∈ I, a contradiction.

Assume m is suitable in the Claim and let r = rm. Fix a k ∈ Xm\

I(spb)∪

⋃

Im(n) :n ∈ [`p

ξ, m]

. Then there is a k such that e(k) = k. Let g be the canonical Na-name

for the constant function with value k. Let p′ ∈ NQ be the condition which extends pby putting g into F p

a (this is really a condition extending p because of our assumption|wp

a | ≥ |Fpa |+ 1). We know that p b = p′ b because a /∈Qb so r ≤ p′ b.

Using Remark 6.2.8 for Qb ⊆ Q<ξ, r ∈ Nb, and p′ ξ ∈ Nξ we can find a q′ ∈ Nξwith q′ ≤ r, p′ ξ. Let p′′ = q′ ∪ p′ [ξ,∞)≤ p.

1This is the only point in the proof where we used tallness of the ideal. As we mentioned in Remark6.3.2, it would be enough to assume that

⋃

m≥M Xm = k ∈ω : ϑ(k)< 2−M /∈ I.

6.4. AN APPLICATION: CODING WITH SPECTRUM 57

Finally, using Lemma 6.3.5 we can extend p′′ to a q such that Dqξ= Dp′′

ξ(= Dp

ξ) and

q ξ Nξ ∀ n ∈ [`pξ, m] sq

b(n) =

f (n) : f ∈⋃


≤b

.Because q b ≤ r we obtain that

q NQI(sq

b)⊆ I(spb)∪

⋃

Im(n) : n ∈ [`pξ, m]

.

By the choice of k and p′ it is clear that k ∈ sqa(m) and ϑ(e(k)) = ϑ(k) < 2−m so

k ∈ I(sqa) which implies that q NQ

k ∈ I(Sa)\N .

To show that q NQk /∈ I(Sb) we know that k /∈ I(sq

b) and if there would be a

q ≤ q such that k ∈ I(sqb), then there would be an n > m and a k′ ∈ sq

b(n) such that

k ∈ e(k′)⊆ I(sqb) but then 2−n > ϑ(e(k′))≥ ϑ(k)≥ 2−m−1 would give a contradiction

because n≥ m+ 1. The proof of Lemma 6.3.6 is done.

Now we have finished the proof of Theorem 6.3.1.

Problem 6.3.7. Are there any connections between known versions of Hechler’s theo-rem, more precisely: can we (easily) deduce one of them from another?

6.4 An application: coding with spectrum

J. D. Hamkins asked the following:

Question 6.4.1. Is it true that for each set x there is a cardinal preserving forcingnotion P such that x is (first order) definable in VP without any parameters?

L. Soukup noticed that spectrums of cardinal invariants and versions of Hechler’stheorem can be used for coding reals in generic extensions. For a given pre-orderedset (P,≤) S. Fuchino and L. Soukup defined (unpublished note) the unbounded chainspectrum of (P,≤): Sp↑(P,≤) is the set of all regular cardinals κ such that there is anunbounded increasing chain of length κ in (P,≤).

We refer the reader to [11] and [41] for other notions of spectra, their closednessproperties, and for related consistency results.

Let (B,v) be a σ-directed Borel pre-ordered set, that is, B ⊆ωω and v as a subsetof ωω ×ωω are Borel-sets. We say that Hechler’s theorem holds for (B,v), if for allσ-directed partially ordered set (Q,≤) there is a ccc forcing notion P such that in VP acofinal subset of (B,v) is order isomorphic to (Q,≤).

Theorems presented in this chapter say that Hechler’s theorem holds for (ωω,≤∗),(N,⊆), (M,⊆), and for (I,⊆∗) if I is a tall analytic P-ideal.

Theorem 6.4.2. (L. Soukup) Assume Hechler’s theorem holds for (B,v) and X ⊆ω is areal. Then there is a ccc forcing notion P such that

VP |= X =

n ∈ω : ℵn+1 ∈ Sp↑(B,v)

.


Proof. Applying Hechler’s theorem to (Q,≤) =∏

n∈X (ℵn+1,≤) we have a ccc forcingnotion P such that VP |=“(Q,≤) can be cofinally embedded into (B,v)”. We will workin this extension.

Assume k ∈ X . For each α < ℵk+1 let f kα ∈

∏

n∈X ℵn+1, f kα (k) = α, and f k

α (`) = 0 if` 6= k. Clearly, f k

α : α < ℵk+1 shows that ℵk+1 ∈ Sp↑(B,v).Assume k ∈ ω\X and assume on the contrary that a (gα)α<ℵk+1

is a ≤-increasingand ≤-unbounded sequence in

∏

n∈X ℵn+1. If n ∈ X ∩ k then (gα(n))α<ℵk+1is a nonde-

creasing sequence ℵn+1 so it is constant from an αn < ℵk+1.If n ∈ X\k then n> k and so gα(n) : α < ℵk+1 is bounded in ℵn+1, let αn < ℵn+1

be an upper bound of this set. The function g ∈∏

n∈X ℵn+1, g(n) = αn shows that(gα)α<ℵk+1

≤-bounded, a contradiction.

If (B,v) = (ωω,≤∗), (N,⊆), (M,⊆) or (I,⊆∗) for some Σ11 tall analytic P-ideal, then

Sp↑(B,v) is first order definable without parameters so we obtain the following:

Corollary 6.4.3. If X ⊆ ω is a real, then there is a ccc forcing extension in which X isfirst order definable without parameters.

It seems that this method (using spectrums of pre-ordered sets) does not work forcoding subsets of uncountable cardinals because it turned out (see [41]) that somenotions of spectrum have certain closedness properties so they cannot be arbitrary.

Chapter 7

Generalizations of thepseudo-intersection number

7.1 Intersection numbers of ideals

In the definition of the pseudo-intersection number one can consider other ideals thanFin. It was done in various ways in many papers:

• add∗(I) is the minimal cardinality of a family of elements of the dual filter I∗

which does not have a pseudo-intersection in I∗ (other notation: p(I∗), see [13]);

• cov∗(I) isthe minimal cardinality of a family of elements of the dual filter I∗

which does not have a pseudo-intersection (other notation: χp(I∗), see [13]);

• the minimal cardinality of a family A with the I-strong finite intersection prop-erty (every finite subfamily has an intersection outside I) without a set outside I

which is almost included (in the sense of I) in every member of A (pI, see [31];p(I), see [12]).

The aim of this chapter is to present another way of generalizing the pseudo-intersection number. This generalization is quite natural in the context of topologicalmotivations. All results in this chapter are from [8].

We will use one more pre-order on ideals:

Definition 7.1.1. (One-to-on order) For ideals I and J let I≤1-1 J if there is a one-to-onefunction f : ω→ω such that A∈ I⇒ f −1[A] ∈ J.

Just like classical pre-orders on ideals (see Section 2.3), the one-to-one order onBorel ideals is also absolute between transitive models M ⊆ N with ωN

1 ⊆ M .One can think about one more natural pre-order here, defined in the same way as

≤1-1 but with “bijection” instead of “one-to-one function”. However, the following factshows that it would not give us anything new.

Proposition 7.1.2. If J strictly extends Fin then I ≤1-1 J if, and only if there is a permu-tation g : ω→ω such that A∈ I⇒ g−1[A] ∈ J.

59

60 CHAPTER 7. GENERALIZATIONS OF THE PSEUDO-INTERSECTION NUMBER

Proof. The “if” part is trivial. Conversely, assume f is one-to-one and A∈ I⇒ f −1[A] ∈J. We can modify f on an infinite element B of J to be a permutation g such thatg (ω \ B)≡ f (ω \ B) and g[B] = f [B]∪ (ω \ ran( f )). Then g is as required.

Using the above proposition, I ≤1-1 J(6= Fin) means that I can be permuted into J

(by g−1). Clearly, Fin ≤1-1 J for each J, and J is maximal in this pre-order iff J is aprime ideal. There is no largest element in this pre-order because there are 2c manyprime ideals but only c many permutations. Furthermore,≤1-1 is c+-downward directedbecause the proof of the first part of Proposition 2.3.2 gives this stronger result.

Definition 7.1.3. For a pre-order (or simply a relation) v on ideals on ω the v-intersection number of an ideal I on ω is

pv(I) =min

χ(J): J 6v I

provided there is an ideal J such that J 6v I.The v-intersection number of a filter F is pv(F) = pv(F∗).

We will be interested in the Katetov-intersection number (which we will denote forsimplicity by pK(I)), the Katetov-Blass-intersection number (pKB(I)), and the one-to-one-intersection number (p1-1(I)).

We list some immediate facts.

Fact 7.1.4.

(a) p≤ p1-1(I)≤ pKB(I)≤ pK(I);

(b) if Iv J, then pv(I)≤ pv(J) for any pre-order v;

(c) pK(I) = p for any I which is not tall.

The last part of the above proposition explains why pK(I) and the cardinal invari-ants defined above generalize the pseudo-intersection number. In fact, we can indicatealso the generalization of the notion of pseudo-intersection itself in this context. As-sume I is an ideal on ω. For an injective sequence x = (xn) ∈ωω, the copy of I on x isthe ideal

I(x) =

A⊆ω: n ∈ω: xn ∈ A ∈ I

.

Let F be a family with SFIP (or simply a filter) and assume that I is an ideal. Wesay that an injective sequence x = (xn) ∈ ωω is an I-intersection of F if ω \ F ∈ I(x)for each F ∈ F. In other words, a set X = ran(x) is an I-intersection of F if we canreorder the elements of X in such a way that elements of F are in the copy of I∗ on therearranged X . Notice that x is a Fin-intersection of F iff ran(x) is a pseudo-intersectionof F. Plainly, p1-1(I) is the minimal cardinality of a family with SFIP and without anI-intersection.

The cardinal suppK(I): I is an ideal on ω ≤ c is the smallest cardinal κ such thatthere is no ≤K-upper bound of all ideals generated by at most κ many elements. Firstwe show that this supremum is determined by cardinal exponentiation.

7.2. THE CONVEX FRÉCHET-URYSOHN PROPERTY 61

Proposition 7.1.5. pK(I)≤ κ for each ideal I if and only if 2κ > 2ω.

Proof. First, we prove the “if” part. Let (A0α, A1

α): α < κ ⊆

[ω]ω2 be an indepen-

dent system, that is A1α = ω \ A0

α for each α, and if D ∈ [κ]<ω and f : D → 2 then

|⋂

Af (α)α : α ∈ D|=ω (see [7, Proposition 8.9]).

For an F ∈ 2κ let IF be the ideal generated by AF(α)α : α < κ. Observe that

χ(IF ) = κ for every F . Suppose for a contradiction that there is an ideal I such thatIF ≤K I for each F witnessed by gF ∈ ωω. Since 2κ > c there are distinct F0, F1 ∈ 2κ

such that gF0= gF1

. Let α be such that F0(α) 6= F1(α). Then AF0(α)α ∪ AF1(α)

α = ω.Consequently

g−1F0[AF0(α)α ]∪ g−1

F1[AF1(α)α ] = g−1

F0[AF0(α)α ]∪ g−1

F0[AF1(α)α ] = g−1

F0[AF0(α)α ∪ AF1(α)

α ] =ω

so ω ∈ I, a contradiction.Now, we prove the converse implication. Assume that 2κ = 2ω. Then the family of

ideals generated by at most κ elements has cardinality c. Using Proposition 2.3.2, thisfamily has a ≤K-upper bound I and so pK(I)> κ, a contradiction.

At last we show an easy upper bound for pKB(I) if I is meager:

Proposition 7.1.6. If I is meager, then pKB(I)≤ b.

Proof. It is enough to show that if I is meager and J ≤KB I, then J is also meagerbecause then we can use Theorem 2.1.4 (there is a nonmeager ideal of character b).

Assume that the partition (Pn) witnesses that I is meager, i.e. ∀ A ∈ I ∀∞ n ∈ ωPn * A (see Theorem 2.1.3), and assume f : ω → ω is finite-to-one and witnessesJ≤KB I. We can define a partition (P ′n) of ran( f ) into finite sets by recursion on n suchthat ∀ n ∈ ω ∃ k ∈ ω Pk ⊆ f −1[P ′n]. Then (P ′n) witnesses that J ran( f ) is meagerand it clearly implies that J is meager too because of the natural homeomorphismP(ω)→ P(ran( f ))×P(ω \ ran( f )).

The assumption on meagerness of the ideal and the use of finite-to-one functionsare necessary in the previous proposition because of part (a) and part (b) of Theorem7.3.4.

7.2 The convex Fréchet-Urysohn property

In this section we will show that the Katetov-intersection number has applications incertain topological and analytical considerations. All topological spaces in what followsare Hausdorff. The weight of a space X (denoted by w(X )) is the minimal cardinalityof a base of topology of X . Recall that a topological space is Fréchet-Urysohn (FU) if forevery subset A of this space and every x ∈ A there is a sequence in A converging to x .The definition of the pseudo-intersection number can be reformulated in topologicalterms: it is the smallest weight of a (locally) countable (even completely regular ornormal) space which is not FU (it is a special case of Theorem 7.2.2).

We can generalize the FU property for ideals using the notion of I-convergency. Asequence (xn) in a space X I-converges to x if

∀ U ⊆ X open

x ∈ U ⇒ n ∈ω: xn /∈ U ∈ I

.


Definition 7.2.1. Let I be an ideal on ω. A space X satisfies the I-Fréchet-Urysohn (I-FU) condition if for every A⊆ X and every x ∈ A there is a sequence in A I-convergingto x .

Clearly, if I is not tall, then the I-FU condition is equivalent to the (Fin-)FU condi-tion.

Theorem 7.2.2. pK(I) is the smallest weight of a countable space which is not I-FU.

Proof. Let F be a filter, χ(F) = pK(I), and F K I∗. Let X = ω∪ F be equipped with

the topology inherited from the Stone space of the Boolean algebra generated by F,that is

(a) subsets of ω are open;

(b) U ∪ F is an open neighborhood of F iff U ∩ω ∈ F.

Clearly, w(X ) = χ(F) and F ∈ω. We claim that there is no sequence (xn) in ω whichI-converges to F. Assume (xn) is a sequence inω and let f ∈ωω, f (n) = xn. Using theassumption F K I

∗ we deduce that there is a V ∈ F such that f −1[V ] /∈ I∗. Considerthe open neighborhood U = V ∪ F of F. Then n: xn /∈ U=ω \ f −1[V ] /∈ I so (xn)does not I-converge to F.

Conversely, let X be a countable space with w(X )< pK(I), let A⊆ X , and x ∈ A\ A.Then the family A∩ U : U is an open neighborhood of x forms a filter-base on A. LetF be the generated filter. Since χ(F) ≤ w(X ) we know that F ≤K I

∗ is witnessed by afunction f ∈ωω.

We claim that the sequence defined by xn = f (n) I-converges to x . Let U be anopen neighborhood of x . Then n: xn /∈ U=ω\ f −1[A∩U] ∈ I and we are done.

We can give another characterization of pK(I) in the special case when I= Z. Recallthat a completely regular space X can be seen as a closed subspace of the space of Borelprobability measures P(X ) on X with the weak∗ topology.

The subbase of this topology is given by the following sets:

U f ,ε(µ) =

(

ν ∈ P(X ):

∫

X

f dµ−∫

X

f dν

< ε

)

where µ ∈ P(X ), f ∈ Cb(X ) = bounded continuous real-valued functions on X , andε > 0. Recall that in this (so called weak∗) topology (µn) converges to µ if, and only if

∫

X

f dµn→∫

X

f dµ for every f ∈ Cb(X ).

The embedding X → P(X ) is given by x 7→ δx where δx is the Dirac-measureconcentrated on x . We will use the notation Aδ = δy : y ∈ A for A⊆ X .

Denote by conv(M) the convex hull of M for M ⊆ P(X ). We will be interested inconv(Xδ), i.e. in the probability measures with finite support.

Definition 7.2.3. We say that X satisfies the convex Fréchet-Urysohn condition if forevery A⊆ X , if x ∈ A then there is a sequence in conv(Aδ) which converges to δx .

7.2. THE CONVEX FRÉCHET-URYSOHN PROPERTY 63

In [39, Theorem 1] the following result was proved:

Theorem 7.2.4. Assume X is compact. A measure µ ∈ P(X ) is a weak∗ limit of measuresof finite support if and only if µ has a uniformly distributed sequence (xk) in X , that is

µ= limn→∞

1

n

∑

k<n

δxk.

Remark 7.2.5.

(1) In [39] this theorem was formulated only for compact spaces but the proof pre-sented there does not use the assumption of compactness and the assertion istrue for every (completely regular) topological space.

(2) It is clear from the proof of this theorem that if µn ∈ conv(Xδ) and µn→ µ, thenthe sequence (xk) can be chosen from

⋃

n<ω supp(µn).

(3) Using this theorem, a space X is convex FU iff if for every A ⊆ X , if x ∈ A thenthere is a sequence (xk) in A such that δx = limn→∞

1n

∑

k<nδxk.

Theorem 7.2.6. Assume X is completely regular. Then X satisfies the convex FU conditionif, and only if X satisfies the Z-FU condition.

Proof. Assume X satisfies the convex FU condition and let A⊆ X , x ∈ A\ A. Then, ac-cording to Remark 7.2.5 there is a sequence (xk) in A such that δx = limn→∞

1n

∑

k<nδxk.

We claim that (xk) Z-converges to x . Assume on the contrary that there is an openneighborhood U of x such that H = n: xn /∈ U /∈ Z. Using complete regularity of X ,there is a continuous f : X → [0, 1] such that f (x) = 0 but f xn : n ∈ H ≡ 1. Thenby the assumption on (xk) we have

∫

X

f d

1

n

∑

k<n

δxk

!

=1

n

∑

k<n

f (xk)→∫

X

f dδx = f (x) = 0

but 1n

∑

k<n f (xk)≥|H∩n|

nfor each n, a contradiction because H /∈ Z.

Conversely, assume that X is Z-FU and let A⊆ X , x ∈ A\A. Then there is a sequence(xk) in A which Z-converges to x .

We claim that δx = limn→∞1n

∑

k<nδxk. Let f ∈ Cb(X ), | f | ≤ c, ε > 0 and x ∈ U

be an open set such that | f (x)− f (y)| < ε for each y ∈ U . If H = k : xk ∈ U ∈ Z∗,then

f (x)−1

n

∑

k<n

f (xk)

=1

n

∑

k∈n∩H

f (x)− f (xk)

+∑

k∈n\H

f (x)− f (xk)

≤

≤1

n

|H ∩ n| · ε+ |n \H| · 2c

→ ε if n→∞.

Because ε was arbitrary, we are done.


So, in a sense we can call pK(Z) the convex pseudo-intersection number.The idea of the cardinal invariant pK(I) came from certain analytic considerations

contained in [9], where authors were exploring a problem if there is a Mazur spacewithout the Gelfand-Phillips property. The Gelfand-Phillips condition is widely usedin functional analysis. The Mazur property is a certain condition weaker than reflex-ivity used in the theory of Pettis integrability (for the detailed discussion about theseproperties, confront [9]). It is known that there is a Gelfand-Phillips space without theMazur property. It is still open if every Mazur space is Gelfand-Phillips.

In [9] the following question connected to the above considerations was raised.

Problem 7.2.7. Is there a minimally generated Boolean algebra A ⊆ P(ω) such thatno ultrafilter on A has a pseudo-intersection but for every ultrafilter F on A we haveF ≤K Z

∗?

In [9] it was shown that if there is such a Boolean algebra and this Boolean algebrais dense in P(ω), then there is a space which is Mazur but not Gelfand-Phillips 1. Brieflyspeaking, the minimal generation implies that every measure on A is in the sequentialclosure of measures of finite support. Since F ≤K Z

∗ for every ultrafilter F on A, everymeasure of finite support is a limit of measures finitely supported on ω (cf. Remark7.2.5 and Theorem 7.2.6 above) and so every measure is in the sequential closure ofmeasures finitely supported on ω. This property simplifies the form of functionals onthe space of measures on A and in this way it can be used to achieve Mazur property.In [9] it was also proved that if pK(Z)> h, then there is a Boolean algebra as describedabove.

In the next section we will show that consistently there is an ideal (unfortunately,not Z) with the above property (see Remark 7.3.8).

Question 7.2.8. Do there exist reasonable topological characterizations of pKB(I) orp1-1(I)?

7.3 Consistency results

Let us denote Sym(ω) the set of permutations on ω with the usual Polish space topol-ogy and let R be the following relation on ([ω]ω × [ω]ω)× Sym(ω): ⟨⟨X , Y ⟩, f ⟩ ∈ Riff | f [X ]∩ Y |=ω. We will need the following lemma:

Lemma 7.3.1.

[ω]ω × [ω]ω, R, Sym(ω)

BGT(Sym(ω),∈,M)⊥. In particular, after

adding a Cohen real there is an f ∈ Sym(ω) such that | f [X ] ∩ Y | = ω for all X , Y ∈[ω]ω ∩ V .

Proof. If X , Y ∈ [ω]ω and f ∈ Sym(ω), then let φ(X , Y ) =

f ∈ Sym(ω) : | f [X ]∩Y |<ω

and ψ( f ) = f . Then φ(X , Y ) is meager because its complement

Sym(ω)\φ(X , Y ) =⋂

n∈ω

f ∈ Sym(ω) : | f [X ]∩ Y | ≥ n

1In fact, in [9] it was shown for the case of one-to-one order but it can be immediately generalized.

7.3. CONSISTENCY RESULTS 65

and it is a countable intersection of dense open sets. It implies that (φ,ψ) :

[ω]ω ×[ω]ω, R, Sym(ω)

BGT(Sym(ω),∈,M)⊥.

Furthermore, it is easy to see that (Sym(ω),∈,M) ≡BGT(2ω,∈,M) so if a forcing

notion P adds a Cohen real, then P is (Sym(ω),∈,M)⊥-dominating and so it is

[ω]ω×[ω]ω, R, Sym(ω)

-dominating as well.

Corollary 7.3.2. d

[ω]ω × [ω]ω, R, Sym(ω)

≤ non(M) and cov(M) ≤ b

[ω]ω ×[ω]ω, R, Sym(ω)

.

Question 7.3.3. Are there any other inequalities between these and classical cardinalinvariants of the continuum?

It is well-known that if κ > ω, then VCκ |= b=ω1.

Theorem 7.3.4. Assume GCH. Then in VCω2 the following hold:

(a) there is a filter F with p1-1(F) =ω2;

(b) there is a meager filter G with pK(G) =ω2;

(c) pK(I) =ω1 for all Fσ ideals and analytic P-ideals.

Proof. (a): We interpret Cω2as the ω2 stage finite support iteration of C. Notice that

for every subfamily of [ω]ω in VCω2 of size ω1, a nice name of it appears already insome VCα for α < ω2. Additionally, there are ωω1

2 = ω2 such families in VCω2 so wecan fix an enumeration Fα : α < ω2 of all names of bases of filters of cardinality ω1in VCω2 in such a way that Fα ∈ VCα for each α.

Using Lemma 7.3.1 we know that in VCω2 there is a sequence ( fα)α<ω2of per-

mutations on ω such that fα ∈ VCα+1 for each α < ω2 and | fα[X ] ∩ Y | = ω for allX , Y ∈ [ω]ω ∩ VCα .

We show that

VCω2 |= “⋃

f ′′α [Fα]: α <ω2

forms a base of a filter.”

Indeed, consider α < β < ω2 and F ∈ Fα, G ∈ Fβ . Since fα[F] ∈ VCα+1 ⊆ VCβ

is infinite, the set fβ[G]∩ fα[F] is also infinite. By induction we can show that everyfinite subfamily of this family has infinite intersection.

Clearly, the filter F generated by this family satisfies p1-1(F) =ω2.(b) follows from part (a), Fact 7.1.4 and Proposition 2.3.3.(c): Now let Cω2

be the set of finite functions from ω2 ×ω to 2 ordered by re-verse inclusion. Let J be the ideal generated by the first ω1 Cohen reals, i.e. byc−1α [1]: α <ω1 where cα : ω→ 2 is the αth Cohen real. We show that J witnesses

part (c), i.e. for each I as in the theorem JK I.Case 1: Let I = Fin(ϕ) be an Fσ ideal. Assume G is a (V,Cω2

)-generic filter andf ∈ ωω ∩ V[G]. We can assume that ϕ( f −1[E]) < ∞ for each E ∈ [ω]<ω, becauseelse f cannot show any Katetov-reduction. There is a countable H ⊆ ω2 such thatboth ϕ Fin and f are in V[G∩CH] where CH is the Cohen forcing which adds Cohen


reals indexed by elements of H. If α ∈ω1 \H then cα is Cohen over V[G ∩CH] so it isenough to show that

Dn =¦

p ∈ C: ϕ

f −1[p−1[1]]

> n©

is dense in C for each n ∈ ω because then f cannot witness J ≤K I in the extension.Assume q ∈ C is defined on an initial segment. Because of our assumption on fwe know that f −1[|q|] ∈ Fin(ϕ) so we can choose a large enough ` > |q| such thatϕ( f −1[` \ |q|]) > n. Define p ∈ C by p |q| = q and p [|q|,`) ≡ 1. Then p ≤ q andp ∈ Dn.

Case 2: Assume I = Exh(ϕ) is an analytic P-ideal, ‖ω‖ϕ = 1. Similarly to theprevious case, we have an f ∈ ωω, we can assume that ‖ f −1[E]‖ϕ = 0 for eachE ∈ [ω]<ω, and it is enough to show that

Dn =¦

p ∈ C: ϕ

f −1[p−1[1]] \ n

> 0.5©

is dense in C for each n. Assume q ∈ C is defined on an initial segment. We can choosea large enough ` > |q| such that ϕ( f −1[` \ |q|] \ n) > 0.5. Let p be chosen as in Case1. Then p ≤ q and p ∈ Dn.

We list here some related questions:

Problem 7.3.5.

• Is pK(I)≤ b for each analytic (P-)ideal I?

• Is p1-1(I) = pKB(I) for each ideal I?

• Is p< p1-1(I) (or at least p< pKB(I)) consistent for some meager (or even analytic(P-)) ideal I? Also, for the purposes described in Section 7.2, the consistency ofh< pK(Z) is particularly interesting.

• Is pKB(I)< b (or at least p1-1(I)< b) consistent for some tall ideal I?

In Proposition 7.1.5 we showed that 2κ > 2ω implies pK(I)≤ κ and thus pKB(I)≤ κfor each ideal I, and that the converse implication for the Katetov-order also holds.Now, we show that ∀ I pKB(I)≤ω1 does not imply 2ω1 > 2ω.

Theorem 7.3.6. It is consistent with ZFC that 2ω1 = 2ω is arbitrary large and theKatetov-Blass-order is not upward directed on filters generated by ω1 sets. In particular,pKB(F)≤ω1 for each filter F.

Proof. Let 2ω = 2ω1 be arbitrary. We will construct two filters F and G generatedby ⊆∗-descending sequences Xα : α < ω1 and Yα : α < ω1 inductively in a modelobtained by anω1 stage finite-support iteration of σ-centered forcing notions (Pα, Qβ :α≤ω1,β < ω1).

It is well-known that a ≤ c-step finite-support iteration of σ-centered forcing no-tions is σ-centered, in particular ccc, so it does not collapse cardinals. It will be trivialthat |Pω1

|= c so VPω1 |=“2ω = (2ω)V and 2ω1 = (2ω1)V .”

7.3. CONSISTENCY RESULTS 67

At the αth stage (in VPα) we have initial segments Xξ : ξ < α and Yξ : ξ < α.Let X ′α and Y ′α be pseudo-intersections of these sequences. We want to add Xα ∈ [X ′α]

ω

and Yα ∈ [Y ′α]ω such that | f −1[Xα] ∩ g−1[Yα]| < ω for each pairs ( f , g) ∈ VPα of

finite-to-one functions from ω to ω. Then in the final extension F and G cannot havea common upper bound in ≤KB.

Let X = X ′α and Y = Y ′α, and let us denote FO the set of all finite-to-one functionsfrom ω to ω. Let Q= Qα be the following forcing notion: (s, t,%) ∈Q iff

(1) s ∈ [X ]<ω and t ∈ [Y ]<ω;

(2) % is a finite partial function from FO× FO to ω such that

∀ ( f , g) ∈ dom(%) | f −1[s]∩ g−1[t]| ≤ %( f , g).

Define the order in the following way: (s, t,%)≤ (s′, t ′,%′) iff s ⊇ s′, t ⊇ t ′, dom(%)⊇dom(%′), and %( f , g)≤ %′( f , g) for each ( f , g) ∈ dom(%′).

Clearly, it is a partial order. It is also σ-centered: fix s ∈ [X ]<ω, t ∈ [Y ]<ω, andconsider conditions (s, t,%i) ∈Q for i < n ∈ω. Let % be the following partial function:dom(%) =

⋃

dom(%i): i < n and %( f , g) = min%i( f , g): i < n, ( f , g) ∈ dom(%i).Then (s, t,%)≤ (s, t,%i) for every i < n.

Let A and B be Q-names for the union of the first and respectively second coordi-nates of the conditions in the generic filter. We claim that these sets can serve as Xαand Yα.

Q“A is infinite”: The set Dn = p ∈ Q: |sp| > n is dense in Q for each n (wherep = (sp, t p,%p)) because if p ∈Q is arbitrary, then sp can be extended by any elementsof the infinite set ω \

⋃

f ′′[g−1[t p]]: ( f , g) ∈ dom(%p)

.Similarly, Q“B is infinite.”At last, we have to show that E f ,g = p ∈ Q: ( f , g) ∈ dom(%p) is dense in Q for

each pair f , g ∈ FO. Any p ∈ Q can be extended by adding ( f , g) to dom(%p) andchoosing %( f , g) to be large enough.

Problem 7.3.7. Is it consistent with ZFC that 2ω1 = 2ω and pKB(I) ≤ ω1 (or at leastp1-1(I) ≤ ω1) for each ideal I but ≤KB (respectively ≤1-1) is upward directed on idealsgenerated by ω1 elements?

Note that if it is possible for ≤1-1, then in such a model c ≥ ω3: It is easy to seethat if ≤1-1 is upward directed on ideals generated by ω1 sets, then any ω2 ideals withcharacter ω1 have an ≤1-1-upper bound. If there would be only 2ω = 2ω1 = ω2 manyideals with character ω1, then they would form a ≤1-1-bounded set, i.e. there wouldbe an ideal I with p1-1(I)>ω1.

Remark 7.3.8. In [9, Theorem 7.4] the authors proved that consistently there is aBoolean algebra A ⊆ P(ω) such that all ultrafilters on A are meager but none of themhas a pseudo-intersection. Using Theorem 7.3.4 (a) and the fact that h = ω1 in theCohen model, we can mimic this proof to show a similar result. Namely, we can provethat, consistently, there is an ideal I and a Boolean algebra A ⊆ P(ω) such that foreach ultrafiler F on A, there is no pseudo-intersection of F but F ≤K I

∗.


7.4 Permuting MAD families into ideals

The pseudo-intersection number is not the only cardinal invariant which can be gen-eralized in the way presented above. We can easily define the analog of the towernumber:

t1-1(I) =min

χ(⟨T⟩fr): T is a tower and ⟨T⟩fr 1-1 I∗,

where I is an ideal and ⟨T⟩fr is the filter generated by T. As in the case of p1-1, we havet1-1(Fin) = t. However, in general this cardinal may not be well-defined, i.e. maybe thefamily of all (filters generated by) towers are ≤1-1-bounded. E.g. consider the filter Ffrom Theorem 7.3.4 (a). Since in the Cohen-model there are no towers of characterω2 [Kunen, unpublished] and every filter generated by ω1 sets is ≤1-1-below F, everytower is ≤1-1-below F, and t1-1(F∗) in undefined.

Similarly, we can define the cardinal invariant a1-1(I) analogous to the almost-disjointness number a:

Definition 7.4.1. Let I be an ideal on ω. An almost disjoint family A is I-maximal ifid(A)1-1 I.

Using Proposition 7.1.2, if I 6= Fin then an AD family is I-maximal iff it cannot bepermuted into I.

Clearly, an AD family is Fin-maximal iff it is a MAD family. Furthermore, if I ≤1-1 J

and an AD family is J-maximal, then it is I-maximal as well. In particular, each I-maximal AD family is a MAD family. From now on we will use the phrase “I-maximalMAD family.” It is trivial that if I is not tall, then each MAD family is I-maximal.

As before, leta1-1(I) =min

χ(⟨A⟩id): A is I-maximal

.

This section is devoted to study when this cardinal invariant is well-defined, i.e. whenthere is an I-maximal MAD family.

Proposition 7.4.2. add∗(I) = c implies that there is an I-maximal MAD family.

Proof. Fix an enumeration ( fα : ω ≤ α < c) of injective sequences of natural numbers.We will construct the desired MAD family inductively. Start with a disjoint partition(An) of ω into infinite sets and assume we have constructed all Aξ’s for ξ < α < c.

If for some ξ < α we have f −1α [Aξ] /∈ I, then take Aα = Aξ.

If not, then consider the family f −1α [Aξ]: ξ < α ⊆ I. Using the assumption

add∗(I) = c we can find a set B ∈ I such that f −1α [Aξ] ⊆

∗ B for every ξ < α. LetAα = f ′′α [ω \ B]. Then Aξ : ξ≤ α is an AD family, and f −1

α [Aα] ∈ I∗.In this way we will construct an AD family A = Aα : α < c such that id(A) 6≤1-1

I.

We will show that under Martin’s Axiom for σ-centered posets (i.e. p = c, see [6]or [7, Theorem 7.12]) there are I-maximal MAD families for each Fσ ideal and analyticP-ideal I. Recall that this axiom does not imply that add(N) = c (see [25, 522S]) andadd∗(I) = add(N) for a lot of tall analytic P-ideals (e.g. for tall summable and talldensity ideals, see Corollary 4.2.5 and Remark 4.2.6).

7.4. PERMUTING MAD FAMILIES INTO IDEALS 69

Theorem 7.4.3. Let I be a tall Fσ ideal or a tall analytic P-ideal. Then MA(σ-centered)implies that there is an I-maximal MAD family.

Proof. Let I be a tall analytic P-ideal and fix an enumeration ( fα : ω ≤ α < c) ofinjective sequences of natural numbers. As in the proof of Proposition 7.4.2, we willconstruct the desired MAD family inductively. Start with a disjoint partition (An) of ωinto infinite sets and assume we have constructed all Aξ’s for ξ < α < c.

As before, if for some ξ < α we have f −1α [Aξ] /∈ I, then let Aα = Aξ. If not, consider

the almost disjoint family A of sets f −1α [Aξ] ∈ I for ξ < α.

We claim that MA(σ-centered) implies that A can be extended to an AD family bya set C from I+. It is enough because then we can choose Aα = f ′′α [C] and proceed asin the proof of Proposition 7.4.2.

Let I be a tall Fσ ideal or a tall analytic P-ideal and assume that A ⊆ I is an ADfamily. Then we have to find a σ-centered forcing notion P(A) such that in VP(A) thefamily A can be extended by an I-positive set.

Let P(A) be the natural forcing notion that extends A with a new element. Namely,let p = (np, sp,Bp) ∈ P(A) iff np ∈ ω, sp ⊆ n, and Bp ∈ [A]<ω. We say that p ≤ q iffnp ≥ nq, sp ∩ nq = sq, and (sp \ sq)∩

⋃

Bq = ;.It is easy to see that P(A) is σ-centered and that the sets Dk = p ∈ P(A): |sp| >

k and DA = p ∈ P(A): A ∈ Bp are dense in P(A) for each k ∈ ω and A ∈ A.Consequently, if S is a P(A)-name such that P(A) S =

⋃

sp : p ∈ G (where G is thecanonical name of the generic filter), then P(A) S ∈ [ω]ω and P(A) |S ∩ A| < ω foreach A∈A.

We have to show that P(A) S ∈ I+. We have two cases:Case 1: I = Fin(ϕ) is a tall Fσ ideal. We will show that P(A) ϕ(S) = ∞. It is

enough to prove that Ek = p ∈ P(A): ϕ(sp) > k is dense in P(A) for each k ∈ ω.Fix a p ∈ P(A). Since

⋃

Bp ∈ I we have ϕ

ω \

np ∪⋃

Bp = ∞ so by the lscproperty of ϕ we can find a finite F ⊆ ω \

np ∪⋃

Bp such that ϕ(F) > k. If q =(max(F) + 1, sp ∪ F,Bp), then q ∈ Ek and q ≤ p. We are done.

Case 2: I = Exh(ϕ) is a tall analytic P-ideal, ‖ω‖ϕ = 1. We will show that P(A)‖S‖ϕ = 1. It is enough to prove that Hεk = p ∈ P(A): ϕ(s

p \ k) > ε is dense in P(A)for each ε < 1 and k ∈ω. To see this, fix a condition p ∈ P(A). Since A ⊆ I and ‖ · ‖ϕis subadditive we conclude that

ϕ

ω \

np ∪ k ∪⋃

Bp

≥

ω \

np ∪ k ∪⋃

Bp

ϕ= 1

so by the lsc property of ϕ we can find a finite F ⊆ ω \

np ∪ k ∪⋃

Bp such thatϕ(F) > ε. If q = (max(F) + 1, sp ∪ F,Bp), then q ∈ Hεk and q ≤ p. The proof iscomplete.

We finish with some related questions:

Question 7.4.4.

• Does there exist an I-maximal MAD family for a tall (analytic) ideal I in ZFC?

• Is it consistent with ZFC that there is no I-maximal MAD family for some (nice)I?


• Is it consistent with ZFC+¬CH that there are I-maximal MAD families for eachideal I?

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