Covering Property Axiom CPA - Department of Mathematicskcies/prepF/BookCPA/00DecCPA/00DecCPA.pdf · Covering Property Axiom CPA KrzysztofCiesielski DepartmentofMathematics, WestVirginiaUniversity,

Covering Property Axiom CPA

Krzysztof CiesielskiDepartment of Mathematics, West Virginia University,

Morgantown, WV 26506-6310, USA

e-mail: K [email protected]

web page: http://www.math.wvu.edu/~kcies

Janusz PawlikowskiDepartment of Mathematics, University of Wroc+law

pl. Grunwaldzki 2/4, 50-384 Wroc+law, Poland

e-mail: [email protected]

and

Department of Mathematics, West Virginia University,

Morgantown, WV 26506-6310, USA

e-mail: [email protected]

Draft of December 18, 2000.

We still plan to make considerable changes to this manuscript.

Comments and list of typos are welcome!

2 Covering Property Axiom CPA December 19, 2000

Contents

I CPA for Sacks forcing: a combinatorial core of theiterated perfect set model 5

Preface to Part I 7Overview of Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1 Axiom CPAcube and its consequences: properties (A)–(E) 151.1 Perfectly meager sets, universally null sets, and continuous im-

ages of sets of cardinality continuum . . . . . . . . . . . . . . . . 181.2 cof(N ) = ω1 and total failure of Martin’s Axiom . . . . . . . . . 201.3 Some consequences of cof(N ) = ω1: Blumberg theorem, strong

measure zero sets, magic set, and cofinality of Boolean algebras . 221.4 Selective ultrafilters and the reaping numbers r and rσ . . . . . . 271.5 On a convergence of subsequences of real-valued functions . . . . 291.6 Remarks on a form and consistency of the axiom CPAcube. . . . 32

2 Games and axiom CPAgamecube 35

2.1 CPAgamecube and disjoint coverings . . . . . . . . . . . . . . . . . . . 36

2.2 MAD families and numbers a and r . . . . . . . . . . . . . . . . . 372.3 Remark on a form of CPAgame

cube . . . . . . . . . . . . . . . . . . . . 39

3 Prisms and axioms CPAgameprism and CPAprism 41

3.1 Fusion Lemma for prisms . . . . . . . . . . . . . . . . . . . . . . 443.2 Basic duality lemma for prisms . . . . . . . . . . . . . . . . . . . 483.3 CPAprism, additivity of s0, and more on (A) . . . . . . . . . . . . 503.4 Remarks on a form and consistency of axioms CPAgame

prism andCPAprism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 CPAprism and coverings with smooth functions 554.1 Chapter overview; properties (H∗) and (Q) . . . . . . . . . . . . 564.2 Proof of Proposition 4.1.3 . . . . . . . . . . . . . . . . . . . . . . 604.3 Proposition 4.2.1: a generalization of a theorem of Morayne . . . 634.4 Theorem 4.1.6: on cov (Dn, Cn) < c . . . . . . . . . . . . . . . . . 664.5 Examples related to cov operator . . . . . . . . . . . . . . . . . . 67

3


5 Applications of the axiom CPAgameprism 71

5.1 Nice Hamel basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Selective and crowded ultrafilters; numbers i and u . . . . . . . . 75

6 CPA and properties (F∗) and (G) 876.1 cov(s0) = c and ω1-intersections of open sets . . . . . . . . . . . . 896.2 Surjections onto nice sets must be continuous on big sets . . . . . 896.3 Sums of Darboux and continuous functions . . . . . . . . . . . . 916.4 Remark on a form of CPAsec

cube. . . . . . . . . . . . . . . . . . . . 96

7 CPA in the Sacks model 97

II CPA for other forcings 101

Notation 103

Bibliography 107

Index 112

Part I

CPA for Sacks forcing: acombinatorial core of theiterated perfect set model

5

Preface to Part I

Many interesting mathematical properties, especially concerning real analysis,are known to be true in the iterated perfect set (Sacks) model, while they arefalse under the continuum hypothesis. However, the proofs that these facts areindeed true in this model are usually very technical and involve heavy forcingmachinery. In this manuscript we extract a combinatorial principle, an axiomsimilar to the Martin axiom, that is true in the model and show that this axiomimplies the above mentioned properties in a simple “mathematical” way. Theproofs are essentially simpler than the original arguments.

To follow all but the last chapter of this manuscript only a moderate knowl-edge of set theory is required. No forcing knowledge is necessary.

Overview of Part I

The iterated perfect set model, known also as the iterated Sacks model, is amodel of the set theory ZFC in which continuum c = ω2 and many of theconsequences of the continuum hypothesis CH fail. In this paper we will describea combinatorial axiom of the form similar to Martin axiom which holds in theiterated perfect set model and represents a combinatorial core of this model –it implies all the “general mathematical statements” which are known (to us)to be true in this model.

It should be mentioned here that our axiom is more an axiom schema withthe perfect set forcing being a “build-in” parameter. A similar axioms hold alsofor several other forcings and their description will be a subject of Part II ofthis manuscript. In this text, however, we will concentrate only on the axiomassociated with the iterated perfect set model. This is dictated by two reasons:the axiom has the simplest form in this particular model; the iterated perfect setmodel was the most studied from the class of forcing models we are interestedin, we had a good supply of statements against which we could test the power ofour axiom. In particular, we used for this purpose the statements listed belowas (A)–(H). The citations in the parentheses refer to the proofs that a givenproperty holds in the iterated perfect set model. For the definitions see the endnext section.

7


(A) (Miller [67]) For every subset S of R of cardinality c there exists a (uni-formly) continuous function f : R → [0, 1] such that f [S] = [0, 1].

(B) (Miller [67]) Every perfectly meager set S ⊂ R has cardinality less than c.

(C) (Laver [59]) Every universally null set S ⊂ R has cardinality less than c.

(D) (Folklore, see e.g. [69] or [4, p. 339]) The cofinality of the measure idealN is less than c.

(E) (Baumgartner, Laver [6]) There exist selective ultrafilters on ω and anysuch ultrafilter is generated by less than c many sets.

(F) (Balcerzak, Ciesielski, Natkaniec [2]) There is no Darboux Sierpinski-Zygmund function f : R → R, that is, for every Darboux function f : R → R

there is a subset Y of R of cardinality c such that f � Y is continuous.

(G) (Steprans [84]) For every Darboux function g: R → R there exists a con-tinuous nowhere constant function f : R → R such that f + g is Darboux.

(H) (Steprans [85]) The plane R2 can be covered by less than c many sets eachof which is a graph of a partial function defined and differentiable on aperfect subset of either horizontal or vertical axis.

The counterexamples, under CH, for (B) and (C) are Luzin and Sierpinski sets.They have been constructed in [62]1 and [77], respectively. The negation of (A)is witnessed by either Luzin or Sierpinski set, as noticed in [77, 78]. The coun-terexamples, under CH, for (F) and (G) can be found in [2] and [55], respectively.The fact that (D), (E), and (H) are false under CH is obvious.

The manuscript is organized as follows. Since our main axiom, which wecall the Covering Property Axiom and denote by CPA, requires some extradefinitions which are unnecessary for most of applications, we will introduce theaxiom in several approximations, from the easiest to state and use to the mostpowerful but more complicated. All the versions of the axiom will be formulatedand discussed in the main body of the chapters. The sections that follow containonly the consequences of the axioms. In particular, most of the sections can beomitted in the reading without causing the difficulty in following the rest of thematerial.

Thus, we start in Chapter 1 with a formulation of the simplest form ofour axiom, CPAcube, which is based on a natural notion of a cube in a Polishspace. In Section 1.1 we show that CPAcube implies the properties (A)–(C),while Section 1.2 is devoted to the proofs that CPAcube implies the property(D) (i.e., cof(N ) = ω1) and the following fact, known as the total failure ofMartin’s Axiom

(I) c > ω1 and for every non-trivial ccc forcing P there exists ω1-many densesets in P such that no filter intersects all of them.

1Constructed also a year earlier by Mahlo [63].

Covering Property Axiom CPA December 19, 2000 9

The consistency of (I) was first proved by Baumgartner [5] in a model obtainedby adding Sacks reals side-by-side. In Section 1.3 we will present some con-sequences of cof(N ) = ω1 that seems to be related to the iterated perfect setmodel. In particular we prove that cof(N ) = ω1 implies that

(J) c > ω1 and there exists a Boolean algebra B of cardinality ω1 which is nota union of strictly increasing ω-sequence of subalgebras of B.

The consistency of (J) was first proved by Just and Koszmider [51] in a modelobtained by adding Sacks reals side-by-side, while Koppelberg [56] showed thatMartin’s Axiom contradicts (J). In Section 1.4 we show that CPAcube impliesthat every selective ultrafilter is generated by ω1 sets (i.e., the second part ofthe property (E)) and that

(K) r = ω1,

where r is the reaping (or refinement) number, that is,

r = min {|W|:W ⊂ [ω]ω & ∀A ∈ [ω]ω ∃W ∈ W (W ⊂ A or W ⊂ ω \A)} .

In Section 1.5 we will prove that CPAcube implies the following version of atheorem of Mazurkiewicz [65].

(L) For each Polish spaceX and uniformly bounded sequence 〈fn:X → R〉n<ωof Borel measurable functions there are the sequences: 〈Pξ: ξ < ω1〉 ofcompact subsets of X and 〈Wξ ∈ [ω]ω: ξ < ω1〉 such that X =

⋃ξ<ω1

Pξand for every ξ < ω1:

〈fn � Pξ〉n∈Wξis a monotone uniformly convergent sequence of

uniformly continuous functions.

We will also show that CPAcube+“∃ selective ultrafilter on ω” implies the fol-lowing variant of (L):

(M) Let X an arbitrary set and fn:X → R be a sequence of functions suchthat the set {fn(x):n < ω} is bounded for every x ∈ X. Then there arethe sequences: 〈Pξ: ξ < ω1〉 of subsets of X and 〈Wξ ∈ F : ξ < ω1〉 suchthat X =

⋃ξ<ω1

Pξ and for every ξ < ω1:

〈fn � Pξ〉n∈Wξis monotone and uniformly convergent.

It should be noted here that a result essentially due to Sierpinski (see Exam-ple 1.5.2) implies that (M) is false under the Martin’s Axiom. The last sectionof this chapter consists of the remarks on a form and consistency of CPAcube.In particular, we note there that CPAcube is false in a model obtained by addingSacks reals side-by-side.

In Chapter 2 we revise slightly the notion of a cube and introduce a cube-game GAMEcube, a covering game of length ω1 which is a foundation for ournext (stronger) variant of the axiom, CPAgame

cube . In Section 2.1, as its application,we will show that CPAgame

cube implies that


(N) c > ω1 and for every Polish space there exists a partition of X into ω1

disjoint closed nowhere dense measure zero sets.

In Section 2.2 we will show that CPAgamecube implies that

(O) c > ω1 and there exists a family F ⊂ [ω]ω of cardinality ω1 which issimultaneously maximal almost disjoint and reaping.

Chapter 3 begins with a definition of a prism, which is a generalization ofa notion of cube in a Polish space. This notion, perhaps the most importantnotion of this text, is then used to our next generation of the axioms, CPAgame

prism

and CPAprism, which are prism (stronger) counterparts of axioms CPAgamecube and

CPAcube. Since the notion of a prism is rather unknown, in the first two sectionsof Chapter 3 we develop the tools that will help to deal with them (Section 3.1)and prove for them the main duality property that distinguishes them formcubes (Section 3.2). In the last section of this chapter we discuss some ap-plication of CPAprism. In particular, we will prove that CPAprism implies thefollowing generalization of the property (A)

(A∗) there exists a family G of uniformly continuous functions from R to [0, 1]such that |G| = ω1 and for every S ∈ [R]c there exists a g ∈ G withg[S] = [0, 1].

We will also show that CPAprism implies that

(P) add(s0), the additivity of the ideal s0, is equal to ω1 < c.

Chapters 4 and 5 deal with the applications of the axioms CPAprism andCPAgame

prism, respectively. Chapter 4 contains a deep discussion of a problem ofcovering R2 and Borel functions from R to R by continuous functions of differ-ent smooth levels. In particular, we show that CPAprism implies the followingstrengthening of the property (H)

(H∗) there exists a family F of less then continuum many C1 functions from R

to R (i.e., differentiable functions with continuous derivatives) such thatR2 is covered by functions from F and their inverses

as well as the following covering property for the Borel functions

(Q) for every Borel function f : R → R there exists a family F of less thancontinuum many “C1” functions (i.e., differentiable functions with contin-uous derivatives, where derivative can be infinite) whose graphs cover thegraph of f .

We will also examine which functions can be covered by less than c many Cnfunctions for n > 1 and give the examples showing that all covering theoremsdiscussed are the best possible.

Chapter 5 concentrates on several specific applications of CPAgameprism. Thus

in Section 5.1 we will show that CPAgameprism implies


(R) there is a family H of ω1 pairwise disjoint perfect subsets of R such thatH =

⋃H is a Hamel basis, that is, a linear basis of R over Q

and show that the following two properties are the consequences of (R):

(S) there exists a non-measurable subset X of R without the Baire propertywhich is N ∩M-rigid, that is, such that X�(r +X) ∈ N ∩M for everyr ∈ R;

(T) there exists a function f : R → R such that for every h ∈ R the differencefunction ∆h(x) = f(x + h) − f(x) is Borel; however, for every α < ω1

there is an h ∈ R such that ∆h is not of Borel class α.

Implication CPAgameprism=⇒(S) answers a question related to the work of Cichon,

Jasinski, Kamburelis, and Szczepaniak [19]. The implication CPAgameprism=⇒(T)

shows that the resent construction of such a function from CH due to Filipow andRecRlaw [40] (and answering a question of Laczkovich [58]) can be also repeatedwith a help of our axiom. In Section 5.2 we deduce from CPAgame

prism that everyweakly selective ideal on ω can be extended to a maximal selective ideal. Inparticular, the first part of the condition (E) holds and u = rσ = ω1, where u isthe smallest cardinality of the base for a non-principal ultrafilter on ω. We willalso prove that CPAgame

prism implies that

(U) there exists a family F ⊂ [ω]ω of cardinality ω1 which is simultaneouslyindependent and splitting.

In particular i = ω1, where i is the smallest cardinality of an infinite maximalindependent family. We will finish this chapter with the prove that CPAgame

prism

implies that

(V) there exists a non-principal ultrafilter on Q which is crowded.

In Chapter 6 we formulate the most general form of our axiom, CPA, andshow that it implies all the other versions of the axiom. In Section 6.1 we willconclude from CPA that

(W) cov(s0) = c

and

(X) if U = {Uξ: ξ < ω1} is a family of open sets in C and |⋂U| = c then

⋂U

contains a perfect set.

In Section 6.2 we show that CPA implies the following two generalizations ofthe property (F):

(F∗) for an arbitrary function h from a subset S of a Polish space X ontoa Polish space Y there exists a uniformly continuous function f from asubset of X into Y such that |f ∩ h| = c

and


(F′) for any function h from a subset S of R onto a perfect subset of R thereexists a function f ∈ “C∞perf” such that |f ∩ h| = c and f can be extendedto a function f ∈ “C1(R)” such that either either f ∈ C1 or f is anautohomeomorphism of R with f−1 ∈ C1.

In Section 6.3 that (A)&(F∗)=⇒(G). In particular, (G) follows from CPA.In Chapter 7 we show that CPA holds in the iterated perfect set model.

Preliminaries

Our set theoretic terminology is standard and follows that of [4], [21], or [57].If a, b ∈ R and a < b then (b, a) = (a, b) will stand for the open interval{x ∈ R: a < x < b}. Similarly [b, a] = [a, b] is an appropriate closed interval. ACantor set 2ω will be denoted by a symbol C. In this paper we use term Polishspace for a complete separable metric space without isolated points. For aPolish space X symbol Perf(X) will stand for a collection of all subsets of Xhomeomorphic to a Cantor set C; as usual C(X) will stand for the family of allcontinuous functions from X into R.

A function f : R → R is Darboux if a conclusion of the intermediate valuetheorem holds for f or, equivalently, when f maps every interval onto an interval;f is a Sierpinski-Zygmund function if its restriction f � Y is discontinuous forevery subset Y of R of cardinality c; f is nowhere constant if it is not constanton any non-trivial interval.

A set S ⊂ R is perfectly meager if S ∩P is meager in P for every perfect setP ⊂ R, and S is universally null provided for every perfect set P ⊂ R the setS ∩ P has measure zero with respect to every probability measure on P .

For an ideal I on a set X its cofinality is defined by

cof(I) = min{|B|:B generates I}

and its covering as

cov(I) = min{|B|:B ⊂ I &⋃B = X}.

Symbol N will stand for the ideal of Lebesgue measure zero subsets of R. Fora fixed Polish space X the ideal of its meager subsets will be denoted by M,and we will use the symbol s0 (or s0(X)) to denote the σ-ideal of Marczewski’ss0-sets, that is,

s0 = {S ⊂ X: (∀P ∈ Perf(X))(∃Q ∈ Perf(X)) Q ⊂ P \ S}.

For an ideal I on a set X we use the symbol I+ to denote its coideal, that is,I+ = P(X) \ I.

For an ideal I on ω containing all finite subsets of ω we will use the followinggeneralized selectivity terminology. We say (see Farah [38]) that an ideal I isselective provided for every sequence F0 ⊃ F1 ⊃ · · · of sets from I+ there exists


an F∞ ∈ I+ (called a diagonalization of this sequence) with the property thatF∞ \ {0, . . . , n} ⊂ Fn for all n ∈ F∞. Notice that this definition agrees with thedefinition of selectivity given by Grigorieff in [44, p. 365]. (The ideals selectivein the above sense Grigorieff calls inductive but he also proves [44, cor. 1.15] thatthe inductive ideals and the ideals selective in his sense are the same notions.)

For A,B ⊂ ω we will write A ⊆∗ B when |A\B| < ω. A set D ⊂ I+ is densein I+ provided for every B ∈ I+ there exists an A ∈ D such that A ⊆∗ B; setD is open in I+ if B ∈ D provided there is an A ∈ D such that B ⊆∗ A. ForD = 〈Dn ⊂ I+:n < ω〉 we say that F∞ ∈ I+ is a diagonalization of D providedF∞ \{0, . . . , n} ∈ Dn for every n < ω. Following Farah [38] we say that an idealI on ω is semiselective provided for every sequence D = 〈Dn ⊂ I+:n < ω〉 ofdense and open subsets of I+ the family of all diagonalizations of D is dense inI+.

Following Grigorieff [44, p. 390] we say that I is weakly selective (or weakselective) provided for every A ∈ I+ and f :A → ω there exists a B ∈ I+ suchthat f � B is either one-to-one or constant. (Farah in [38, sec. 2] terms suchideals as having Q+-property. Note also that Baumgartner and Laver in [6]call such ideals selective, despite the fact that they claim to use Grigorieff’sterminology from [44].)

We have the following implications between these notions. (See Farah [38,sec. 2].)

I is selective =⇒ I is semiselective =⇒ I is weakly selective

All these notions represent different generalizations of the properties of the ideal[ω]<ω. In particular, it is easy to see that [ω]<ω is selective.

We say that an ideal I on a countable set X is selective (weakly selective)provided it is such upon an identification of X with ω via an arbitrary bijection.A filter F on a countable set X is selective (semiselective, weakly selective)provided so is its dual ideal I = P(X) \ F .

It is important to note that a maximal ideal (or an ultrafilter) is selective ifand only if it is weakly selective. This follows, for example, directly from thedefinitions of these notions as in Grigorieff [44]. Recall also that the existenceof selective ultrafilters cannot be proved in ZFC. (This is the case since everyselective ultrafilter is a p-point, while Shelah proved that there are models withno p-points, see e.g. [4, thm. 4.4.7].)


Chapter 1

Axiom CPAcube and itsconsequences: properties(A)–(E)

For a Polish space X we will consider Perf(X), the family of all subsets ofX homeomorphic to a Cantor set C, as ordered by inclusion. Thus, a familyE ⊂ Perf(X) is dense in Perf(X) provided for every P ∈ Perf(X) there exists aQ ∈ E such that Q ⊂ P .

All different versions of our axiom will be more or less of the form

if E ⊂ Perf(X) is appropriately dense in Perf(X) then some portionE0 of E covers almost all of X in a sense that |X \

⋃E0| < c.

If the word “appropriately” in the above is ignored, then it implies the followingstatement.

Naıve-CPA: If E is dense in Perf(X) then |X \⋃E| < c.

It is a very good candidate for our axiom in a sense that it implies all theproperties we are interested in. It has, however, one major flaw – it is false!This is the case since S ⊂ X \

⋃E for some dense set E in Perf(X) provided

for each P ∈ Perf(X) there is a Q ∈ Perf(X) such that Q ⊂ P \ S.

This means that the family G of all sets of the form X \⋃E , where E is dense

in Perf(X), coincides with the σ-ideal s0 of Marczewski’s sets, since G is clearlyhereditary. Thus we have

s0 ={X \

⋃E : E is dense in Perf(X)

}. (1.1)

However, it is well known (see e.g. [68, thm. 5.10]) that there are s0-sets ofcardinality c. Thus, our Naıve-CPA “axiom” cannot be consistent with ZFC.

15


In order to formulate the real axiom CPAcube we need the following termi-nology and notation. A subset C of a product Cη of the Cantor set is said tobe a perfect cube (or just cube) if C =

∏n∈η Cn, where Cn ∈ Perf(C) for each

n. For a fixed Polish space X let Fcube stand for the family of all continuousinjections from a perfect cube C ⊂ Cω onto a set P from Perf(X). We considereach function f ∈ Fcube from C onto P as a coordinate system imposed on P .We say that P ∈ Perf(X) is a cube if we consider it with (implicitly given)witness function f ∈ Fcube onto P , and Q is a subcube of a cube P ∈ Perf(X)provided Q = f [C], where f ∈ Fcube is a witness function for P and C is asubcube of the domain of f .

We say that a family E ⊂ Perf(X) is Fcube-dense (or cube-dense) in Perf(X)provided every cube P ∈ Perf(X) contains a subcube Q ∈ E . More formally,E ⊂ Perf(X) is Fcube-dense provided

∀f ∈ Fcube ∃g ∈ Fcube (g ⊂ f & range(g) ∈ E). (1.2)

It is easy to see that the notion of Fcube-density is a generalization of a notionof density as defined in the first paragraph of this chapter:

if E is Fcube-dense in Perf(X) then E is dense in Perf(X). (1.3)

On the other hand, the converse implication is not true, as shown by the fol-lowing simple example.

Example 1.0.1 Let X = C×C and let E be the family of all P ∈ Perf(X) suchthat either

• all vertical sections Px = {y ∈ C: 〈x, y〉 ∈ P} of P are countable, or

• all horizontal sections P y = {x ∈ C: 〈x, y〉 ∈ P} of P are countable.

Then E is dense in Perf(X), but it is not Fcube-dense in Perf(X).

With these notions in hand we are ready to formulate our axiom CPAcube.

CPAcube: c = ω2 and for every Polish space X and every Fcube-dense familyE ⊂ Perf(X) there is an E0 ⊂ E such that |E0| ≤ ω1 and |X \

⋃E0| ≤ ω1.

The proof that CPAcube is consistent with ZFC (it holds in the iteratedperfect set model) will be presented in the next chapters. In the reminder ofthis chapter we will take a closer look at CPAcube and its consequences.

In what follows we consider Fcube as ordered by the inclusion. ThereforeF ⊂ Fcube is dense in Fcube provided for every f ∈ Fcube there exists a g ∈ Fwith g ⊂ f .

Proposition 1.0.2 CPAcube is equivalent to the following.

CPA0cube: c = ω2 and for every Polish space X and every dense subfamily F ofFcube there is an F0 ∈ [F ]≤ω1 such that |X \

⋃g∈F0

range(g)| ≤ ω1.


Proof. First notice that

E ⊂ Perf(X) is Fcube-dense if and only if Fcube(E) is dense in Fcube, (1.4)

where Fcube(E) = {g ∈ Fcube: range(g) ∈ E}.Now, to see that CPA0

cube implies CPAcube take an Fcube-dense family E ⊂Perf(X). Then, using CPA0

cube with F = Fcube(E), we can find an appropri-ate F0 ∈ [F ]≤ω1 . Clearly E0 = {range(g): g ∈ F0} satisfies the conclusion ofCPAcube.

To see the other implication, take a dense subfamily F of Fcube and letE = {range(g): g ∈ F}. Then, by (1.4), E is Fcube-dense in Perf(X), since F ⊂Fcube(E). By CPAcube we can find an E0 ∈ [E ]≤ω1 such that |X \

⋃E0| < c. For

each E ∈ E0 take an fE ∈ F such that range(fE) = E. Then F0 = {fE :E ∈ E0}satisfies the conclusion of CPA0

cube.

Condition CPA0cube has a form which closer to our main axiom CPA than

CPAcube. Also CPA0cube does not require a new notion of Fcube-density. So why

bother with CPAcube at all? The reason is that the consequences presented inthis chapter follows more naturally from CPAcube than from CPA0

cube. More-over, from CPAcube it is also clearer that the axiom describes, in fact, a propertyof perfect subsets of X, rather than of some coordinate functions.

Note that if Fall stands for the family of all injections from perfect subsetsof Cω into X then we can use schema (1.2) to define a notion of F-density ofE ⊂ Perf(X) for any family F ⊂ Fall: E ⊂ Perf(X) is F-dense provided

∀f ∈ F ∃g ∈ F (g ⊂ f & range(g) ∈ E). (1.5)

In particular, it is easy to see that a family E ⊂ Perf(X) is dense in Perf(X) ifand only if E is Fall-dense in Perf(X).

It is also worth to notice that in order to check that E is Fcube-dense itis enough to consider in condition (1.2) only functions f defined on the entirespace Cω, that is

Fact 1.0.3 E ⊂ Perf(X) is Fcube-dense if and only if

∀f ∈ Fcube, dom(f) = Cω, ∃g ∈ Fcube (g ⊂ f & range(g) ∈ E). (1.6)

Proof. To see this, let Φ be the family of all bijections h = 〈hn〉n<ω be-tween perfect subcubes

∏n∈ωDn and

∏n∈ω Cn of Cω such that each hn is a

homeomorphism between Dn and Cn. Then

f ◦ h ∈ Fcube for every f ∈ Fcube and h ∈ Φ with range(h) ⊂ dom(f). (1.7)

Now take an arbitrary f :C → X from Fcube and choose an h ∈ Φ mapping Cω

onto C. Then f = f ◦ h ∈ Fcube maps Cω into X and, using (1.6), we can findg ∈ Fcube such that g ⊂ f and range(g) ∈ E . Then g = f � h[dom(g)] satisfies(1.2).


Next, let us consider

scube0 =

{X \

⋃E : E is Fcube-dense in Perf(X)

}(1.8)

= {S ⊂ X:∀ cube P ∈ Perf(X) ∃ subcube Q ⊂ P \ S} .

It can be easily shown, in ZFC, that scube0 forms a σ-ideal. However, we will

not use this fact in this text in that general form. This is the case, since we willusually assume that CPAcube holds while CPAcube implies the following strongerfact.

Proposition 1.0.4 If CPAcube holds then scube0 = [X]≤ω1 .

Proof. It is obvious that CPAcube implies scube0 ⊂ [X]<c. The other inclusion

is always true and it follows from the following simple fact.

Fact 1.0.5 [X]<c ⊂ scube0 ⊂ s0 for every Polish space X.

Proof. Choose S ∈ [X]<c. In order to see that S ∈ scube0 note that the

family E = {P ∈ Perf(X):P ∩ S = ∅} is Fcube-dense in Perf(X). Indeed, iffunction f :Cω → X is from Fcube then there is a perfect subset P0 of C which isdisjoint with the projection π0(f−1(S)) of f−1(S) into the first coordinate. Thenf

[∏i<ω Pi

]∩S = ∅, where Pi = C for all 0 < i < ω. Therefore, f

[∏i<ω Pi

]∈ E .

Thus, S ⊂ X \ E ∈ scube0 .

The inclusion scube0 ⊂ s0 follows immediately from (1.1), (1.8), and (1.3).

1.1 Perfectly meager sets, universally null sets,and continuous images of sets of cardinalitycontinuum

An important quality of the ideal scube0 , and so the power of the assumption

scube0 = [X]<c, is well depicted by the following fact.

Proposition 1.1.1 If X is a Polish space and S ⊂ X does not belong to scube0

then there exist a T ∈ [S]c and a uniformly continuous function h from T onto C.

Proof. Take an S as above and let f :Cω → X be a continuous injection suchthat f [C]∩S %= ∅ for every cube C. Let g:C→ C be a continuous function suchthat g−1(y) is perfect for every y ∈ C. Then h0 = g ◦ π0 ◦ f−1: f [Cω] → C isuniformly continuous. Moreover, if T = S ∩ f [Cω] then h0[T ] = C since

T ∩ h−10 (y) = T ∩ f [π−1

0 (g−1(y))] = S ∩ f [g−1(y)× C× C× · · ·] %= ∅

for every y ∈ C.


Corollary 1.1.2 Assume scube0 = [X]<c for a Polish space X. If S ⊂ X has

cardinality c then there exists a uniformly continuous function f :X → [0, 1]such that f [S] = [0, 1].

In particular, CPAcube implies property (A).

Proof. If S is as above then, by CPAcube, S /∈ scube0 . Thus, by Proposi-

tion 1.1.1 there exists a uniformly continuous function h from a subset of Sonto C. Let h:X → [0, 1] be an uniformly continuous extension of h and let gmaps continuously C onto [0, 1]. Then f = h ◦ g is as desired.

For more on the property (A) see also Corollary 3.3.5.It is worth to note here that the function f in Corollary 1.1.2 cannot be

required to be either monotone or in the class “D1” of all functions havingfinite or infinite derivative at every point. This follows immediately from thefollowing proposition, since each function which is either monotone or “D1”belongs to the Banach class

(T2) ={f ∈ C(R): {y ∈ R: |f−1(y)| > ω} ∈ N

}.

(See [41] or [75, p. 278].)

Proposition 1.1.3 There exists, in ZFC, an S ∈ [R]c such that [0, 1] %⊂ f [S]for every f ∈ (T2).

Proof. Let {fξ: ξ < c} be an enumeration of all functions from (T2) whichrange contains [0, 1]. Construct by induction a sequence 〈〈sξ, yξ〉: ξ < c〉 suchthat for every ξ < c

(i) yξ ∈ [0, 1] \ fξ[{sζ : ζ < ξ}] and |f−1ξ (yξ)| ≤ ω.

(ii) sξ ∈ R \({sζ : ζ < ξ} ∪

⋃ζ≤ξ f

−1ζ (yζ)

).

Then the set S = {sξ: ξ < c} is as required since yξ ∈ [0, 1] \ fξ[S] for everyξ < c.

Theorem 1.1.4 If S ⊂ R is either perfectly meager or universally null thenS ∈ scube

0 .In particular, CPAcube =⇒ “scube

0 = [R]<c” =⇒ “(B) & (C).”

Proof. Take an S ⊂ R which is either perfectly meager or universally null andlet f : Cω → R be a continuous injection. Then S∩f [Cω] is either meager or nullin f [Cω]. Thus G = Cω \ f−1(S) is either comeager or of full measure in Cω.Hence the theorem follows immediately from the following fact.

Claim 1.1.5 If G ⊂ Cω is either comeager or of full measure in Cω then itcontains a perfect cube

∏i<ω Pi.

Claim 1.1.5 follows easily, by induction on coordinates, from the followingwell known fact.


For every comeager (full measure) subset H of C×C there are perfectset P ⊂ C and a comeager (full measure) subset H of C such thatP × H ⊂ H.

The category version is easy and can be found in [53, Exercise 19.3]. (Its versionfor R2 is also proved, for example, in [32, condition ('), p. 416].) The measureversion follows easily form the fact that

for every full measure subset H of [0, 1]× [0, 1] there are perfect setP ⊂ C and a positive inner measure subset H of [0, 1] such thatP × H ⊂ H

which is proved by Eggleston [37] and, independently, by Brodskiı [10].

1.2 cof(N ) = ω1 and total failure of Martin’s Ax-iom

We will start with the proof that CPAcube implies that for every non-trivial cccforcing P there exists ω1-many dense sets in P such that no filter intersects allof them. This follows immediately from the theorem below. The consistency ofthis fact with c > ω1 was first proved by Baumgartner [5] in a model obtainedby adding Sacks reals side-by-side.

Theorem 1.2.1 If CPAcube holds then every compact ccc topological space isa union of ω1 nowhere dense sets.

Proof. Let cBA stand for complete Boolean algebra. First notice that in orderto prove the theorem it is enough to show that for every ccc cBA B there is afamily of ω1-many maximal antichains in B such that no filter in B intersectsall of them.

Next notice that it is enough to prove this fact only for countable generatedccc cBA’s. Indeed, every ccc cBA B contains a complete countably generatedBoolean subalgebra B0, which is still ccc. Now, every maximal antichain in B0

is also a maximal antichain in B. So, if A = {Aξ: ξ < ω1} is a family of maximalantichains in B0 such that no filter in B0 intersects all of them, then A witnessthe same fact for B.

Next let B be the algebra of Borel subsets of C = 2ω and note that it is a freecountably generated cBA, with free generators Ai = {s ∈ C: s(i) = 0}. Thus,by Loomis-Sikorski theorem (see [82, p. 117] or [61]), every countably generatedccc cBA B0 (which is just a ccc sigma complete Boolean algebra) is isomorphicto the quotient algebra B/I, where I is some σ-ideal of Borel sets.

It follows that we need only to consider cBA’s of the form B/I, where I issome σ-ideal of Borel sets containing all singletons. To prove that such algebrahas ω1 maximal antichains as desired, it is enough to prove that

• C is a union of ω1 sets from I.


But this follows easily from CPAcube, since any cube P in C contains a subcubeQ ∈ I: any cube P can be partitioned into c-many disjoint subcubes and, bythe ccc property of I, only countable many of them can be outside I.

Next we show that CPAcube implies that cof(N ) = ω1. So, under CPAcube,all cardinals from Cichon’s diagram (see e.g. [4]) are equal to ω1.

Let CH be the family of all subsets∏n<ω Tn of ωω such that Tn ∈ [ω]≤n+1

for all n < ω. We will use the following characterization.

Proposition 1.2.2 (Bartoszynski [4, thm. 2.3.9])

cof(N ) = min{|F|:F ⊂ CH &

⋃F = ωω

}.

Lemma 1.2.3 Family CH is Fcube-dense in Perf(ωω).

Proof. Let f :Cω → ωω be a continuous injection. By (1.6) it is enough to finda cube C in Cω such that f [C] ∈ CH . So, let {〈ni,mi〉: i < ω} be an enumerationof ω × ω and for i < ω let

Si = {s ∈ Cω: s(nj)(mj) = 0 for all i ≤ j < ω}.

Notice that |Si| = 2i for every i < ω. By induction on i < ω we will define setsXi = {xs ∈ Cω: s ∈ Si}. For the unique s ∈ S0 we choose xs ∈ Cω arbitrarily.Now, if Xi−1 is constructed for some 0 < i < ω we extend it to Xi as follows.

For s′ ∈ Si \ Si−1 let s ∈ Si−1 be such that s(nj)(mj) = s′(nj)(mj) for allj %= i. We choose xs′ %= xs such that

(i) xs′(n) = xs(n) for all n %= ni, and

(ii) f(xs) � 2i = f(xs′) � 2i.

The choice satisfying (ii) can be made since f is continuous.Note that we have defined a uniformly continuous mapping g, s (→ xs, from

S =⋃i<ω Si onto X = {xs: s ∈ S}. So g can be extended continuously to a

function g from Cω = cl(S) onto C = cl(X). Note that, by (i), C is a cubesince s(n) = s′(n) implies g(s)(n) = g(s′)(n) for all s, s′ ∈ Cω and n < ω. Also,by (ii),

f(x)(k) ∈ Tk = {f(xs)(k): s ∈ Si}for all x ∈ C and 2i ≤ k < 2i+1. So, f [C] ⊂

∏n<ω Tn ∈ CH .

Corollary 1.2.4 If CPAcube holds then cof(N ) = ω1.

Proof. By CPAcube and Lemma 1.2.3 there exists an F ∈ [CH ]≤ω1 such that|ωω \

⋃F| ≤ ω1. This and Proposition 1.2.2 imply cof(N ) = ω1.

As a last application in this section let us consider the following coveringnumber connected to a Blumberg’s theorem (see next section) and studied byF. Jordan in [49]. Here B1 stands for the class of all Baire class 1 functionsf : R → R.


• cov(B1,Perf(R)) is the smallest cardinality of F ⊂ B1 such that for eachP ∈ Perf(R) there is an f ∈ F with f � P not continuous.

Jordan proves also [49, thm. 7(a)] that cov(B1,Perf(R)) is equal to the coveringnumber of the space Perf(R) (considered with the Hausdorff metric) by theelements of some σ-ideal Zp and notices [49, thm. 17(a)] that every compact setC ∈ Perf(R) contains a denseGδ subset which belongs to Zp. So, by Claim 1.1.5,the elements of Perf(R) ∩ Zp are Fcube-dense in Perf(R). Thus

Corollary 1.2.5 CPAcube implies that cov(B1,Perf(R)) = cov(Zp) = ω1.

1.3 Some consequences of cof(N ) = ω1: Blum-berg theorem, strong measure zero sets, ma-gic set, and cofinality of Boolean algebras

In this section we will show some consequences of cof(N ) = ω1 (so also ofCPAcube) which are related to the Sacks model.

In 1922 Blumberg [9] proved that for every f : R → R there exists a densesubset D of R such that f � D is continuous. This theorem sparked a lot of dis-cussion and generalizations, see e.g. [22, pp. 147–150]. In particular, Shelah [76]showed that there is a model of ZFC in which for every f : R → R there is anowhere meager subset D of R such that f � D is continuous. The dual mea-sure result, that is the consistency of a statement for every f : R → R there is asubset D of R of positive outer Lebesgue measure such that f � D is continuous,has been also recently established by RosRlanowski and Shelah [73]. Below wenote that each of these properties contradicts CPAcube. (See also in [28].)

Theorem 1.3.1 Let I ∈ {N ,M}. If cof(I) = ω1 then there exists an f : R → R

such that f � D is discontinuous for every D ∈ P(R) \ I.

Proof. We will assume that I = N , the proof for I = M being essentiallyidentical.

Let {Nξ ⊂ R2: ξ < ω1} be a family cofinal in the ideal of null subsets of R2

and for each ξ < ω1 let

N∗ξ = {x ∈ R: (Nξ)x /∈ N},

where (Nξ)x = {y ∈ R: 〈x, y〉 ∈ Nξ}. By Fubini theorem each N∗ξ is null. For

each x ∈ N∗ξ \

⋃ζ<ξN

∗ζ we choose f(x) so that

f(x) /∈⋃ζ<ξ

(Nζ)x.

Then function f is as desired.Indeed, if f � D is continuous for some D ⊂ R then f � D is null in R2. In

particular, there exists a ξ < ω1 such that f � D ⊂ Nξ. But this means thatD ⊂

⋃ζ≤ξN

∗ζ .


Note that essentially the same proof works if we assume only that cof(I) isequal to the covering number cov(I) of I.

Corollary 1.3.2 Assume CPAcube. Then there exists an f : R → R such thatif f � D is continuous then D ∈ N ∩M.

Proof. By CPAcube we have cof(N ) = cof(M) = ω1. Let fN and fM be fromTheorem 1.3.1 constructed for the ideals N and M, respectively. Let G ⊂ R bea dense Gδ of measure zero and put f = [fM � G]∪ [fN � (R \G)]. Then this fis as desired.

Although CPAcube implies that all cardinals from Cichon’s diagram are assmall as possible, this does not extend to all possible cardinal invariants. Forexample we note that CPAcube refutes Borel conjecture.

Corollary 1.3.3 If CPAcube holds then there exists an uncountable strong mea-sure zero set.

Proof. This follows from the fact (see e.g. [4, thm. 8.2.8]) that cof(N ) = ω1

implies the existence of an uncountable strong measure zero set.

Recall that a set M ⊂ R is a magic set (or set of range uniqueness) if forevery different nowhere constant functions f, g ∈ C(R) we have f [M ] %= g[M ].It has been proved by Berarducci and Dikranjan [7, thm. 8.5] that a magic setexists under CH, while Ciesielski and Shelah [31] constructed a model with nomagic set. It is relatively easy to see that if M is magic then M is not meagerand f [M ] %= [0, 1] for every f ∈ C(R). (See [17, cor. 5.15 and thm. 5.6(5)].)So CPAcube implies there is no magic set of cardinality c. On the other handit was noticed by Todorcevic (see [17, p. 1097]) that in the iterated perfect setmodel there is a magic set (clearly of cardinality ω1). We like to note here thatthe same is implied by CPAcube. (See also in [28].)

Proposition 1.3.4 If the cofinality cof(M) of the ideal M of meager sets isequal ω1 then there exists a magic set.

In particular, CPAcube implies that there exists a magic set.

Proof. An uncountable set L ⊂ R is a 2-Lusin set provided for every disjointsubsets {xξ: ξ < ω1} and {yξ: ξ < ω1} of L, where the enumerations are one-to-one, the set of pairs {〈xξ, yξ〉: ξ < ω1} is not a meager subset of R2. In [17, prop.4.8] it was noticed that every ω1-dense 2-Lusin set is a magic set. It is also astandard and easy diagonal argument that cof(M) = ω1 implies the existenceof a ω1-dense 2-Lusin set. (The proof presented in [86, prop. 6.0] works alsounder the assumption cof(M) = ω1.) So,

CPAcube =⇒ cof(N ) = ω1 =⇒ cof(M) = ω1 =⇒ “there is a magic set.”


Recall also that the existence of a magic set for the class “D1” can be provedin ZFC. This follows from [18, thm. 3.1], since every function from “D1” belongsto the class (T2). (Compare also [18, cor. 3.3 and 3.4].)

For an infinite Boolean algebra B its cofinality cof(B) is defined as the leastinfinite cardinal number κ such that B is a union of strictly increasing sequenceof type κ of subalgebras of B. Koppelberg [56] proved that Martin’s Axiomimplies that cof(B) = ω for every Boolean algebra B with |B| < c while Justand Koszmider [51] proved that in the model obtained by adding at least ω2

Sacks reals side-by-side there exists a Boolean algebra B of cardinality ω1 suchthat cof(B) = ω1 < c. (See also [36].)

We like to notice here that this follows also form CPAcube, since we have:

Theorem 1.3.5 cof(N ) = ω1 implies that there exists a Boolean algebra B ofcardinality ω1 such that cof(B) = ω1.

The proof of Theorem 1.3.5 presented below follows the argument givenin [27]. It will be based on the following lemma.

Lemma 1.3.6 If cof(N ) = ω1 then for every infinite countable Boolean algebraA there exists a family {aξn ∈ A:n < ω & ξ < ω1} with the following properties.

(i) aξn ∧ aξm = 0 for every n < m < ω and ξ < ω1.

(ii) For every increasing sequence 〈An:n < ω〉 of proper subalgebras of A withA =

⋃n<ω An there exists a ξ < ω1 such that aξn /∈ An for all n < ω.

Proof. In the argument that follows every sequence A = 〈An:n < ω〉 as in (ii)will be identified with a function fA = f ∈ ωA for which f−1(n) = An\

⋃i<nAi.

We will denote the set of all such functions by X. Also, let {bn:n < ω} be anenumeration of A and for each n < ω let Bn be a finite algebra generated by{bi: i < n}. Thus, A =

⋃i<nBi.

Since cof(N ) = ω1, the dominating number

d = min {|K|:K ⊂ ωω & (∀f ∈ ωω)(∃g ∈ K)(∀n < ω) f(n) < g(n)}

is equal to ω1. (See e.g. [4].) So, there exists a dominating family K ⊂ ωω

of cardinality ω1. We can also assume that the sequences in K are strictlyincreasing and that for every g ∈ K function g defined by g(n) =

∑i≤n g(i) also

belongs to K.Next notice that for every f ∈ X there exist d = 〈dk: k < ω〉 ∈ K and

r = 〈rk: k < ω〉 ∈ K such that for every k < ω

(a) f(b) < rk for all b ∈ Bk; and

(b) there are disjoint b0, . . . , b2k ∈ Bdkwith rdk−1 < f(b0) < · · · < f(b2k).

Indeed, the existence of r satisfying (a) follows directly from the definition ofa dominating family. Moreover, since all algebras An = f−1({0, . . . , n}) are


proper, for every number d < ω there exist disjoint b0, . . . , b2d ∈ A such thatrd < f(b0) < · · · < f(b2d). Let h ∈ ωω be such that b0, . . . , b2d ∈ Bd+h(d)

for every d < ω and let g ∈ K be a function dominating h. Then d = g is asrequired.

The above implies, in particular, that for every f ∈ X there are d, r ∈ K suchthat f satisfies (b) and the sequence f r = 〈f � (Bdk

\Bdk−1): k < ω〉 belongs to

X(d, r) =∏k<ω

(rdk)Bdk

\Bdk−1 .

Now, since cof(N ) = ω1, by Proposition 1.2.2 (applied to∏k<ω ω

Bdk\Bdk−1

in place of ωω) we can find an ω1-covering of X(d, r) by sets T of the form∏k<ω Tk, where Tk ∈

[ωBdk

\Bdk−1

]≤k+1

for all k < ω. Since the total number

of these sets T (for different d, r ∈ K) is equal to ω1, to finish the proof it isenough to show that for any such T there is one sequence 〈an:n < ω〉 satisfying(i) and such that (ii) holds for every for every A = 〈An:n < ω〉 for which f rAbelongs to T and fA satisfies (b).

So, let T be as above and let T ∗ be the set of all functions fA satisfying(b) for which f rA ∈ T . By induction on k < ω we will construct a sequence〈ck ∈ Bdk

\Bdk−1 : k < ω〉 such that

(∗) f(ck) > rdk≥ k for every k < ω and f ∈ T ∗.

So fix a k < ω and let {fi: i < k} be such that

{fi � Bdk\Bdk−1 : i < k} = {f � Bdk

\Bdk−1 : f ∈ T ∗} ⊂ Tk.

We show inductively that for every m < k

there is a c ∈ Bdksuch that fj(c) > rdk

for all j ≤ m. (1.9)

So, fix an m < k and let a ∈ Bdksuch that fj(ac) = fj(a) > rdk

for allj < m. If fm(a) > rdk

then c = a satisfies property (1.9). Thus, assumethat fm(ac) = fm(a) ≤ rdk

. By (b) we can find b0, . . . , b2k ∈ Bdksuch that

rdk−1 < fm(b0) < · · · < fm(b2k). By Pigeon Hole Principle we can find anI ∈ [{0, . . . , 2k}]k+1 and a b ∈ {a, ac} such that fm(b∧ bi) = fm(bi) for all i ∈ I.Without loss of generality we can assume that I = {0, . . . , k} and b∧ bi = bi forall i ≤ k. Then

fm(bc ∨ bi) > rdkfor all i ≤ k.

Moreover, for every j < m there is at most one ij ≤ k for which

fj(bc ∨ bij ) ≤ rdk

since for different i, i′ ≤ k we have fj((bc ∨ bi)∧ (bc ∨ bi′)) = fj(bc) > rdk. Thus,

by Pigeon Hole Principle, there is an i ≤ k such that c = bc ∨ bi satisfies (1.9).This finishes the proof of (∗).

Clearly the sequence 〈ck ∈ Bdk\Bdk−1 : k < ω〉 satisfies (ii) for every A with

fA ∈ T ∗. Thus, we need only to modify it to get also the condition (i).To do it, use the fact that


rdk< f(ck) < rdk+1 for every k < ω and f ∈ T ∗

to construct the sequences: ω = I0 ⊃ I1 ⊃ · · · of infinite subsets of ω, increasing〈kj ∈ Ij : j < ω〉, and 〈c∗kj

∈ {ckj, cckj

}: j < ω〉 such that for every j < ω

• f(akj∧ cl) > rdl

for every f ∈ T ∗ and l > kj with l ∈ Ij ,

where akj = c∗k0∧· · ·∧c∗kj

. Then the sequence 〈akj : j < ω〉 is a strictly decreasingsequence satisfying (ii) and it is now easy to see that by putting aj = akj ∧ ackj+1

we obtain the desired sequence.

Proof of Theorem 1.3.5. Algebra B we construct will be a subalgebra ofthe algebra P(ω) of all subsets of ω. First, let K ⊂ ωω be a dominating familywith |K| = ω1 and fix a partition {Dk: k < ω} of ω into infinite subsets.

For every sequence a = 〈an:n < ω〉 of pairwise disjoint subsets of ω andk < ω put a∗k =

⋃{an:n ∈ Dk}. In addition, for every h ∈ K we put

ah =⋃{anh(k): k < ω}

where nh(k) = min{n ∈ Dk:n > max{h(k), k}}. We also put

F (a) = {a∗k: k < ω} ∪ {ah:h ∈ K} ∈ [P(ω)]≤ω1 .

Next, we will construct an increasing sequence 〈Bξ ∈ [P(ω)]ω1 : ξ ≤ ω1〉 ofsubalgebras of P(ω) aiming for B = Bω1 . Thus, we choose B0 as an arbitrarysubalgebra of P(ω) with |B0| = ω1 and for limit ordinal numbers λ ≤ ω1 we putBλ =

⋃ξ<λBξ. The algebra Bξ+1 is formed from Bξ in the following way.

Let {bη: η < ω1} be an enumeration of Bξ and for η < ω let Aξη be asubalgebra of Bξ generated by {bζ : ζ < η}. For each such algebra we applyLemma 1.3.6 to find the sequences aγ = 〈aγn:n < ω〉, γ < ω1, satisfying (i) and(ii) and let

G(Aξη) =⋃γ<ω1

F (aγ).

Bξ+1 is defined as the algebra generated by Bξ ∪⋃η<ω1

G(Aξη). This finishesthe construction of B.

Clearly, |B| = ω1. To prove that cof(B) = ω1 it is enough to show that B isnot a union of an increasing sequence B = 〈Bn:n < ω〉 of proper subalgebras.So, by way of contradiction, assume that such a sequence B exists. For everyn < ω choose bn ∈ B \Bn and find ξ, η < ω1 such that {bn:n < ω} ⊂ Aξη. Thenthe algebras An = Bn ∩ Aξη form an increasing sequence of proper subalgebrasof A = Aξη. Thus, one of the sequences aγ satisfies (ii) for A. So, if we putaγ = a = 〈an:n < ω〉 we conclude that {a∗k: k < ω} ∪ {ah:h ∈ K} ⊂ B. Letf(k) = min{n < ω: a∗k ∈ Bn} and let h ∈ K be such that f(k) < h(k) for allk < ω.

The final contradiction is obtained by noticing that ah cannot belong toany Bk. Indeed, if ah ∈ Bk for some k, then ah ∩ a∗k = anh(k) belongs to


Bmax{f(k),k}, since a∗k ∈ Bf(k). But max{f(k), k} ≤ max{h(k), k} < nh(k) sowe get anh(k) ∈ Bnh(k) contradicting the fact that anh(k) belongs to A \ Anh(k),which is disjoint with Bnh(k).

1.4 Selective ultrafilters and the reaping num-bers r and rσ

In this section, which in part is based on [29], we will show that CPAcube impliesthat every selective ultrafilter is generated by ω1 sets and that the reapingnumber r is equal to ω1. The actual construction of a selective ultrafilter willrequire a stronger version of the axiom and will be done in Theorem 5.2.3.

We will use here the terminology introduced in the Preliminaries Section. Inparticular, recall that the ideal [ω]<ω of finite subsets of ω is semiselective.

The most important combinatorial fact for us concerning semiselective idealsis the following property. (See theorem 2.1 and remark 4.1 in [38].) This is ageneralization of a theorem of Laver [60] who proved this fact for the idealI = [ω]<ω.

Proposition 1.4.1 (Farah [38]) Let I be a semiselective ideal on ω. For everyanalytic set S ⊂ Cω × [ω]ω and every A ∈ I+ there exist a B ∈ I+ ∩ P(A)and a perfect cube C in Cω such that C × [B]ω is either contained in or disjointwith S.

With this fact in hand we can prove the following theorem.

Theorem 1.4.2 Assume that CPAcube holds. If I is a semiselective ideal thenthere is a family W ⊂ I+, |W| ≤ ω1, such that for every analytic set A ⊂ [ω]ω

there is a W ∈ W for which either [W ]ω ⊂ A or [W ]ω ∩A = ∅.

Proof. Let S ⊂ C × [ω]ω be a universal analytic set, that is such that thefamily {Sx:x ∈ C} (where Sx = {y ∈ [ω]ω: 〈x, y〉 ∈ S}) contains all analyticsubsets of [ω]ω. (See e.g. [48, lemma 39.4].) In fact, we will take S such thatfor any analytic set A in [ω]ω

|{x ∈ C:Sx = A}| = c. (1.10)

For this particular set S consider the family E of all Q ∈ Perf(C) for which thereexists a WQ ∈ I+ such that

Q× [WQ]ω is either contained in or disjoint with S. (1.11)

Note that, by Proposition 1.4.1, family E is Fcube-dense in Perf(C). So, byCPAcube, there exists an E0 ⊂ E , |E0| ≤ ω1, such that |C \

⋃E0| < c. Let

W = {WQ:Q ∈ E0}.

It is enough to see that this W is as required.


Clearly |W| ≤ ω1. Also, by (1.10), for an analytic set A ⊂ [ω]ω there exista Q ∈ E0 and an x ∈ Q such that A = Sx. So, by (1.11), {x} × [WQ]ω is eithercontained in or disjoint with {x} × Sx = {x} ×A.

Recall (see e.g. [4] or [89]) that a family W ⊂ [ω]ω is a reaping familyprovided

∀A ∈ [ω]ω ∃W ∈ W (W ⊂ A or W ⊂ ω \A).The reaping (or refinement) number r is defined as

r = min {|W|:W ⊂ [ω]ω & ∀A ∈ [ω]ω ∃W ∈ W (W ⊂ A or W ⊂ ω \A)} .Also, a number rσ is defined as the smallest cardinality of a family W ⊂ [ω]ω

such that for every sequence 〈An ∈ [ω]ω:n < ω〉 there exists a W ∈ W such thatfor every n < ω either W ⊆∗ An or W ⊆∗ ω \ An. (See [15] or [89].) Clearlyr ≤ rσ.

Corollary 1.4.3 If CPAcube holds then for every semiselective ideal I thereexists a family W ⊂ I+, |W| ≤ ω1, such that for every A ∈ [ω]ω there is aW ∈ W for which either W ⊆∗ A or W ⊆∗ ω \A.

In particular, CPAcube implies that r = ω1 < c.

Proof. Family W from Theorem 1.4.2 works: since [A]ω is analytic in [ω]ω

there exists a W ∈ W such that either [W ]ω ⊂ [A]ω or [W ]ω ∩ [A]ω = ∅.Note also that CPAcube implies the second of the property (E).

Corollary 1.4.4 If CPAcube holds then every selective ultrafilter F on ω isgenerated by a family of size ω1 < c.

Proof. If F is a selective ultrafilter on ω then I = P(ω)\F is a selective idealand I+ = F . Let W ⊂ I+ = F be as in Corollary 1.4.3. Then W generates F .

Indeed, if A ∈ F then there exists a W ∈ W such that either W ⊂ A orW ⊂ ω \ A. But it is impossible that W ⊂ ω \ A since then we would have∅ = A ∩W ∈ F .

As mentioned above, in Theorem 5.2.3 we will prove that some version ofour axiom implies that there exists a selective ultrafilter on ω. In particular,the assumption of the next corollary are implied by such a version of our axiom.

Corollary 1.4.5 If CPAcube holds and there exists a selective ultrafilter F onω then rσ = ω1 < c.

Proof. Let W ∈ [F ]≤ω1 be a generating family for F . We will show that itjustifies rσ = ω1. Indeed, take a sequence 〈An ∈ [ω]ω:n < ω〉. For every n < ωlet A∗

n belongs to F ∩ {An, ω \An}. Since F is selective, there exists an A ∈ Fsuch that A ⊆∗ A∗

n for every n < ω. Let W ∈ W be such that W ⊂ A. Thenfor every n < ω either W ⊆∗ An or W ⊆∗ ω \An.

We got some special interest in number rσ since it is related to differentvariants of sets of uniqueness coming from harmonic analysis, as described ina survey paper [15]. In particular, from [15, thm. 12.6] it follows that anappropriate version of our axiom implies that all covering numbers described inthe paper are equal to ω1.


1.5 On a convergence of subsequences of real-valued functions

This section can be viewed as an extension of the discussion around Egorov’stheorem presented in [54, Ch. 9]. In 1932 Mazurkiewicz [65] proved the followingvariant of Egorov’s theorem, where a sequence 〈fn〉n<ω of real-valued functions isuniformly bounded provided there exists an r ∈ R such that range(fn) ⊂ [−r, r]for every n.

Mazurkiewicz’ Theorem For every uniformly bounded sequence〈fn〉n<ω of real-valued continuous functions defined on a Polish spaceX there exists its subsequence which is uniformly convergent on someperfect set P .

Theorem 1.5.1 If CPAcube holds then for every Polish space X and uniformlybounded sequence 〈fn:X → R〉n<ω of Borel measurable functions there are thesequences: 〈Pξ: ξ < ω1〉 of compact subsets of X and 〈Wξ ∈ [ω]ω: ξ < ω1〉 suchthat X =

⋃ξ<ω1


〈fn � Pξ〉n∈Wξis a monotone uniformly convergent sequence of uni-

formly continuous functions.

Proof. We first note that the family E of all P ∈ Perf(X) for which thereexists a W ∈ [ω]ω such that

the sequence 〈fn � P 〉n∈W is monotone and uniformly convergent

is Fcube-dense in Perf(X).Indeed, let g ∈ Fcube, g:Cω → X, and consider the functions hn = fn ◦ g.

Since h = 〈hn:n < ω〉:Cω → Rω is Borel measurable, there is a dense Gδ subsetG of Cω such that h � G is continuous. So, we can find a perfect cube C ⊂ G ofCω, and for this C function h � C is continuous. Thus, identifying the coordinatespaces of C with C, without loss of generality we can assume that C = Cω, thatis, that each function hn:Cω → R is continuous. Now, by [87, thm. 6.9], thereis a perfect cube C in Cω and a W ∈ [ω]ω such that the sequence 〈hn � C〉n∈Wis monotone and uniformly convergent.1 So P = g[C] is in E .

Now, by CPAcube, there exists an E0 ∈ [E ]≤ω1 such that |X \⋃E0| ≤ ω1.

Then {Pξ: ξ < ω1} = E0 ∪ {{x}:x ∈ X \⋃E0} is as desired: if Pξ ∈ E0 then

the existence of an appropriate Wξ follows from the definition of E . If Pξ is asingleton, then the existence of Wξ follows from Ramsey theorem.

Of course neither Theorem 1.5.1 nor Mazurkiewicz’ theorem can be provedif we do not assume some regularity of the functions fn even if X = R. But isit at least true that

1Actually [87, thm. 6.9] is stated for functions defined on [0, 1]ω . However, the proofpresented there for works also for functions defined on Cω .


(∗) for every uniformly bounded sequence 〈fn: R → R〉n<ω the conclusion ofMazurkiewicz’ theorem holds for some P ⊂ R of cardinality c?

The consistency of the negative answer follows from the next example, whichis essentially due to Sierpinski [79].2 (See [54, pp. 193-194], where it is provedunder the assumption of the existence of ω1-Luzin set. The same proof worksalso for our more general statement.)

Example 1.5.2 Assume that there exists a κ-Luzin set.3 Then for every Polishspace X there exists a sequence 〈fn:X → {0, 1}〉n<ω with the property that forevery W ∈ [ω]ω the subsequence 〈fn〉n∈W converges pointwise for less thanκ-many points x ∈ X.

In particular, under Martin’s Axiom the above sequence exists for κ = c.

Note also that under MA the above example can hold only for κ = c, since MAimplies that

for every set S of cardinality less than c every uniformly boundedsequence 〈fn:S → R〉n<ω has a pointwise convergent subsequence.

(See [54, p. 195].)Our main result of this section is the proof that (∗) is consistent with (so, by

the example, also independent from) the usual axioms of set theory ZFC. Thisfollows from Corollaries 1.2.4, 1.4.5, and 1.5.4 and the fact that the existenceof a selective ultrafilter on ω follows from a stronger version of our axiom. (SeeTheorem 5.2.3.) The proof presented below follows the argument given in [28].

Theorem 1.5.3 Assume that cof(N ) = ω1 and that there exists a selectiveω1-generated ultrafilter on ω.

Let X be an arbitrary set and 〈fn:X → R〉n<ω be a sequence of functionssuch that the set {fn(x):n < ω} is bounded for every x ∈ X. Then thereare sequences: 〈Pξ: ξ < ω1〉 of subsets of X and 〈Wξ ∈ F : ξ < ω1〉 such thatX =

⋃ξ<ω1


the sequence 〈fn � Pξ〉n∈Wξis monotone and uniformly convergent.

The conclusion of Theorem 1.5.3 is obvious for sets X with cardinality ≤ ω1,since sets Pξ can be chosen just as singletons. Thus, we will be interested in thetheorem only for the sets X of cardinality greater than ω1. In particular, we getthe following corollary which, under the additional set theoretical assumptions,generalizes Mazurkiewicz’ theorem and implies (∗).

Corollary 1.5.4 Assume that cof(N ) = ω1 < c and that there exists a selectiveω1-generated ultrafilter on ω.

For every Polish space X and uniformly bounded sequence 〈fn:X → R〉n<ωthere exist sequences: 〈Pξ: ξ < ω1〉 of subsets of X and 〈Wξ ∈ [ω]ω: ξ < ω1〉such that X =

⋃ξ<ω1


2Sierpinski constructed this example under the assumption of the Continuum Hypothesis.3A set L ⊂ R is a κ-Luzin set if |L| = κ but |L ∩ N | < κ for every nowhere dense subset

N of R. Recall that Martin’s Axiom implies the existence of a c-Luzin set.


the sequence 〈fn � Pξ〉n∈Wξis monotone and uniformly convergent.

In particular, there exists a ξ < ω1 such that |Pξ| = c.Moreover, if functions fn are continuous then we can additionally require

that all sets Pξ are compact.

Proof. The main part follows immediately from the discussion above and thePigeon Hole Principle. To see the additional part it is enough to note that forcontinuous functions sets Pξ can be replaced by their closures, since for anysequence 〈fn:P → R〉n<ω of continuous functions if 〈fn � D〉n<ω is monotoneand uniformly convergent for some dense subset D of P then so is 〈fn〉n<ω.

It is worth to notice here that a stronger version of our axiom implies thatthe last part of Corollary 1.5.4 holds also for Borel measurable functions fn.This follows immediately from Theorem 4.1.1(a).

Recall that a non-principal filter F on ω is said to be Ramsey provided forevery B ∈ F and h: [B]2 → {0, 1} there exist i < 2 and A ∈ F such that A ⊂ Band h

[[A]2

]= {i}. It is well known that an ultrafilter on ω is Ramsey if and

only if it is selective. (See e.g. [38] or [44].)

Proof of Theorem 1.5.3. Let F be a selective ultrafilter on ω for which thereexists a familyW ∈ [F ]ω1 generating it. For every x ∈ X define hx: [ω]2 → {0, 1}by putting for every n < m < ω

h(n,m) = 1 if and only if fn(x) ≤ fm(x).

Since F is Ramsey and W generates F we can find a Wx ∈ W and an ix < 2such that hx[[Wx]2] = {ix}. Thus, the sequence Sx = 〈fn(x)〉n∈Wx is monotone.It is increasing when ix = 1 and it is decreasing for ix = 0.

For W ∈ W and i < 2 let P iW = {x ∈ X:Wx = W & ix = i}. Then{P iW :W ∈ W & i < 2} is a partition of X and for every W ∈ W and i < 2 thesequence 〈fn � P iW 〉n∈W is monotone and pointwise convergent to some functionf :P iW → R.

To get uniform convergence note that for every x ∈ P iW there exists ansx ∈ ωω such that

(∀k < ω) (∀n ∈W \ sx(k)) |fn(x)− f(x)| < 2−k.

Since cof(N ) = ω1 implies that the dominating number d is equal to ω1 (seee.g. [4]) there exists a T ∈ [ωω]ω1 dominating ωω. In particular, for everyx ∈ P iW there exists a tx ∈ T and an nx < ω such that sx(n) ≤ tx(n) for allnx ≤ n < ω. For t ∈ T and n < ω let

P iW (t, n) = {x ∈ P iW : tx = t & nx = n}.

Then sets {P iW (t, n): i < 2, W ∈ W, t ∈ T, n < ω} form a desired covering{Pξ: ξ < ω1} of X, since every sequence 〈fk � P iW (t, n)〉k∈W is monotone anduniformly convergent.


1.6 Remarks on a form and consistency of theaxiom CPAcube.

First notice that the equation c = ω2 is, in some sense, a consequence of thesecond part of the axiom CPAcube.

Remark 1.6.1 Each of the following statements is equivalent to CPAcube.

(i) c ≥ ω2 and for every Polish space X and Fcube-dense family E ⊂ Perf(X)there is an E0 ⊂ E such that |E0| ≤ ω1 and |X \

⋃E0| ≤ ω1.

(ii) c ≤ ω2 and for every Polish space X and Fcube-dense family E ⊂ Perf(X)there is an E0 ⊂ E such that |E0| < c and |X \

⋃E0| < c.

Proof. Clearly (ii) implies that either CPAcube or CH hold. But (ii) implies,as in Proposition 1.1.1, that scube

0 = [X]<c so, by Corollary 1.1.2, property (A)holds, which contradicts CH. So, (ii) implies CPAcube. It is also obvious thatCPAcube implies (i). To see that (i) =⇒ (ii) it is enough to notice that (i)implies that c ≤ ω2. This follows from Proposition 1.1.1 and the fact that theconclusion of Proposition 1.0.4 follows already from (i). (These two facts implythat every subset of X of cardinality greater than ω1 can be mapped onto C, soc ≤ ω2.)

Notice also, that if CPAcube[X] stands for CPAcube for a fixed Polish spaceX then

Remark 1.6.2 For any Polish space X axiom CPAcube[X] implies the full ax-iom CPAcube.

Proof. Let X be a Polish space. First notice the following two facts.

(F1) If Y is a Polish subspace of X then CPAcube[X] implies CPAcube[Y ].

Indeed, let E be an Fcube-dense subset of Perf(Y ) and for every cube Q in Y letϕ(Q) ∈ E be a subcube of Q. Next, for each cube P in X let QP be its subcubesuch that either QP ∩ Y = ∅ or QP ⊂ Y . Such a subcube can be found byClaim 1.1.5, since Y is a Gδ subset of X. If QP ⊂ Y put ψ(P ) = ϕ(QP ) ∈ E ;otherwise we put ψ(P ) = QP . Now E = {ψ(P ):P is a cube in X} is Fcube-dense subset of Perf(X). So, by CPAcube[X], there exists an E0 ∈ [E ]≤ω1 suchthat |X \

⋃E0| ≤ ω1. Then E0 = E ∩ E0 ∈ [E ]≤ω1 and |Y \

⋃E0| ≤ ω1. So (F1)

is proved.

(F2) If a Polish space Y is a one-to-one continuous image ofX then CPAcube[X]implies CPAcube[Y ].

Indeed, let f be a continuous bijection from X onto Y and let E be an Fcube-dense subset of Perf(Y ). Put E = {f−1(P ):P ∈ E} and notice that E is anFcube-dense subset of Perf(X). So, by CPAcube[X], there is an E0 ∈ [E ]≤ω1 such


that |X \⋃E0| ≤ ω1. Then E0 = {f [Q]:Q ∈ E0} ∈ [E ]≤ω1 and |Y \

⋃E0| ≤ ω1.

So (F2) is proved.To finish the proof take a Polish space X for which CPAcube[X] holds and

recall that the Baire space ωω is homeomorphic to a subspace of X (since Xcontains a copy of C and C contains a copy of ωω). Thus, by (F1), CPAcube[Z]holds for an arbitrary Polish subspace Z of ωω. Now, if Y is an arbitrary Polishspace then there exists a closed subset F of ωω such that Y is a one-to-onecontinuous image of F . (See e.g. [53, thm. 7.9].) So, by (F2), CPAcube[Y ] holdsas well.

Remark 1.6.3 scube0 %= [X]≤ω1 in a model obtained by adding Sacks numbers

side-by-side. In particular CPAcube (and all other versions of CPA axiom con-sidered in this paper) are false in this model.

Proof. This follows from the fact that scube0 = [X]≤ω1 implies the property (A)

(see Corollary 1.1.2) while it is false in the model mentioned above, as noticedby Miller in [67, p. 581]. (In this model the set X of all Sacks generic numberscannot be mapped continuously onto [0, 1].)


Chapter 2

Games and axiom CPAgamecube

Before we get to the formulation of our next version of the axiom it is goodto note that in many applications we would prefer to have a full covering ofa Polish space X rather that the almost covering as claimed by CPAcube. Toget a better access to the missing singletons1 we will extend the notion of cubeby allowing also the constant cubes: a family Ccube(X) of constant “cubes” isdefined as a family of all constant functions from a cube C ⊂ Cω into X. Wedefine also F∗

cube(X) asF∗

cube = Fcube ∪ Ccube. (2.1)

Thus, F∗cube is the family of all continuous functions from a cube C ⊂ Cω into

X which are either one-to-one or constant. Now the range of every f ∈ F∗cube

belongs to the family Perf∗(X) of all sets P such that either P ∈ Perf(X) or Pis a singleton. The terms “P ∈ Perf∗(X) is a cube” and “Q is a subcube of acube P ∈ Perf∗(X)” are defined in a natural way.

Consider also the following game GAMEcube(X) of length ω1. The game hastwo players, Player I and Player II. At each stage ξ < ω1 of the game Player Ican play an arbitrary cube Pξ ∈ Perf∗(X) and Player II must respond with asubcube Qξ of Pξ. The game 〈〈Pξ, Qξ〉: ξ < ω1〉 is won by Player I provided⋃

ξ<ω1

Qξ = X;

otherwise the game is won by Player II.Recall also that a strategy for Player II is any function S with the property

that S(〈〈Pη, Qη〉: η < ξ〉, Pξ) is a subcube of Pξ, where 〈〈Pη, Qη〉: η < ξ〉 is anypartial game. A game 〈〈Pξ, Qξ〉: ξ < ω1〉 is played according to a strategy S forPlayer II provided Qξ = S(〈〈Pη, Qη〉: η < ξ〉, Pξ) for every ξ < ω1. A strategyS for Player II is a winning strategy for Player II provided Player II wins anygame played according to the strategy S.

Here is our new version of the axiom.1The logic for accessing the singletons in such a strange is justified by the versions of the

axiom which will be presented in Chapter 6.

35


CPAgamecube : c = ω2 and for any Polish space X Player II has no winning strategyin the game GAMEcube(X).

Notice that

Proposition 2.0.4 Axiom CPAgamecube implies CPAcube.

Proof. Let E ⊂ Perf(X) be Fcube-dense. Thus for every cube P ∈ Perf(X)there exists a subcube s(P ) ∈ E of P . Now, for a singleton P ∈ Perf∗(X)put s(P ) = P and consider the following strategy (in fact, it is a tactic) S forPlayer II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = s(Pξ).

By CPAgamecube it is not a winning strategy for Player II. So there exists a game

〈〈Pξ, Qξ〉: ξ < ω1〉 in which Qξ = s(Pξ) for every ξ < ω1 and Player II loses,that is, X =

⋃ξ<ω1

Qξ. Now, let E0 = {Qξ: ξ < ω1 & Qξ ∈ Perf(X)}. Then|X \

⋃E0| ≤ ω1, so CPAcube is justified.

2.1 CPAgamecube and disjoint coverings

Theorem 2.1.1 Assume that CPAgamecube holds and let X be a Polish space. If

D ⊂ Perf(X) is Fcube-dense and it is closed under perfect subsets then thereexists a partition of X into ω1 disjoint sets from D ∪ {{x}:x ∈ X}.

In the proof we will use the following easy lemma.

Lemma 2.1.2 Let X be a Polish space and let P = {Pi: i < ω} ∈ Perf∗(X).For every cube P ∈ Perf(X) there exists a subcube Q of P such that eitherQ ∩

⋃i<ω Pi = ∅ or Q ⊂ Pi for some i < ω.

Proof. If P ∩⋃i<ω Pi is meager in P then, by Claim 1.1.5, we can find a

subcube Q of P such that Q ⊂ P \⋃i<ω Pi.

If P ∩⋃i<ω Pi is not meager in P then there exists an i < ω such that P ∩Pi

if of second category in P . Thus, there exists a subcube Q0 of P (clopen subsetof P ) such that Q0 ∩ Pi is residual in Q0. Then, by Claim 1.1.5, we can find asubcube Q of Q0 such that Q ⊂ Q0 ⊂ Pi.Proof of Theorem 2.1.1. For a cube P ∈ Perf(X) and a countable familyP ∈ Perf∗(X) let D(P ) ∈ D be a subcube of P and Q(P, P ) ∈ D be as inLemma 2.1.2 used with D(P ) in place of P . For a singleton P ∈ Perf∗(X) wejust put Q(P, P ) = P .

Consider the following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q({Qη: η < ξ}, Pξ).

By CPAgamecube strategy S is not a winning strategy for Player II. So there exists

a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which Player II loses, thatis, X =

⋃ξ<ω1

Qξ.


Notice that for every ξ < ω1 either Qξ ∩⋃η<ξ Qη = ∅ or there is an η < ω1

such that Qξ ⊂ Qη. Let

F =

Qξ: ξ < ω1 & Qξ ∩

⋃η<ξ

Qη = ∅

.

Then F is as desired.

Since a family of all measure zero perfect subsets of X is Fcube-dense we getthe following corollary.

Corollary 2.1.3 CPAgamecube implies that for every Polish space there exists a

partition of X into ω1 disjoint closed nowhere dense measure zero sets.

Note that the conclusion of Corollary 2.1.3 does not follow from the fact thatX can be covered by ω1 perfect measure zero subsets. (See [66, thm. 6].)

2.2 MAD families and numbers a and r

Recall that a family A ⊂ [ω]ω is almost disjoint provided |A ∩ B| < ω and itis maximal almost disjoint, MAD, provided it is not a proper subfamily of anyother almost disjoint family. The cardinal a is defined as follows:

a = min{|A|:A is infinite and MAD}.

According to Andreas Blass [8, sec. 11.5] the fact that the equation a = ω1 holdsin the iterated perfect set model was first proved by Spinas. (See also [29].)

Theorem 2.2.1 CPAgamecube implies that a = ω1.

Our proof of Theorem 2.2.1 is based on the following lemma.

Lemma 2.2.2 For every countable infinite almost disjoint family W ⊂ [ω]ω

and a cube P ∈ Perf([ω]ω) there exist a W ∈ [ω]ω and a perfect subcube Q ofP such that W ∪ {W} is almost disjoint but W ∪ {W,x} is not almost disjointfor every x ∈ Q.

Proof. Let W = {Wi: i < ω}. For every i < ω choose Vi ⊂ Wi such that:sets Vi’s are pairwise disjoint, each set Wi \ Vi is finite, but Vω = ω \

⋃i<ω Vi is

infinite. LetB = {x ∈ P : (∀i ≤ ω) |x ∩ Vi| < ω}

and notice that B is a Borel subset of P . (In fact, B is an Fσδ-set.) So, byClaim 1.1.5, there is a perfect subcube P ∗ of P such that either P ∗ ⊂ B orP ∗ ∩B = ∅.

If P ∗ ∩ B = ∅ then W = Vω and Q = P ∗ satisfy the conclusion of thelemma. So, suppose that P ∗ ⊂ B and let µ be a Borel measure on P ∗ vanishingon points and such that µ(P ∗) = 1.


For i, n < ω letPni = {x ∈ P ∗:x ∩ Vi ⊂ n}.

Then all sets Pni are Borel (in fact, they are closed) and P ∗ =⋃n<ω P

ni for

every i < ω. Thus for each i < ω there exists an n(i) < ω such that

µ(Pn(i)i

)> 1− 2−i.

Then the set T =⋃j<ω

⋂j<i<ω P

n(i)i has a µ-measure 1 so, by Claim 1.1.5,

there is a subcube Q of P ∗ which is a subset of T . Let

W =⋃i<ω

[Vi ∩ n(i)] .

We claim that W and Q satisfy the lemma.It is obvious that W is almost disjoint with each Wi. So, fix an x ∈ Q. To

finish the proof it is enough to show that

x ⊆∗ W.

But x ∈ Q ⊂⋃j<ω

⋂j<i<ω P

n(i)i . Therefore, there exists a j < ω such that

x ∈⋂j<i<ω P

n(i)i . So, x∩

⋃j<i<ω Vi =

⋃j<i<ω(x∩Vi) ⊂

⋃j<i<ω(Vi∩n(i)) ⊂W

and the set

x \W ⊂ x ∩

Vω ∪ ⋃

i≤jVi

= (x ∩ Vω) ∪

⋃i≤j

(x ∩ Vi)

is finite.

Proof of Theorem 2.2.1. For a countable infinite almost disjoint familyW ⊂ [ω]ω and a cube P ∈ Perf([ω]ω) let W (W, P ) ∈ [ω]ω and a subcubeQ(W, P ) of P be as in Lemma 2.2.2. For P = {x} ∈ Perf∗([ω]ω) we putQ(W, P ) = P and define W (W, P ) as an arbitrary W almost disjoint with eachset from W and such that A ∩ x is infinite for some A ∈ W ∪ {W}.

Let A0 ⊂ [ω]ω be an arbitrary infinite almost disjoint family and considerthe following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q(A0 ∪ {Wη: η < ξ}, Pξ),

where sets Wη are defined inductively by Wη = W (A0 ∪ {Wζ : ζ < η}, Pη). Inother words, Player II remembers (recovers) sets Wη associated with the setsPη played so far, and he uses them (and Lemma 2.2.2) to get the next answerQξ = Q(A0 ∪ {Wη: η < ξ}, Pξ), while remembering (or recovering each time)the set Wξ = W (A0 ∪ {Wη: η < ξ}, Pξ).

By CPAgamecube strategy S is not a winning strategy for Player II. So there

exists a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which and Player IIloses, that is, [ω]ω =

⋃ξ<ω1

Qξ.


Now, notice that the family A = A0 ∪ {Wξ: ξ < ω1} is a MAD family. It isclear that A is almost disjoint, since every set Wξ was chosen as almost disjointwith every set from A0 ∪ {Wζ : ζ < ξ}. To see that A is maximal it is enoughto note that every x ∈ [ω]ω belongs to a Qξ for some ξ < ω1, and so there is anA ∈ A0 ∪ {Wη: η ≤ ξ} such that A ∩ x is infinite.

By Theorem 2.2.1 we see that CPAgamecube implies the existence of MAD family

of size ω1. Next we will show that such a family can be simultaneously a reapingfamily. This result is similar in flavor to that from Theorem 5.2.10.

Theorem 2.2.3 CPAgamecube implies that there exists a family F ⊂ [ω]ω of cardi-

nality ω1 which is simultaneously MAD and reaping.

Proof. The proof is just a slight modification of that for Theorem 2.2.1.For a countable infinite almost disjoint family W ⊂ [ω]ω and a cube P ∈

Perf([ω]ω) let W0 ∈ [ω]ω and a subcube Q0 of P be as in Lemma 2.2.2. Let A ∈[ω]ω be almost disjoint with every set from W ∪{W0}. By Laver’s theorem [60]we can also find a subcube Q1 of Q0 and a W1 ∈ [A]ω such that

• either W1 ∩ x = ∅ for every x ∈ Q1,

• or else W1 ⊂ x for every x ∈ Q1.

Let Q(W, P ) = Q1 andW(W, P ) = {W0,W1}. If P ∈ Perf∗([ω]ω) is a singletonthen we put Q(W, P ) = P and we can easy findW0 andW1 satisfying the aboveconditions.

Let A0 ⊂ [ω]ω be an arbitrary infinite almost disjoint family and considerthe following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q(A0 ∪

⋃{Wη: η < ξ}, Pξ

),

where Wη’s are defined inductively by Wη =W(A0 ∪⋃{Wη: η < ξ}, Pη).

By CPAgamecube strategy S is not a winning strategy for Player II. So there


⋃ξ<ω1

Qξ. Then the family F = A0 ∪⋃{Wξ: ξ < ω1} is

MAD and reaping.

2.3 Remark on a form of CPAgamecube .

Notice also, that if CPAgamecube [X] stands for CPAgame

cube for a fixed Polish space Xthen, similarly as in Remark 1.6.2 we can also prove

Remark 2.3.1 For any Polish space X axiom CPAgamecube [X] implies the full

axiom CPAgamecube .


Chapter 3

Prisms and axioms CPAgameprism

and CPAprism

The axioms CPAgamecube and CPAcube dealt with the notion of Fcube-density, where

Fcube is the family of all injections f :C → X with C being a subcube of Cω. Inthe applications of these axioms we were using the facts that different subfamiliesof Perf(X) were Fcube-dense. Unfortunately, in many cases the notion of Fcube-density is too weak to do the job – in the applications that follow the familiesE ⊂ Perf(X) will not be Fcube-dense, but they will be dense in a weaker sensedefined below. Luckily, this weaker notion of density still leads to the consistentaxioms.

To define this weaker notion of density, let us first take another look at thenotion of cube. Let A be a non-empty countable set of ordinal numbers. Thenotion of a cube in CA can be defined the same way as it was done for Cω.However, it will be more convenient for us to define it as follows. Let Φcube bethe family of all continuous injection f :CA → CA such that

f(x)(α) = f(y)(α) for all α ∈ A and x, y ∈ CA with x(α) = y(α).

In other words Φcube is the family of all functions of the form f = 〈fα〉α∈A,where each fα is an injection from C into C. Then the family of all cubes in CA

for an appropriate A is equal to

CUBE = {range(f): f ∈ Φcube}

and Fcube is the family all continuous injections f :C → X with C ⊂ Cω andC ∈ CUBE.

In the definitions that follow the notion of “cube” will be replaced by that ofa “prism.” So, let Φprism(A) be the family of all continuous injection f :CA → CA

with the property that

f(x) � α = f(y) � α for all α ∈ A and x, y ∈ CA with x � α = y � α (3.1)

41


or, equivalently, such that

f �� α def= {〈x � α, y � α〉: 〈x, y〉 ∈ f}

is a function from CA∩α into CA∩α for every α ∈ A. Functions f from Φprism(A)were first introduced, in more general setting, in [52] where they are calledprojection-keeping homeomorphisms. Note that

Φprism(A) is closed under the compositions (3.2)

and that for every ordinal number α > 0

if f ∈ Φprism(A) then f �� α ∈ Φprism(A ∩ α). (3.3)

LetSA = {range(f): f ∈ Φprism(A)}.

We will write Φprism for⋃

0<α<ω1Φprism(α) and define

PRISM = {range(f): f ∈ Φprism} =⋃

0<α<ω1Sα.

We also let Fprism(X) (or just Fprism, where X is clear from the context) to bethe family all continuous injections f :P → X where P ∈ PRISM and X is afixed Polish space.

Following the schema presented in (1.5) we say that a family E ⊂ Perf(X)is Fprism-dense provided

∀f ∈ Fprism ∃g ∈ Fprism (g ⊂ f & range(g) ∈ E).

Similarly as in Fact 1.0.3 using (3.2) we can also prove that

Fact 3.0.2 E ⊂ Perf(X) is Fprism-dense if and only if

∀α < ω1 ∀f ∈ Fprism, dom(f) = Cα ∃g ∈ Fprism (g ⊂ f & range(g) ∈ E) (3.4)

Notice also that Φcube ⊂ Φprism, so every cube is also a prism. From this andFact 3.0.2 it also easy to see that

if E ⊂ Perf(X) is Fcube-dense then E is also Fprism-dense. (3.5)

The converse of (3.5), however, is false. (See Remark 3.2.2.)We also adopt the shortcuts similar to that for cubes. Thus, we say that

P ∈ Perf(X) is a prism if we consider it with an (implicitly given) witnessfunction f ∈ Fprism onto P , and Q is a subprism of a prism P ∈ Perf(X)provided Q = f [C], where f ∈ Fprism is as above and C ∈ PRISM. Alsosingletons {x} in X will be identified with constant functions from P ∈ PRISMto {x}, and these functions will be considered as elements of F∗

prism, similarlyas in (2.1).

Now we are ready to state the next version of our axiom, in which the gameGAMEprism(X) is an obvious generalization of GAMEcube(X).


CPAgameprism: c = ω2 and for any Polish space X Player II has no winning strategyin the game GAMEprism(X).

Notice that if a prism P ∈ Perf(X) is considered with a canonical witnessthen it is also a cube and any subcube of P is also a subprism of P . Thus, anyPlayer II strategy in a game GAMEcube(X) can be translated to a strategy ina game GAMEprism(X). (You need to appropriately identify Cα with Cω: firstyou identify Cα with Cω×Cα\{0}, which is important for a finite α, and then thissecond space identify with Cω coordinatewise.) In particular, CPAgame

prism impliesCPAgame

cube . Also, essentially the same argument as used for Proposition 2.0.4gives also the following.

Proposition 3.0.3 Axiom CPAgameprism implies the following prism version of the

axiom CPAcube:

CPAprism: c = ω2 and for every Polish space X and every Fprism-dense familyE ⊂ Perf(X) there is an E0 ⊂ E such that |E0| ≤ ω1 and |X \

⋃E0| ≤ ω1.

By (3.5) it is also obvious that CPAprism implies CPAcube. All these implicationscan be summarized by a graph.

CPAgameprism

✏✏✏✏✶

��

CPAgamecube

CPAprism

��

✏✏✏✏✶ CPAcube

Chart 1.

We will prove the consistency of CPAgameprism in Chapter 7. For the reminder of

this chapter we will concentrate on some basic consequences of CPAprism. Theapplications of the axioms CPAprism and CPAgame

prism, respectively, will be left forthe following two chapters.

We finish here with few simple remarks on prisms. First note that for every0 < α < ω1

if f ∈ Φprism(α) and P ∈ Sα then f [P ] ∈ Sα. (3.6)

Indeed, if P = g[Cα] for some g ∈ Φprism(α) then, by condition (3.2), we havef [P ] = f [g[Cα]] = (f ◦ g)[Cα] ∈ Sα.

In what follows for a fixed 0 < α < ω1 and 0 < β ≤ α symbol πβ will standfor the projection from Cα onto Cβ .

From now on we will agree that Cα, for 0 < α < ω1, is equipped with thefollowing metric ρ. First fix a metric d on C such that the set D of all distanceswithin C is nowhere dense. For each γ < α choose bγ ∈ (0, 1] ∩D such that forevery ε > 0 the set {γ < α: bγ > ε} is finite. Then ρ is defined by

ρ(x, y) = maxγ<α

min{d(x(γ), y(γ)), bγ}.


Note that for this metric any open ball Bα(z, ε) with center at z ∈ Cα andradius ε ∈ (0,∞) \D is a clopen cube in Cα. Also for every γ < α

πγ [Bα(s, ε)] = πγ [Bα(t, ε)] for every s, t ∈ Cα with s � γ = t � γ. (3.7)

In addition, ifBα def= {B ∈ Sα: B is clopen in Cα} (3.8)

then Bα(z, ε) ∈ Bα for every z ∈ Cα and ε ∈ (0,∞) \D.Notice also that if P ∈ Sα then

P ∩ π−1β (P ′) ∈ Sα for every P ′ ∈ Sβ with P ′ ⊂ πβ [P ]. (3.9)

Indeed, let f ∈ Φprism(β) and g ∈ Φprism(α) be such that f [Cβ ] = P ′ andg[Cα] = P . Let Q = (g � β)−1[P ′] = (g � β)−1 ◦ f [Cβ ] ∈ Sβ . Then π−1

β (Q)belongs to Sα and P ∩ π−1

β (P ′) = g[π−1β (Q)] ∈ Sα.

3.1 Fusion Lemma for prisms

One of the main technical tools used to prove that a family of perfect sets isdense is the so called fusion lemma. It tells that for an appropriately chosendecreasing sequence {Pn:n < ω} of perfect sets its intersection P =

⋂n<ω Pn,

called fusion, is still a perfect set. A simple structure of perfect cubes makes itquite easy to formulate a “cube fusion lemma” in which the fusion set P is alsoa cube. However, so far we did not have any need for such a lemma (at least inan explicit form), since its use was always hidden in the proofs of the results wequoted, like Claim 1.1.5 or Proposition 1.4.1. On the other hand, the new andmore complicated structure of prisms does not leave us an option of avoidingfusion argument any longer – we have to face it up front.

Since the fusion argument for prisms requires a very unpleasant bookkeeping,we will reduce the appearance of this part of the argument to just one placein this text. This will be done by formulating a general fusion lemma whichsubsequently will be used in all applications in which fusion argument is needed.

To make our terminology and notation easier to follow we will start with thesimple fusion argument needed to prove the following fact.

(∗) Let h be a continuous function from a perfect set P into a Polish spaceX. If h is not constant on any perfect subset of P then there is a perfectsubset Q of P such that h � Q is one-to-one.

To prove this, construct by induction a sequence 〈Ek ∈ [Perf(P )]<ω: k < ω〉 ofpairwise disjoint sets starting with E0 = {P} and such that for every k < ω wehave

(B) Ek+1 ≺ Ek, where E ≺ E ′ for some pairwise disjoint families E and E ′ ={Ei: i < n} from [Perf(P )]<ω provided

E = {Eji : i < n & j < 2} and Eji ⊂ Ei for all i < n and j < 2;


(I) h[E] ∩ h[E′] = ∅ for every distinct E,E′ ∈ Ek, and diam(E) < 2−k forevery E ∈ Ek+1.

It is easy to see that Q =⋂k<ω

⋃Ek is as desired.

If we try to prove (∗) for prisms (which is very close to what we will actuallydo in Lemma 3.2.1) we have a problem with this schema even if we assume thatall elements of Ek’s are subprisms of P = Cα – there is no reason for Q to be aprism in Cα. This, however, can be guaranteed with choosing the sets Ek withmore care.

Thus, in our prism-fusion schema we will be repeating the basic step (B)of the above construction assuming additionally that the elements of Ek’s areprisms in Cα, i.e., Ek ∈ [Sα]<ω. Sets Ek+1 will be chosen from the families D(Ek)of all E satisfying (B) and some construction-specific conditions like (I) above.However, in order to guarantee that the fusion Q is a prism, we will need tohave some freedom of choice of sets Ek+1. Thus, we will need also to assumethat the families D(Ek) satisfy some additional simple condition (stated in thelemma below) which gives us enough freedom to carry out this task.

Since, in general, the construction-specific conditions may depend on allpreviously chosen sets E0, . . . , Ek, we will describe our procedure in a form ofa two-player game, FusionGameα(A), of length ω, where 0 < α < ω1 andA ∈ {Bα,Sα}. The game starts with Player I playing E0 = {Cα}. At everyk-th stage 〈E0,D0, . . . , Ek〉 of the game Player II responds with a non-empty setDk ⊂ {E ∈ [A]<ω: E ≺ Ek}. At the stage k + 1 Player I chooses Ek+1 from Dk.The game 〈〈Ek,Dk〉: k < ω〉 is won by Player I provided Q =

⋂k<ω

⋃Ek belongs

to Sα; otherwise the game is won by Player II.

Lemma 3.1.1 (Game-Fusion Lemma) For each 0 < α < ω1 and A ∈{Bα,Sα} there exists a strategy S for Player I in FusionGameα(A) such thatif a game 〈〈Ek,Dk〉: k < ω〉 is played according to S and for every k < ω thesets Dk satisfy the following conditions:

(P1) (Dk is A-open) if {E0, . . . , En} ∈ Dk and E′0, . . . , E

′n ∈ A are such that

E′i ⊂ Ei for every i ≤ n then {E′

0, . . . , E′n} ∈ Dk;

(∗) (split) if Ek = {E0, . . . , En} then for every γ < α there exists a family{Eji ∈ A: i ≤ n & j < 2} ∈ Dk such that for every i ≤ n: E0

i and E1i are

disjoint subsets of Ei and πγ [E0i ] = πγ [E1

i ];

then the game is won by Player I, that is, Q =⋂k<ω

⋃Ek belongs to Sα.

Before we prove the lemma let us note the following fact that will be usedin the proof.

Claim 3.1.2 Let β < α < ω1, h0, h1 ∈ Φprism(α), and Ei = hi [Cα] for i < 2.If πβ(E0) = πβ(E1) then there exists an autohomeomorphism f ∈ Φprism(α) ofCα such that

h0 �� β = (h1 ◦ f) �� β. (3.10)


Proof. For i < 2 let fi = hi �� β. Then each fi ∈ Φprism(β) is a homeo-morphism between Cβ and πβ(E0) = πβ(E1) so f = f−1

1 ◦ f0 is an autohome-omorphism of Cβ with f ∈ Φprism(β). Define an autohomeomorphism f ofCα extending f in a sense that f(x) � β = f(x � β) and f(x)(γ) = x(γ) forβ ≤ γ < α. Then f ∈ Φprism(α) and f �� β = f . So f satisfies (3.10) since

(h1 ◦ f) �� β = (h1 �� β) ◦ (f �� β) = f1 ◦ f = f0 = h0 �� β.

This finishes the proof.

Proof of Lemma 3.1.1. We will describe the strategy for the choice of setsEk+1 ∈ Dk for which there exists an h ∈ Φprism(α) with Q = h [Cα] ∈ Sα. Forthis proof we will identify Cα = (2ω)α with 2ω×α, so function h will be in factdefined from 2ω×α into 2ω×α.

Let {〈nk, βk〉: k < ω} be an enumeration of ω × α and for k < ω let usdefine Ak = {〈ni, βi〉: i < k}. We put E0 = {E∅} = {2ω×α} and by inductionon 0 < k < ω we will define the families Ek =

{Ps: s ∈ 2Ak

}∈ Dk−1 such that

Ek satisfies the following inductive conditions for every 0 < k < ω, where fors ∈ 2Ak−1 and j < 2 symbol s j stands for s ∪ {〈〈nk−1, βk−1〉, j〉} ∈ 2Ak .

(i) The diameter of Ps is less than or equal to 2−k for every s ∈ 2Ak .

(ii) Psˆj ⊂ Ps for every s ∈ 2Ak−1 and j < 2.

(iii) For every s, t ∈ 2Ak and β < α

– if s � (β × ω) = t � (β × ω) then πω×β [Ps] = πω×β [Pt];

– if s � (β × ω) %= t � (β × ω) then πω×β [Ps] ∩ πω×β [Pt] = ∅.

So, assume that for some k < ω a family Ek satisfying (i)–(iii) has beenalready constructed. We have to construct an appropriate Ek+1.

Let γ = max{β0, . . . , βk} < α. Using (∗) for every s ∈ 2Ak we can finddisjoint Esˆ0, Esˆ1 ∈ A such that Esˆ0 ∪ Esˆ1 ⊂ Ps, πγ(Esˆ0) = πγ(Esˆ1), andthe family {Esˆj : s ∈ 2Ak & j < 2} belongs to Dk. We will construct Ek+1 ={Psˆj ∈ A: s ∈ 2Ak & j < 2} such Psˆj ⊂ Esˆj for all s ∈ 2Ak and j < 2. Thischoice will clearly imply (ii) and, by (P1), that Ek+1 ∈ Dk. Thus, we just needto guarantee satisfaction of conditions (i) and (iii).

Let{si: i < 2k+1

}be an enumeration of 2Ak+1 . By induction on i < 2k+1

we will choose xi ∈ Cα and hi ∈ Φprism(α) such that for every j < i

(I) hi[Cα] = Esi and xi extends si;

(II) xj � βij = xi � βij and hj �� βij = hi �� βij , where

βij = max{β: sj � (β × ω) = si � (β × ω)}.


To find such xi and hi let β = max{βij : j < i} and choose a j0 < i whichwitnesses it, that is such that β = βij0 .

First choose an xi ∈ Cα extending xj0 � β and si. Such a choice can be madesince xj0 � β extends sj0 � (β × ω) = si � (β × ω). Condition (II) is satisfiedsince for any j < i we have βij ≤ β so that sj � βij = si � βij = sj0 � βij . Thus,by the inductive assumption, xj � βij = xj0 � βij = xi � βij .

Function hi is chosen via Claim 3.1.2 such that hi �� β = hj0 �� β. Thiscan be done since, by inductive assumption (iii) and the choice of sets Esˆl weclearly have πβ(Ei) = πβ(Ej0). As above we can see that the second part of (II)is satisfied. This finishes the construction of xi’s and hi’s.

Next, to finish the construction of Ek+1 let D be as in the condition (3.7)and choose a δ ∈ (0,∞) \D small enough such that

• hi[B(z, δ)] has diameter less than 2−k for every i < 2k+1 and z ∈ Cα;

• hj [B(xj , δ)] � (βij + 1) ∩ hi[B(xi, δ)] � (βij + 1) = ∅, where βij is as above.

Define Psi= hi[B(xi, δ)] and note that it belongs to A. Indeed, it follows

from (3.6) that Psi∈ Sα. Thus, if A = Sα we are done. If, on the other

hand, A = Bα then the sets hi[Cα] = Esi and B(z, δ) belong to Bα. ThereforePsi = hi[B(xi, δ)] ⊂ Esi is clopen as well.

It is easy to see that the choice of Psi’s guarantees (i)–(iii). This finishes the

inductive construction.

Now, to finish the proof define h: 2ω×α → 2ω×α by

h(z) = r ⇐⇒ {r} =⋂k<ω

Pz�Ak

and notice that h ∈ Φprism(α).Indeed, the diameters of Ps’s and the disjointness of the elements of Ek’s

implies that h is a continuous injection. To show that h is projection-keepingfix β < α and take x, y ∈ 2ω×α with x � (ω×β) = y � (ω×β). We have to showthat h(x) � (ω × β) = h(y) � (ω × β). But, by (iii),

{h(x) � (ω × β)} = πω×β[⋂

{Px�Ak: k < ω}

]=

⋂{πω×β [Px�Ak

]: k < ω}

=⋂{πω×β [Py�Ak

]: k < ω}= {h(y) � (ω × β)}

showing that h ∈ Φprism(α).It is clear that Q =

⋂k<ω

⋃Ek = h [Cα] ∈ Sα.

The game approach used in Game-Fusion Lemma 3.1.1 makes sense onlywhen the definitions of the sets Dk are dependent on the previous stages ofthe fusion. In majority of the applications, however, to obtain the desired


conclusion it is enough to use the sets Dk that depend only of the stage levelk. In particular, the following lemma follows immediately from Lemma 3.1.1,since Player I wins in the game FusionGameα(A) in which Player II plays thesets Dk from Lemma 3.1.3. (Notice the shift of indexing in Ek’s: each Ek fromLemma 3.1.1 is replaced here by Dk−1.)

Lemma 3.1.3 (Prism-Fusion Lemma) Let 0 < α < ω1, A ∈ {Bα,Sα}, andlet 〈Dk ⊂ [A]<ω: k < ω〉 be such that for every k < ω the following holds, whereD−1 = {{Cα}}.

(P1) (Dk is A-open) If {E0, . . . , En} ∈ Dk and E′0, . . . , E



0, . . . , E′n} ∈ Dk.

(∗) (split) For every γ < α and {E0, . . . , En} ∈ Dk−1 there exists a family{Eji : i ≤ n & j < 2} ∈ Dk such that for every i ≤ n: E0

i and E1i are

disjoint subsets of Ei and πγ [E0i ] = πγ [E1

i ].

Then there exists a sequence 〈Ek ∈ Dk: k < ω〉 with the property that its fusionQ =

⋂k<ω

⋃Ek belongs to Sα.

In most of our applications we will use the following version of fusion lemma,which is an easy corollary from Lemma 3.1.3. It will be easier to use it in severalof our applications, since in all these cases the requirement (P2) is triviallysatisfied.

Lemma 3.1.4 (Simple Fusion Lemma) Let 0 < α < ω1, A ∈ {Bα,Sα}, andlet 〈Dk ⊂ [A]<ω: k < ω〉 be such that for every k < ω the following holds.

(P1) (Dk is A-open) If {E0, . . . , En} ∈ Dk and E′0, . . . , E



0, . . . , E′n} ∈ Dk.

(P2) (sequence splits) If {E0, . . . , En} ∈ Dk and {Ei0, Ei1} ∈ Dk+1 for everyi ≤ n is such that Ei0 ∪ Ei1 ⊂ Ei then {Eij : i ≤ n & j < 2} ∈ Dk+1.

(P3) (Dk is nicely A-dense) For every E ∈ A and γ < α there are disjointE0, E1 ∈ A such that E0 ∪ E1 ⊂ E, {E0, E1} ∈ Dk, and πγ [E0] = πγ [E1].

Then there exists a sequence 〈Pk ∈ Dk: k < ω〉 with the property that its fusionP =

⋂k<ω

⋃Pk belongs to Sα.

Proof. It is easy to see that conditions (P3) and (P2) imply the condition (∗)from Lemma 3.1.3.

3.2 Basic duality lemma for prisms

In what follows we will also need the following fact, which is one of the mostimportant properties of prisms and distinguishes them from cubes. (Comparealso [52, thm. 20].)


Lemma 3.2.1 For every 0 < α < ω1, E ∈ Sα, a Polish space X, and a contin-uous function f :E → X there exist 0 < β ≤ α and P ∈ Sα, P ⊂ E, such thatf ◦ π−1

β is a function on πβ [P ] ∈ Sβ which is either one-to-one or constant.

Proof. Let 0 < β ≤ α be the smallest ordinal such that f = f ◦ π−1β is a

function on πβ [P0] ∈ Sβ for some P0 ∈ Sα with P0 ⊂ E. We will show thatthere is a P ′

0 ⊂ πβ [P0], P ′0 ∈ Sβ , such that f is either one-to-one or constant on

P ′0. This will be enough since then P = P0 ∩ π−1

β (P ′0), which belongs to Sα by

(3.8), will have the desired properties.If β = 1 then f is defined just on a perfect subset πβ [P0] of C1 and we

can easily choose a perfect subset P ′0 of πβ [P0], which clearly belongs to S1, on

which f is either one-to-one or constant. So, assume that β > 1. We will find aP ′

0 ∈ Sβ , P ′0 ⊂ πβ [P0], on which f is one-to-one.

Let g ∈ Φprism(β) be such that πβ [P0] = g[Cβ ]. Using Simple FusionLemma 3.1.4 with A = Bβ (we can use also A = Sβ) we will find a Q ∈ Sβ

such that f is one-to-one on P ′0 = g[Q] ∈ Sβ .

Let D ⊂ [Bβ ]<ω be the collection of all families {Pi: i < m} of pairwisedisjoint sets such that

f [g[Pi]] ∩ f [g[Pn]] = ∅ for all i < n < m (3.11)

and for each k < ω let Dk be the family of all {Pi: i < m} ∈ D such that eachPi has the diameter less than 2−k. Notice that the conditions (P1) and (P2)are clearly satisfied for this collection of Dk’s. Thus, we need to check onlycondition (P3).

So fix k < ω, E ∈ Bβ , and 0 < γ < β. It is enough to find disjoint E0, E1 ∈Bβ each of the diameter less than 2−k such that E0∪E1 ⊂ E, πγ(E0) = πγ(E1),and

f [g[E0]] ∩ f [g[E1]] = ∅since then clearly {E0, E1} ∈ Dk. However it is easy to see that there existx0, x1 ∈ E with x0 � γ = x1 � γ for which f(g(x0)) %= f(g(x1)) since otherwisef ◦ π−1

γ would be a function on πγ [g[E]], contradicting the minimality of β.So, by the continuity of f ◦ g, we can find disjoint neighborhoods E0, E1 ⊂ Eof x0 and x1, respectively, each of the diameter less than 2−k and such thatE0, E1 ∈ Bβ , πγ(E0) = πγ(E1), and f [g[E0]] ∩ f [g[E1]] = ∅.

Now, by Lemma 3.1.4, there exist Pk ={P ki : i < mk

}∈ Dk such that

Q =⋂k<ω

⋃i<mk

P ki ∈ Sβ . It is enough to show that f is one-to-one onP ′

0 = g[Q] ∈ Sβ . To see this, take different y0, y1 ∈ P ′0 and let x0, x1 ∈ Q be

such that g(xj) = yj for j < 2. Let k < ω be such that the distance betweenx0 and x1 is greater than 2−k. Then they must belong to different Pi’s from Pkand so, by (3.11), f(y0) = f(g(x0)) %= f(g(x1)) = f(y1).

Remark 3.2.2 Notice that Lemma 3.2.1 is false if we replace prisms Sα withthe family CUBEα of all subcubes of Cα. Indeed, this is obviously the case ifwe take f :C2 → C given by f(x0, x1) = x1.


3.3 CPAprism, additivity of s0, and more on (A)

We will start with noticing that the axiom CPAprism leads in a natural way tothe following generalization of the ideal scube

0 :

sprism0 =

{X \

⋃E : E is Fprism-dense in Perf(X)

}.

Similarly as for Proposition 1.0.4 it can be shown that

Proposition 3.3.1 If CPAprism holds then sprism0 = [X]≤ω1 .

It can be also shown, refining the argument for Fact 1.0.5, that

Fact 3.3.2 [X]<c ⊂ scube0 ⊂ sprism

0 ⊂ s0 for every Polish space X.

However, we will not use these facts in the rest of the paper.

The next lemma and its corollaries represent a very usefull application ofLemma 3.2.1. For a fixed Polish space X and 0 < α < ω1 let Fα denote thefamily of all continuous injections from Cα into X. Note that if we consider Fαwith the topology of uniform convergence then

Fα is a Polish space. (3.12)

To prove (3.12) it is enough to show that Fα is a Gδ subset of the space C =C(Cα, X) of all continuous functions from Cα into X. But Fα is the intersectionof the open sets Gn, n < ω, where sets Gn are constructed as follows. Fix afinite partition Pn of Cα into clopen sets each of the diameter less than 2−n,and let Hn be the family of all mappings h from Pn into the topology of X suchthat h(P ) ∩ h(P ′) = ∅ for distinct P, P ′ ∈ Pn. We put

Gn =⋃

h∈Hn

{f ∈ C: (∀P ∈ Pn)(∀x ∈ P ) f(x) ∈ h(P )}.

This completes the argument for (3.12).

Lemma 3.3.3 Let X be a Polish space and 0 < α < ω1. Then every prismf :Cβ → Fα from Fprism(Fα) contains a subprism f∗ ∈ Fprism(Fα) with the

property that there exists an f ∈ Fprism(X) defined on a subset of Cβ+α suchthat

(a) f(s, t) = f∗(s)(t) for all 〈s, t〉 ∈(Cβ × Cα

)∩ dom(f), and

(b) for each s ∈ dom(f∗) function f(s, ·): {t ∈ Cα: 〈s, t〉 ∈ dom(f)} → X is asubprism of f∗(s).


Proof. Let f :Cβ → Fα, f ∈ Fprism(Fα), and define a function g from a setCβ × Cα = Cβ+α into X by g(s, t) = f(s)(t) for 〈s, t〉 ∈ Cβ × Cα. It is easy tosee that g is continuous.

Apply Lemma 3.2.1 to a prism E = Cβ+α ∈ Sβ+α and the function g to finda γ ≤ β + α and a subprism P of E such that g◦π−1

γ is a function on πγ [P ] ∈ Sγwhich is either one-to-one or constant. Let f∗ = f � πβ [P ]. We will show thatit is as desired.

First note that

γ = β + α and g is one-to-one on P .

Indeed, if z ∈ range(f∗) ∩ Fprism(X) and z = f∗(s) then for every differentt0, t1 ∈ Cα with 〈s, t0〉, 〈s, t1〉 ∈ P we have g(s, t0) = f(s)(t0) = z(t0) %= z(t1) =g(s, t1). So, g cannot be constant and if γ < β + α then we can find t0 and t1such that πγ(〈s, t0〉) = πγ(〈s, t1〉) contradicting the above calculation.

It is easy to see that f = g � P is as desired.

Lemma 3.3.3 implies the following useful fact.

Proposition 3.3.4 CPAprism implies that for every Polish space X there existsa family H of continuous functions from compact subsets of X onto C× C suchthat |H| ≤ ω1 and

• for every prism P in X there are h ∈ H and c ∈ C such that h−1({c}×C)and h−1(〈c, d〉) are subprisms of P for every d ∈ C.

In particular, F = {h−1({c} × C):h ∈ H & c ∈ C} is Fprism-dense in X.

Proof. Let 0 < α < ω1. We use the notation as in Lemma 3.3.3.Since the family of all sets range(f∗) is Fprism-dense in Fα by CPAprism we

can find a family Gα = {f∗ξ : ξ < ω1} such that Rα = Fα \⋃ξ<ω1

range(f∗ξ ) hascardinality less than of equal to ω1.

If f∗ ∈ Gα then f maps injectively a P = Pf ∈ Sβ+α onto Q = Qf ⊂ X.Moreover, for every prism z ∈ Fα \Rα there are f∗ ∈ Gα and s ∈ dom(f∗) suchthat z = f∗(s) and f(s, ·) is a subprism of z.

Now, let Hf ∈ Φprism(β+α) be from Cβ+α onto P and consider the prism f ◦Hf :Cβ+α → Q. Then functions (f◦Hf )−1:Qf → Cβ+α are our desired functionsmodulo some projections. More precisely, let k0:Cβ → C be a homeomorphismand let k1:C → C be such that k−1

1 (c) ∈ Perf(C) for every c ∈ C. Definehαf :Qf → C× C by

hαf (x) = 〈(k0 ◦ πβ)((f ◦Hf )−1(x)), k1([(f ◦Hf )−1(x)](β))〉.Then family H0 = {hαf :α < ω1 & f∗ ∈ Gα} works for all prisms not in R =⋃

0<α<ω1Rα. It is easy to extend H0 by ω1 functions to a family H which is as

desired.

Proposition 3.3.4 implies the following stronger version of the property (A).This can be considered as a version of a remark due to Miller [67, p. 581], whonoticed that in the Sacks model functions coded in the ground model can betaken as a family G.


Corollary 3.3.5 Assume that CPAprism holds. Then

(A∗) there exists a family G of uniformly continuous functions from R to [0, 1]such that |G| = ω1 and for every S ∈ [R]c there exists a g ∈ G withg[S] = [0, 1].

Proof. Let H be as in Proposition 3.3.4 for X = R, k:C→ [0, 1] be continuoussurjection, and for every h = 〈h0, h1〉 ∈ H let gh: R → [0, 1] be a continuousextension of a function h∗: dom(h) → [0, 1] defined by h∗(x) = k(h1(x)). Weclaim that G = {gh:h ∈ H} is as desired.

Since, by Proposition 3.3.1, sprism0 = [R]≤ω1 there exists a prism P in R

such that S intersects every subprism of P . Let h ∈ H and c ∈ C be suchthat h−1({c} × C) and h−1(〈c, d〉) are subprisms of P for every d ∈ C. Then Sintersects each h−1(〈c, d〉) so h[S] contains {c} × C. Thus gh[S] = [0, 1].

Next we will show that CPAprism implies1 that the additivity of the ideal s0:

add(s0) = min{|F |:F ⊂ s0 &

⋃F /∈ s0

}is equal to ω1. This stays in contrast with Proposition 6.1.1 in which we willshow that CPA implies cov(s0) = ω2.

Notice that numbers add(s0), cov(s0), non(s0), and cof(s0) has been inten-sivly studied. (See e.g. [50].) It is known that cof(s0) > c (see [50, thm. 1.3])and that non(s0) = c since there are s0-sets of cardinality c. There are models ofZFC+MA with c = ω2 and cov(s0) = ω1, while Proper Forcing Axiom impliesthat add(s0) = c.

In what follows we need the following useful fact.

Fact 3.3.6 For any open dense subset D of Perf(C) (considered as ordered byinclusion) there exists a maximal antichain A ⊂ D consisting of pairwise disjointsets such that every P ∈ Perf(

⋃A) is covered by less than continuum many sets

from A.

Proof. Let Perf(C) = {Pα:α < c}. We will build inductively a sequence〈〈Aα, xα〉 ∈ D × C:α < c〉 aiming for A = {Aα:α < c}. At step α < c, givenalready 〈〈Aβ , xβ〉:β < α〉 we look at Pα.

Choice of xα: If Pα ⊂⋃β<αAβ we take xα as an arbitrary element of C;

otherwise we pick xα ∈ Pα \⋃β<αAβ .

Choice of Aα: If there is a β < α such that Pα ∩ Aβ is uncountable, we letAα = Aβ ; otherwise pick Aα ∈ D below Pα and notice that we can refine it, ifnecessary, to be disjoint with

⋃β<αAβ ∪ {xβ :β ≤ xα}.

It is easy to see that A = {Aα:α < c} is as required.

Corollary 3.3.7 CPAprism implies that add(s0) = ω1.

1In fact, this follows also from CPAcube.


Proof. Let H = {hξ: ξ < ω1} be as in Proposition 3.3.4 with X = C. For everyξ < ω1 put A0

ξ = {h−1({c} × C): c ∈ C}. Then each A0ξ is a family of pairwise

disjoint sets and A0 =⋃ξ<ω1

A0ξ is dense in Perf(C).

For each ξ < ω1 let A∗ξ be a maximal antichain extending A0

ξ , define Dξ ={P ∈ Perf(C):P ⊂ A for some A ∈ A∗

ξ}, and let Aξ ⊂ Dξ be as in Fact 3.3.6.Then A =

⋃ξ<ω1

Aξ is still dense in Perf(C).For each ξ < ω1 let {Pαξ :α < c} be an enumeration of Aξ. (Note that each

Aξ has cardinality c, since this was the case for sets A0ξ .) Pick xαξ from each

Pαξ and put Aξ = {xαξ :α < c}. Then Aξ ∈ s0 for every ξ < ω1. However,A =

⋃ξ<ω1

Aξ /∈ s0 since it intersects every element of a dense set A.

It can be also shown that CPAprism, with a help of Proposition 3.3.4, impliesthat the Sacks forcing S = 〈Perf(C),⊂〉 collapses c to ω1. However, this alsofollows immediately from a theorem of P. Simon [83] that S collapses c to b

while already CPAcube implies that b ≤ cof(N ) = ω1.Note also that although the family F from Proposition 3.3.4 is Fprism-dense

we certainly cannot repeat the proof of Corollary 3.3.7 to show that, underCPAprism, add(sprism

0 ) = ω1 – this clearly contradicts Proposition 3.3.1. Theplace where proof breaks is Fact 3.3.6, which cannot be proved for simple densitybeing replaced by Fprism-density.

3.4 Remarks on a form and consistency of ax-ioms CPAgame

prism and CPAprism.

Remark 3.4.1 The following weaker version of CPAprism:

CPAcprism: for every Polish space X and every Fprism-dense family E ⊂ Perf(X)there is an E0 ⊂ E such that |E0| < c and |X \

⋃E0| < c

implies that continuum is a cardinal successor.

Proof. First note that CPAcprism implies the following version of part (b) from

Theorem 4.1.1:

there exists an F ⊂ C1 such that |F| < c and R2 =⋃f∈F (f ∪ f−1).

The proof is identical to that for Theorem 4.1.1(b). On the other hand for acardinal number κ we have

c = κ+ if and only if there exists an F ⊂ RR such that

|F| = κ < c and R2 =⋃f∈F

(f ∪ f−1).

This is an easy generalization of the property P1 from [81]. (See also [21, thm.6.1.8].) It is easy to see that the above two conditions imply that c is a cardinalsuccessor.


Note also that CPAcprism can be considered as a prism version of a property

(ii) from Remark 1.6.1. In addition if CPAgameprism[X] and CPAprism[X] stands,

respectively, for the axioms CPAgameprism and CPAprism for a fixed Polish space X

then, similarly as in Remark 1.6.2 we can also prove

Remark 3.4.2 For any Polish space X:

• CPAgameprism[X] implies CPAgame

prism, and

• CPAprism[X] implies CPAprism.

Chapter 4

CPAprism and coverings withsmooth functions

This chapter is based on our paper [30]. Below we will use standard notationfor the classes of differentiable partial functions from R into R. Thus, if Xis an arbitrary subset of R we will write C0(X) or C(X) for the class of allcontinuous functions f :X → R and D1(X) for the class of all differentiablefunctions f :X → R, that is, those for which the limit

f ′(x0) = limx→x0, x∈X

f(x)− f(x0)x− x0

exists and is finite for all non-isolated elements x0 of X. Since for the isolatedpoints x0 ∈ X there is no clear way to define the derivative f ′(x0), we will definefunctions with more smoothness only for the sets X ⊂ R for which isolatedpoints do not have accumulation points in X. For such an X and 0 < n < ωwe will write Dn(X) to denote the class of all functions f :X → R which aren-times differentiable with all derivatives being finite and Cn(X) for the classof all functions f ∈ Dn(X) whose n-th derivative f (n) is continuous. SymbolC∞(X) will be used for all infinitely many times differentiable functions from Xinto R. In addition, we say that a function f :X → R is in the class “Dn(X)”if f ∈ Cn−1(X) and it has the n-th derivative which can be infinite; f is inthe class “Cn(X)” when f is in “Dn(X)” and its n-th derivative is continuouswhen its range [−∞,∞] is considered with the standard topology. “C∞(X)”will stand for all functions f :X → R which are either in C∞(X) or, for some0 < n < ω, they are in “Cn(X)” and f (n) is constant equal to ∞ or −∞. Wewill use these symbols mainly for X from the class Perf(R) of all perfect subsetsof R. In particular, Cnperf will stand for the union of all Cn(P ) for which P ⊂ R

is either perfect or a singleton. The classes Dnperf , C∞perf , and “C∞perf” are defined

the similar way. In all of the above we will drop parameter X if X = R. Inparticular, Dn = Dn(R) and Cn = Cn(R). The relations between these classesfor n < ω are given in a chart below, where arrows −→ indicate the strictinclusions �.

55


Cn ✛ “Dn+1” ✛ “Cn+1”

❄ ❄

Dn+1 ✛ Cn+1

Chart 2.

In addition for F ⊂ R2 we define F−1 = {〈y, x〉: 〈x, y〉 ∈ F} and for F ⊂ P(R2)we put F−1 = {F−1:F ∈ F}.

4.1 Chapter overview; properties (H∗) and (Q)

The main result of this chapter is the following theorem.

Theorem 4.1.1 The following facts follow from CPAprism.

(a) For every Borel measurable function g: R → R there exists a family offunctions {fξ ∈ “C∞perf”: ξ < ω1} such that

g =⋃ξ<ω1

fξ.

Moreover for each ξ < ω1 there exists an extension fξ: R → R of fξ suchthat

(i) fξ ∈ “C1” and

(ii) either fξ ∈ C1 or fξ is a homeomorphism from R onto R such thatf−1ξ ∈ C1.

(b) There exists a sequence {fξ ∈ RR: ξ < ω1} of C1 functions such that

R2 =⋃ξ<ω1

(fξ ∪ f−1ξ ).

Clearly parts (a) and (b) of the theorem imply properties (Q) and (H∗),respectively. In particular we have

Corollary 4.1.2 CPAprism implies properties (Q) and (H∗).

Note also that, by Corollary 5.0.10, under CPAgameprism functions fξ in Theo-

rem 4.1.1(a) may be chosen to have disjoint graphs. Also, R2 can be coveredby ω1 pairwise disjoint sets P such that either P or P−1 is a function in classC1perf ∩ “C∞perf”.

The essence of Theorem 4.1.1 lies in the following real analysis fact. Its proofis combinatorial in nature and uses no extra set-theoretical assumptions.


Proposition 4.1.3 Let g: R → R be Borel, 0 < α < ω1, and E ∈ Sα.

(a) For every continuous injection h:E → R there exists a subprism E1 ofE such that g � h[E1] ∈ “C∞perf” and there is an extension f : R → R

of g � h[E1] such that f ∈ “C1” and either f ∈ C1 or f is an auto-homeomorphism of R with f−1 ∈ C1.

(b) For every continuous injection h:E → R2 there exists a subprism E1 of Esuch that either F = h[E1] ⊂ R2 or its inverse, F−1, is a function whichcan be extended to a C1 function f : R → R.

With Proposition 4.1.3 in hand the proof of Theorem 4.1.1 becomes an easyexercise.

Proof of Theorem 4.1.1. (a) Let g: R → R be a Borel function and let E bethe family of all P ∈ Perf(R) such that

g � P ∈ “C∞perf” and there is an extension f : R → R of g � P suchthat f ∈ “C1” and either f ∈ C1 or f is an autohomeomorphism ofR with f−1 ∈ C1.

By Proposition 4.1.3(a) family E is Fprism-dense. So, by CPAprism, there existsan E0 ∈ [E ]≤ω1 such that |R \

⋃E0| ≤ ω1. Let E1 = E0 ∪ {{r}: r ∈ R \

⋃E0}.

Then the family {g � P :P ∈ E1} satisfies the theorem.(b) Let E be the family of all P ∈ Perf(R2) such that either P or P−1

is a function which can be extended to a C1 function f : R → R. By Propo-sition 4.1.3(b) family E is Fprism-dense, so there exists an E0 ∈ [E ]≤ω1 suchthat |R \

⋃E0| ≤ ω1. Let E1 = E0 ∪ {{x}:x ∈ R2 \

⋃E0}. For every P ∈ E1

let fP : R → R be a C1 function which extends either P or P−1. Then family{fP :P ∈ E1} is as desired.

The proof of Proposition 4.1.3 will be left to the latter sections of this chap-ter. Meanwhile we like to present a discussion of Theorem 4.1.1.

First we like to reformulate Theorem 4.1.1 in a language of a covering numbercov defined below, where X is an infinite set (in our case X ⊂ R2 with |X| = c)and A,F ⊂ P(X):

cov(A,F) = min({κ: (∀A ∈ A)(∃G ∈ [F ]≤κ) A ⊂

⋃G}∪ {|X|+}

).

If A ⊂ X we will write cov(A,F) for cov({A},F). Notice the followingmonotonicity of cov operator: for every A ⊂ B ⊂ X, A ⊂ B ⊂ P(X), andF ⊂ G ⊂ P(X)

cov(A,G) ≤ cov(B,G) ≤ cov(B,F) & cov(A,G) ≤ cov(B,G) ≤ cov(B,F).

In terms of the cov operator Theorem 4.1.1 can be expressed in the followingform, where Borel stands for the class of all Borel functions f : R → R.


Corollary 4.1.4 CPAprism implies that

(a) cov(Borel, “C∞perf”

)= ω1 < c;

(b) cov(Borel, “C1”

)= ω1 < c;

(c) cov(Borel, C1 ∪ (C1)−1

)= ω1 < c;

(d) cov(R2, C1 ∪ (C1)−1

)= ω1 < c.

Proof. The fact that all numbers cov(A,G) listed above are ≤ ω1 followsdirectly from Theorem 4.1.1. The other inequalities follow from Examples 4.5.6and 4.5.8.

Theorem 4.1.1(b) and Corollary 4.1.4(d) can be treated as generalizationsof a result of Steprans [85] who proved that in the iterated perfect set model

we have cov(

R2,(“D1

perf”)∪

(“D1

perf”)−1

)≤ ω1. This clearly follows from

Corollary 4.1.4(d) since C1 � C1perf � D1

perf � “D1perf”. (See survey article [12].

For more information how to “locate” Steprans’ result in [85] see also [25, cor.9].)

The following proposition shows that Theorem 4.1.1 is, in a way, the bestpossible. (Parts (i), (ii), and (iii) relate, respectively, to items (b), (c) and (d),and (a) from Corollary 4.1.4.)

Proposition 4.1.5 The following is true in ZFC.

(i) cov(Borel, C1

)= cov

(“C1”, C1

)= cov

(“C1”, D1

perf

)= c. Moreover,

cov(“Cn”, Cn) = cov(“Cn”, Dnperf) = c for every 0 < n < ω.

(ii) cov(Borel, C2 ∪ (C2)−1

)= cov

(“C2”, D2

perf ∪ (D2perf)

−1)

= c, and

cov(R2, C2 ∪ (C2)−1

)= cov

(“C2”, D2

perf ∪ (D2perf)

−1)

= c.

(iii) cov(Borel, C∞perf

)= cov

(“C1”, C∞perf

)= cov

(“C1”, D1

perf

)= c, and

cov (Borel, “C∞”) = cov(C1, “C∞”

)= cov

(C1, “D2”

)= c. Moreover,

cov(Cn, “Dn+1”

)= c for every 0 < n < ω.

Proof. (i) follows immediately from Examples 4.5.2 and 4.5.3.(ii) follows from monotonicity of cov operator and Example 4.5.1.The first part of (iii) follows from (i). The remaining two parts follow,

respectively, from Examples 4.5.4 and 4.5.5.

Corollary 4.1.4 and Proposition 4.1.5 establish the values of cov operatorfor all classes in Chart 2 except for cov (Dn, Cn) and cov (“Dn”, “Cn”). Theseare established in the following theorem, which proof will be left to the lattersections of this chapter.


Theorem 4.1.6 If CPAprism holds then for every 0 < n < ω

cov (Dn, Cn) = cov (“Dn”, “Cn”) = ω1 < c.

Note also that, by Corollary 5.0.10, under CPAgameprism covering functions in

Theorem 4.1.6 may be chosen to have disjoint graphs.With this theorem in hand we can summarize the values of the cov operator

between the classes from Chart 2 in the following graphical form. Here themark “c” next to the arrow means that the covering of the larger class by thefunctions from the smaller class is equal to c and that this can be proved inZFC. The mark “< c” next to the arrow means that it is consistent with ZFC(and it follows from CPAprism) that the appropriate cov number is < c. (FromExamples 4.5.6, 4.5.7, and 4.5.8 it follows that all these numbers are greater thanor equal to min{cov(M), cov(N )} > ω. So under the continuum hypothesis CHor Martin’s Axiom MA all these numbers are equal to c.)

C0 ✛< c

“D1” ✛< c

“C1”

❄c

❄c

D1 ✛< c

C1

Cn ✛c

“Dn+1” ✛< c

“Cn+1”

❄c

❄c

Dn+1 ✛< c

Cn+1

Chart 3. Values of cov operator: for n = 0 (left) and n > 0 (right).

The values of cov next the vertical arrows are justified by cov(“Cn”, Dn) = c

(Proposition 4.1.5(i)), while marks “< c” below the upper horizontal arrows andthat directly below them follow from Theorem 4.1.6. The remaining arrow of theright part of the chart is the restatement of the last part of Proposition 4.1.5(iii),while its counterpart in the left part of the chart follows from Corollary 4.1.4(b):cov

(C, “C1”

)= cov

(Borel, “C1”

)< c is a consequence of CPAprism. Finally let

us mention that in Corollary 4.1.4(b) there is no chance to increase family Borelin any essential way and keep the result. This follows from the following fact

cov(Sc, C) = cov(RR, C

)≥ cof(c), (4.1)

where symbol Sc stands for the family of all symmetrically continuous functionsf : R → R which are, in particular, continuous outside of some set of measurezero and first category. (See [24, cor. 1.1] and the remarks below on the operatordec.)

Number cov(A,F) is very closely related to the following decompositionnumber

dec(A,F) = min({κ: (∀A ∈ A)(∃G ∈ [F ]κ) G is a partition of A} ∪ {|X|+}

)which was first studied by Cichon, Morayne, Pawlikowski, and Solecki [20] forthe Baire class α functions. (More information on dec(F ,G) can be found in a


survey article [22, sec. 4].) It is easy to see that if A and F are some classesof partial functions and Fr denotes all possible restrictions of functions from Fthen cov(A,F) = dec(A,Fr). In particular, for all situations relevant to ourdiscussion above the operators cov and dec have the same values.

Our number cov is also related to the following general class of problems.We say that the families A,F ⊂ P(X) satisfy Intersection Theorem, what wedenote by

IntTh(A,F),

if for every A ∈ A there exists an F ∈ G such that |A ∩ F | = |X|. Thiskind of theorems have been studied for a big part of this century. In par-ticular, in early 1940’s Ulam asked in the Scottish Book [64, Problem 17.1] ifIntTh(C,Analytic) holds, that is, whether for every f ∈ C there exists a realanalytic function g: R → R which agrees with f on a perfect set. (See [88].)In 1947 Zahorski [91] gave a negative answer to this question by proving thatthe proposition IntTh(C∞,Analytic) is false. At the same paper he also raiseda natural question, which has become known as Ulam-Zahorski Problem: DoesIntTh(C,G) hold for G = C∞ (or G = Cn or G = Dn)? Here is a quick summaryof what is known on this problem. (See [12].)

Proposition 4.1.7 (a) (Zahorski [91]) ¬IntTh(C∞,Analytic).

(b) (Agronsky, Bruckner, Laczkovich, Preiss [1]) IntTh(C, C1).

(c) (Olevskiı [72]) IntTh(C1, C2).

(d) (Olevskiı [72]) ¬IntTh(C, C2) and ¬IntTh(Cn, Cn+1) for n ≥ 2.

We are interested in these problems since for the families A,F ∈ P(Rn) ofuncountable Borel sets

¬IntTh(A,F) =⇒ cov(A,F) = c (4.2)

as, in this situation, if ¬IntTh(A,F) then there exists an A ∈ A, |A| = c,such that |A ∩ F | ≤ ω for every F ∈ F . Thus in the examples relevant toProposition 4.1.5 instead of proving cov(A,F) = c we will be in fact showing astronger fact that ¬IntTh(A0,F) for appropriate A0 ⊂ A ∈ A.

4.2 Proof of Proposition 4.1.3

Proposition 4.1.3 will be deduced from the following fact, which is a generaliza-tion of a theorem of Morayne [70]. (Morayne proved his results for E and E1

being just perfect sets, that is, for α = 1.)

Proposition 4.2.1 Let 0 < α < ω1, E ∈ Sα, h:E → R be a continuousinjection, and G be a function from h[E]2\∆ = {〈x, y〉 ∈ h[E]×h[E]:x %= y} into[0, 1] which is continuous and symmetric, that is, such that G(x, y) = G(y, x)for all x, y ∈ h[E]2 \∆. Then there exists an E1 ∈ Sα, E1 ⊂ E, such that G isuniformly continuous on h[E1]2 \∆.


The proof of Proposition 4.2.1 will be presented in the next section. We willalso need the following lemma, which will be deduced from Proposition 4.2.1.

Lemma 4.2.2 Let g: R → R be Borel, 0 < α < ω1, and E ∈ Sα. For everycontinuous injection h:E → R there exist a subprism E1 of E and a “C1”function f : R → R such that f extends g � h[E1].

In addition we can require that either f ∈ C1 or

(') f ′ � h[E1] is constant equal to ∞ or −∞ and f is an autohomeomorphismof R such that f−1 ∈ C1.

Proof. First note that there exists an E′ ∈ Sα, E′ ⊂ E, such that

g � h[E′] is continuous. (4.3)

Indeed, let h0 ∈ Φprism be such that E = h0 [Cα] and let U be a comeagersubset of h[E] = (h ◦ h0) [Cα] such that the restriction g � U is continuous.Then (h ◦ h0)−1(U) is comeager in Cα and, by Claim 1.1.5, there is a perfectcube Q ⊂ (h ◦ h0)−1(U). The set E′ = h0[Q] ∈ Sα has the desired propertysince h[E′] = h[h0[Q]] ⊂ U .

Now let k: [−∞,∞] → [0, 1] be a homeomorphism and let G be defined onh[E′]2 \∆ by

G(x, y) = k

(g(x)− g(y)x− y

).

Then, by Proposition 4.2.1, there exists an E′1 ∈ Sα, E′

1 ⊂ E′, such that G isuniformly continuous on h[E′

1]2 \ ∆. So, there exists a uniformly continuous

extension of G � h[E′1]

2 \∆ to G � h[E′1]

2. Clearly k−1(G(x, x)) is a derivative(possibly infinite) of g0 = g � h[E′

1] for every x ∈ h[E′1], so g0 ∈ “C1(h[E′

1])”.Now, if (g′0)

−1(R) is non-empty then, as in the argument for (4.3), we canfind an E1 ∈ Sα, E1 ⊂ E′

1, such that h[E1] ⊂ (g′0)−1(R). This obviously implies

g � h[E1] ∈ C1perf . But we also know that the difference quotient function

g(x)−g(y)x−y is uniformly continuous on h[E1]2 \ ∆. So, by Whitney’s extension

theorem [90] (see also Lemma 4.4.1), we can find a C1 extension f : R → R ofg � h[E1].

So, assume that (g′0)−1(R) = ∅. Then either (g′0)

−1(∞) or (g′0)−1(−∞) is

non-empty and open. Assume the former case. Similarly as above we can findan E′′

1 ∈ S(α), E′′1 ⊂ E′

1, such that g′0[h[E′′1 ]] = {∞}. Then, by a version of

Whitney’s extension theorem from [11, thm. 2.1], we can find a “C1” extensionf0: R → R of g � h[E′′

1 ].But then there exists an open interval J in R intersecting h[E′′

1 ] on a clo-sure of which f ′0 is positive. So f1 = f0 � cl(J) is strictly increasing and thederivative of f−1

1 is continuous, non-negative, and bounded. Thus there existsa homeomorphism f2: R → R extending f−1

1 with f2 ∈ C1. Now put f = f−12

and take an E1 ∈ Sα with E1 ⊂ E′′1 ∩ h−1(J). It is easy to see that E1 and f

are as required.


Proof of Proposition 4.1.3(a). By Lemma 4.2.2 we can assume, decreasingE if necessary, that there is an extension f : R → R of g � h[E] such that f ∈ “C1”and either f ∈ C1 or f is an autohomeomorphism of R with f−1 ∈ C1. Thus, itis enough to find an E1 for which g � h[E1] ∈ “C∞perf”.

If there exist E1 ∈ Sα and n < ω such that

f = g � h[E1] ∈ “Cnperf” and f (n) has a constant value ∞ or −∞ (4.4)

then this E1 is as desired. So assume that there is no E1 ∈ Sα satisfyingcondition (4.4). We will use Simple Fusion Lemma 3.1.4 with A = Sα to find asubprism E1 of E for which g � h[E1] ∈ C∞perf .

First notice that we can assume that E = Cα, since we can replace h withh ◦ h0, where h0 ∈ Φprism is such that E = h0 [Cα]. For k < ω let Dk ∈ [Sα]<ω

be the collection of all families{Pi: i < 2k

}of pairwise disjoint prisms, each of

the diameter less than 2−k, and such that

g �⋃i<2k

h[Pi] ∈ Ckperf . (4.5)

We need to show that Dk’s satisfy the assumptions of Lemma 3.1.4.It is obvious that the conditions (P1) and (P2) are satisfied. To see that

(P3) holds for k < ω fix E ∈ Sα and γ < α. Applying Lemma 4.2.2 k-times andusing the fact that (4.4) is false we can find a tower of prisms E = P0 ⊃ · · · ⊃ Pksuch that g � h[Pi] ∈ Ciperf for each i ≤ k. Take disjoint subprisms E0 and E1

of Pk, each of diameter less than 2−k, such that πγ(E0) = πγ(E1). It is easy tosee that E0 and E1 satisfy the requirements of the condition (P3).

Now, by Lemma 3.1.4, there exist Pk ={P ki : i < 2k

}∈ Dk such that E1 =⋂

k<ω

⋃i<2k P ki ∈ Sα. Clearly g � h[E1] ∈ C∞perf for such an E1.

Proof of Proposition 4.1.3(b). Let πx and πy be the projections of R2

onto x-axis and y-axis, respectively, and consider functions hx = πx ◦ h andhy = πy ◦ h. Applying Lemma 3.2.1 two times we can find βx, βy ≤ α andE′ = Py ⊂ Px ⊂ E from Sα such that hx ◦ π−1

βxis a function on πβx(Px) ∈ Sβx ,

hy ◦ π−1βy

is a function on πβx(Py) ∈ Sβy

, and each of these functions is eitherone-to-one or constant. We will assume that βy ≤ βx, the other case beingessentially identical. Note that for every x ∈ E′ we have

h(x) = 〈πx ◦ h(x), πy ◦ h(x)〉 = 〈(hx ◦ π−1βx

)(πβx(x)), (hy ◦ π−1βy

)(πβy (x))〉.

Since h is one-to-one this implies that βx = α. So πx ◦h is a one-to-one functionfrom E′ into πx(E) ⊂ R. In particular, F = h[E′] ⊂ R2 is a function fromπx[h[E′]] into R.

Then, by Lemma 4.2.2 used with g = F and h = πx ◦ h � E′, we can find asubprism E1 of E′ and a function f : R → R extending F such that either f orf−1 belong to C1.


4.3 Proposition 4.2.1: a generalization of a the-orem of Morayne

Our proof of Proposition 4.2.1 is based on the following lemmas, the first ofwhich is a version of a theorem of Galvin [42, 43]. (For the proof see [53, thm.19.7] or [16].) More precisely, Galvin proved his results for α = 1.

Lemma 4.3.1 For every 0 < α < ω1 and every continuous symmetric functionh from Cα × Cα \∆ into {0, 1} there exists a P ∈ Sα such that h is constant onP 2 \∆.

Proof. For j < 2 let Gj be the set of all s ∈ Cα such that

(∀β < α)(∀ε > 0)(∃t ∈ Cα) 0 < ρ(s, t) < ε & s � β = t � β & h(s, t) = j

and notice that

each Gj is a Gδ-set and Cα = G0 ∪G1. (4.6)

Indeed, to see that Gj is a Gδ-set it is enough to note that for every β < α andε > 0 the set

Gβ,εj = {s ∈ Cα: (∃t ∈ Cα) 0 < ρ(s, t) < ε & s � β = t � β & h(s, t) = j}

is open in Cα. So let s ∈ Gβ,εj and take t ∈ Cα witnessing it, that is, such that0 < ρ(s, t) < ε, s � β = t � β, and h(s, t) = j. We can choose basic openneighborhoods U and V of s and t, respectively, such that U × V \∆ ⊂ h−1(j).In addition we can assume that πβ(U) = πβ(V ) and that each of the sets Uand V has the diameter less than δ = (ε− ρ(s, t))/3. Then s ∈ U ⊂ Gβ,εi sincefor every s′ ∈ U there exists a t′ ∈ V , t′ %= s′, with s′ � β = t′ � β (sinceπβ(U) = πβ(V )), h(s′, t′) ∈ h[U × V \∆] = {j} and

0 < ρ(s′, t′) ≤ ρ(s′, s) + ρ(s, t) + ρ(t, t′) ≤ δ + ρ(s, t) + δ < ε.

Thus each Gβ,εj is open and Gj is a Gδ-set.To see the second part of (4.6) assume, by way of contradiction, that there

exists an s ∈ Cα \ (G0 ∪ G1). Let β0, ε0 and β1, ε1 witness that s /∈ G0 ands /∈ G1, respectively. Put ε = min{ε0, ε1} > 0 and β = max{β0, β1} < α andfind t ∈ Cα such that t � β = s � β, ρ(s, t) < ε, and t(β) %= s(β). Then thereexists a j < 2 such that h(s, t) = j and this, together with t � βj = s � βj andρ(s, t) < εj contradicts the choice of βj and εj . This finishes the proof of (4.6).

Next find a j < 2 and a basic clopen set U in Cα such that Gj is residual inU . Replacing Cα with U , if necessary, we can assume that Gj is residual in Cα.Using Simple Fusion Lemma 3.1.4 with A = Bα we will find a P ∈ Sα for whichP 2 \∆ ⊂ h−1(j).

For each k < ω let Dk ∈ [Bα]<ω be the collection of all families {Pi: i < m}of pairwise disjoint sets, each of the diameter less than 2−k, such that

Pi × Pn ⊂ h−1(j) for all i < n < m. (4.7)


It is obvious Dk’s satisfy conditions (P1) and (P2) from Lemma 3.1.4. Thus,we need only to check (P3).

So, take E ∈ Bα and γ < α. It is enough to find disjoint E0, E1 ∈ Bα subsetsof E such that πγ [E0] = πγ [E1] and

E0 × E1 ⊂ h−1(j). (4.8)

For this choose an s ∈ E∩Gj and let ε0 > 0 be such that Bα(s, ε0) ⊂ E. By thedefinition of Gj we can find a t ∈ Cα for which 0 < ρ(s, t) < ε0, s � γ = t � γ,and h(s, t) = j. In particular s, t ∈ E and 〈s, t〉 ∈ h−1(j). Since h is continuouswe can find an ε ∈ (0,∞)\D small enough that E0 = Bα(s, ε) and E1 = Bα(t, ε)are disjoint subsets of E for which (4.8) holds.


}∈ Dk such that

P =⋂k<ω

⋃i<mk

P ki ∈ Sα. It is enough to show that P 2 \∆ ⊂ h−1(j). To seethis, take different s, t ∈ P and let k < ω be such that the distance between sand t is greater than 2−k. Then they must belong to different Pi’s from Pk andso, by (4.7), 〈s, t〉 ∈ h−1(j).

We will also need the following simple fact, which must be well known.

Lemma 4.3.2 There exists a continuous function h:C→ [0, 1] with the follow-ing property. If X is a zero-dimensional Polish space then for every continuousfunction f :X → [0, 1] there exists a continuous g:X → C such that f = h ◦ g.

Proof. Let {Uσ:σ ∈ 2<ω} be an open basis for [0, 1] such that U∅ = [0, 1]and, for every σ ∈ 2k, Uσ = Uσˆ0 ∪ Uσˆ1 and diam(Uσ) ≤ 21−k. For everys ∈ 2ω let h(s) ∈ [0, 1] be such that {h(s)} =

⋂n<ω cl(Us�n). It is clear that h

is continuous.To see that h is as required take X and f as in the lemma. For every σ ∈ 2<ω

choose an open set Vσ ⊂ f−1(Uσ) such that V∅ = X, Vσˆ0 and Vσˆ1 are disjoint,and Vσˆ0 ∪ Vσˆ1 = Vσ. This can be easily done by induction on the length ofσ using zero-dimensionality of X. Thus for every n < ω the sets {Uσ:σ ∈ 2n}form a clopen partition of X.

Define g(x) as the only s ∈ C for which x ∈⋂n<ω Vs�n. Clearly g is contin-

uous. Moreover, if g(x) = s then

x ∈⋂n<ω

Vs�n ⊂ f−1

( ⋂n<ω

cl(Us�n)

)= f−1({h(s)}) = f−1({h(g(x))})

so that f(x) ∈ {h(g(x))}. Hence f = h ◦ g.

The next lemma is already a very close approximation of Proposition 4.2.1.

Lemma 4.3.3 If α < ω1 and H is a continuous symmetric function from aset Cα × Cα \ ∆ into C then there exists an E ∈ Sα such that H is uniformlycontinuous on E2 \∆.


Proof. For n < ω define hn:Cα × Cα \∆ → 2 by hn(s, t) = H(s, t)(n). Thus,H = 〈hn:n < ω〉 and each hn satisfies the assumptions of Lemma 4.3.1.

Using Simple Fusion Lemma 3.1.4 with A = Sα we will find an E ∈ Sα forwhich each hn is uniformly continuous E2 \∆. Then clearly H is also uniformlycontinuous on this set.

For k < ω let Dk ∈ [Sα]<ω be the collection of all families {Pi: i < m} ofpairwise disjoint sets such that

hk is constant on Pi × Pi \∆ for each i < 2k. (4.9)

Clearly sets Dk’s satisfy conditions (P1) and (P2) from Lemma 3.1.4. Thus, weneed to verify only (P3).

So, fix k < ω, E ∈ Sα, and γ < α. It is enough to find disjoint subprismsE0, E1 of E, each of diameter less than 2−k, such that πγ(E0) = πγ(E1) and

hk is constant on Ej × Ej \∆ for each j < 2.

Let f ∈ Φprism(α) be such that E = f [Cα] and let h:Cα × Cα \∆ → {0, 1} bedefined by h(s, t) = hk(f(s), f(t)). Then h satisfies the assumptions of Lemma4.3.1 so there exists a P ∈ Sα such that h is constant on P 2 \ ∆. Choosedisjoint subprisms E0, E1 of P each of diameter less than 2−k and such thatπγ(E0) = πγ(E1). They satisfy (P3).


}∈ Dk such that

E =⋂k<ω

⋃i<mk

P ki ∈ Sα. Notice that if {Pi: i < mk} belongs to Dk thenhk is uniformly continuous on

( ⋃i<mk

Pi

)2

\∆ =

⋃i �=n

Pi × Pn

∪

⋃i<mk

(Pi × Pi \∆) .

So each hk is uniformly continuous E2 \∆ ⊂(⋃

i<mkPi

)2 \∆.

Proof of Proposition 4.2.1. Let f0 ∈ Φprism(α) be such that E = f0 [Cα]and put

F = G ◦ 〈h ◦ f0, h ◦ f0〉:Cα × Cα \∆ → [0, 1]. (4.10)

Note also thatF = h ◦H (4.11)

for some continuous symmetric function fromH:Cα×Cα\∆ → C and continuoush0:C→ [0, 1]. This follows immediately from Lemma 4.3.2 used with a functionf = F � {〈x, y〉 ∈ Cα × C:x < y}, where < is the lexigographical order on Cα.

Now, by Lemma 4.3.3, then there exists an E1 ∈ Sα such that H is uniformlycontinuous on (E1)2 \ ∆. So H can be extended to a uniformly continuousfunction H on (E1)2. Then function

G = h0 ◦ H ◦ 〈h ◦ f0, h ◦ f0〉−1 = h0 ◦ H ◦ 〈(f0)−1 ◦ h−1, (f0)−1 ◦ h−1〉


is also uniformly continuous on (h[f0[E1]])2. Put E1 = f0[E1] and notice thatit is as desired.

Indeed, clearly E1 ∈ Sα and E1 ⊂ E. Moreover, it is not difficult to see thatG � h[E1]2 \∆ = G � h[E1]2 \∆. So G is uniformly continuous on h[E1]2 \∆.

4.4 Theorem 4.1.6: on cov (Dn, Cn) < c

In the proof we will use the following lemma.

Lemma 4.4.1 For n < ω let f ∈ Cn and let P ⊂ R be a perfect set for whichthe function F :P 2 \∆ → R defined by

F (x, y) =f (n)(x)− f (n)(y)

x− y

is uniformly continuous and bounded. Then f � P can be extended to a Cn+1

function.

Proof. This follows from the fact that f � P satisfies the assumptions ofWhitney’s extension theorem. To see this notice first that F naturally extendsto a continuous function on P 2 with F (a, a) = f (n+1)(a). Next, for q = 1, 2, 3, . . .and a ∈ P let

ηq(a) = sup{∣∣∣∣f (n)(x)− f (n)(a)

x− a − f (n+1)(a)∣∣∣∣ : 0 < |x− a| < 1

q

}.

In the second part of the proof of [39, thm. 3.1.15] it is shown that if

limq→∞

sup{ηq(a): a ∈ P} = 0 (4.12)

then f � P satisfies the assumptions of Whitney’s extension theorem. Howeverwe have

f (n)(x)− f (n)(a)x− a − f (n+1)(a) = F (x, a)− F (a, a),

so uniform continuity of F clearly implies (4.12).

Proof of Theorem 4.1.6. The lower bound inequalities cov (Dn, Cn) > ωand cov (“Dn”, “Cn”) > ω follow from Example 4.5.7. So it is enough to proveonly that these numbers are ≤ ω1.

To prove cov (Dn, Cn) ≤ ω1, take an f ∈ Dn and note that, by CPAprism, itis enough to show that the following set

E = {E ∈ Perf(R): (∃h ∈ Cn(R)) h � E = f � E}

is Fprism-dense. So fix a prism P in R. Let k: [−∞,∞]→ [0, 1] be a homeomor-phism. Applying n-times Proposition 4.2.1 in the same way as in the proof of


Lemma 4.2.2 we find a subprism E of P such that for each i < n the functionk ◦ Fi:P 2 \∆ → [0, 1] is uniformly continuous, where Fi:P 2 \∆ → R is definedby

Fi(x, y) =f (i)(x)− f (i)(y)

x− y .

So each Fi can be extended to a continuous function Fi:P 2 → [−∞,∞]. Notealso that since Fi(x, x) = f (i+1)(x) ∈ R, as f ∈ Dn, we in fact have Fi[P 2] ⊂ R.

Next, starting with f0 = f we use Lemma 4.4.1 to prove by induction thatfor every i < n there exists an fi+1 ∈ Ci+1(R) extending fi � E. Then functionh = fn ∈ Cn(R) witnesses that E ∈ E .

To prove cov (“Dn”, “Cn”) ≤ ω1, take an f ∈ “Dn”. As before it is enoughto show that

E ′ = {E′ ∈ Perf(R): (∃h ∈ “Cn(R)”) h � E′ = f � E′}

is Fprism-dense. So fix a prism P in R and find E, Fi’s, and Fi’s as above. Notethat Fk’s are well defined since f ∈ “Dn” ⊂ Cn−1. By the same reason we havethat Fi[P 2] ⊂ R for all i < n. However, Fn can have an infinite values.

Proceeding as in the proof of Lemma 4.2.2, decreasing E if necessary, wecan assume that either the range of Fn is bounded or Fn � P 2 ∩∆ is constantequal to ∞ or −∞. If Fn is bounded then, taking E′ = E, we are done as inthe previous case. So, assume that Fn[P 2 ∩ ∆] = {∞}. (The case of −∞ ishandled by replacing f with −f .) Then f (n−1) and E satisfy the assumptions ofBrown’s version of Whitney’s extension theorem [11, thm. 2.1]. So, we can finda “C1” extension g: R → R of f (n−1) � E such that g′[E] = (f (n−1))′[E] = {∞}and g′[R \ E] ⊂ R. By (n − 1) times integrating g we can find a g: R → R

such that G(n−1) = g. Then G ∈ “Cn”. Next notice that G− f ∈ Cn(E), since(G − f)(n−1) = g − f (n−1) ≡ 0 on E. Now, proceeding as above for the caseof f ∈ Cn we can find a subprism E′ of E and a function h ∈ Cn(R) extendingG−f � E′. Then function h = G− h belongs to “Cn” as a difference of functionsfrom “Cn” and Cn. Moreover, h extends f � E′ since h = G−h = G−(G−f) = fon E′. So, h witness E′ ∈ E ′.

4.5 Examples related to cov operator

We will start with the examples needed for the proof of Proposition 4.1.5 whichgive c as a lower bound for the appropriate numbers cov(A,F).

Example 4.5.1 There exist a homeomorphism h: R → R and a perfect setP ⊂ R such that h, h−1 ∈ “C2”, h′′ � P ≡ ∞, and (h−1)′′ � h[P ] ≡ −∞. In

particular ¬IntTh(h � P,D2

perf ∪ (D2perf)

−1)

and

cov(“C2”, D2

perf ∪ (D2perf)

−1)

= cov(h,D2

perf ∪ (D2perf)

−1)

= c.


Proof. First notice that there exist a strictly increasing homeomorphism h0

from R onto (0,∞) and a perfect set P ⊂ R such that

h0 ∈ “C1” and h′0 � P ≡ ∞. (4.13)

Indeed, let C be an arbitrary nowhere dense perfect subset of [2, 3] with 2 ∈ Cand let d(x) denotes the distance between x ∈ R and C. Let f0: (0,∞)→ [0,∞)be defined by f0(x) = x−2 for x ∈ (0, 1] and f0(x) = d(x) for x ∈ [1,∞).Then f0 is continuous and f0(x) = 0 precisely when x ∈ C. Define a strictlyincreasing function f from (0,∞) onto R by a formula f(x) =

∫ x1f0(t) dt. Then

f ′ = f0 and f(x) = 1 − 1x on (0, 1). It is easy to see that h0 = f−1 and

P = f [C] ⊂ (0,∞) satisfy (4.13).Now put h(x) =

∫ x0h0(t) dt. Then clearly h is strictly increasing since h0

is positive. Also h is onto R as on (−∞, 0) we have h0(x) = 11−x and so

h(x) = − ln(1 − x). It is easy to see that h′ = h0 so, by (4.13), h ∈ “C2” andh′′ � P ≡ ∞. Also, if g = h−1 then g′(x) = 1/h′(g(x)) = 1/h0(g(x)) > 0 isstrictly decreasing and h−1 = g ∈ C1. Thus, to see that h−1 = g ∈ “C2” and(h−1)′′ ≡ −∞ on P = f [C] it is enough to notice that g′′(x) = −h′′(x)[g′(x)]3,which is obtained by differentiating twice h(g(x)) = x. Thus, h and P have thedesired properties.

To see the additional part note first that for every f ∈ D2perf functions f and

h � P may agree on at most countable set S since at any point x of a perfectsubset Q of S we would have

(h � Q)′′(x) =∞ %= (f � Q)′′(x).

Similarly, |f ∩ (h � P )| ≤ ω for every f ∈ (D2perf)

−1. This clearly implies theadditional part.

Example 4.5.2 There exists a perfect set P ⊂ R and a function f ∈ “C1” such

that f ′(x) =∞ for every x ∈ P . In particular ¬IntTh(f � P,D1

perf

)and

cov(Borel, C1

)= cov

(“C1”, C1

)= cov

(“C1”, D1

perf

)= cov

(f,D1

perf

)= c.

Proof. If f is a function h0 from (4.13) then it has the desired properties.For such an f and any function g ∈ D1

perf the intersection f ∩ g must befinite. So

c ≥ cov(Borel, C1

)≥ cov

(“C1”, D1

perf

)≥ cov

(f,D1

perf

)≥ c.

Monotonicity of cov operator gives the other equations.

Example 4.5.3 For every 0 < n < ω there exists an f ∈ “Cn” and a perfect

set P ⊂ R such that ¬IntTh(f � P,Dn

perf

)so that

cov(“Cn”, Cn) = cov(“Cn”, Dnperf) = cov(f,Dn

perf) = c.


Proof. For n = 1 this is a restatement of Example 4.5.2. The general case canbe done by induction: If f is good for some n and F is a definite integral of fthen F ∈ “Cn+1” and cov(F,Dn+1

perf ) = c.

Example 4.5.4 There exists an f ∈ C1 and a perfect set P ⊂ R such that|(f � P ) ∩ g| ≤ ω for every g ∈ “D2”. In particular ¬IntTh

(f � P, “D2”

)and

cov(C1, “D2”

)= cov

(f, “D2”

)= c.

Proof. In [1, thm. 22] the authors construct a perfect set P ⊂ [0, 1] anda function f ∈ C1 which is as required. The argument for this is implicitlyincluded in the proof of [1, thm. 22] and goes like that.

Function f has the property that f ′(x) = 0 for all x ∈ P . Now, assume thatsome g ∈ “D2” agrees with f on a perfect set Q ⊂ P . Then clearly we wouldhave (g � Q)′′ ≡ [(g � Q)′]′ ≡ [(f � Q)′]′ ≡ [0]′ ≡ 0. On the other hand, in [1,thm. 22] it is shown that for such a g1 we would have g′′(x) ∈ {±∞} for everyx ∈ Q, a contradiction.

Example 4.5.5 For every 0 < n < ω there exist an f ∈ Cn and a perfect setP ⊂ R such that ¬IntTh

(f � P, “Dn+1”

)and

cov(Cn, “Dn+1”

)= cov

(f, “Dn+1”

)= c.

Proof. For n = 1 this is a restatement of Example 4.5.4. The general case canbe done by induction: If f is good for some n and F is a definite integral of fthen F ∈ Cn+1 and cov(F, “Dn+2”) = c.

Next we will describe the examples showing that the cov(A,F) numbersconsidered in Corollary 4.1.4 and Theorem 4.1.6 have values greater than ω. Inwhat follows cov(M) (cov(N ), respectively) will stand for the smallest cardi-nality of a family F ⊂ P(R) of measure zero sets (nowhere dense, respectively)such that R =

⋃F .

Example 4.5.6 There exists a function f ∈ D1 such that

cov(f, “C1” ∪ (D1)−1

)≥ cov(M) > ω.

In particular

cov(Borel, “C1”

)≥ cov

(D1, “C1”

)≥ cov(M) > ω

and

cov(Borel, C1 ∪ (C1)−1

)≥ cov

(D1, C1 ∪ (C1)−1

)≥ cov(M) > ω.

1Actually, the calculation in [1, thm. 22] is done under the assumption that g ∈ C2, butit works also under our weaker assumption that g ∈ “D2”.


Proof. We will construct function f only on [0, 1]. It can be easily modifiedto a function defined on R.

Let E ⊂ [0, 1] be an Fσ-set of measure 1 such that Ec = [0, 1] \E is dense in[0, 1]. It is well known that there exists a derivative g: [0, 1] → [0, 1] such thatg[E] ⊂ (0, 1] and g[Ec] = {0}. (See e.g. [14, p. 24].) Let f : [0, 1] → R be suchthat f ′ = g. We claim that this f is as desired.

Indeed, by way of contradiction assume that for some κ < cov(M) thereexists a family {hξ ∈ RR: ξ < κ} ⊂ “C1” ∪ (D1)−1 such that f ⊂

⋃ξ<κ hξ.

Since hξ are closed subsets of R2 and the graph of f is compact, we see thatthe x-coordinate projections Pξ = πx(f ∩ hξ) are closed. So, [0, 1] is covered byless than cov(M) closed sets Pξ. Thus, there exists an η < κ such that Pη hasnon-empty interior U = int(Pη).

Now, if hη ∈ “C1” then h′η = f ′ = g on U , which is impossible, sinceh′η is continuous, while g is not continuous on any non-empty open set. Soassume that hη ∈ (D1)−1. Note that f is strictly increasing as an integral offunction g which is strictly positive a.e. So f−1 is a strictly increasing andagrees with h = h−1

η ∈ D1 on an open set f [U ]. But then if x ∈ U \ E thenh′(f(x)) = (f−1)′(f(x)) = 1

f ′(x) =∞ which contradicts h ∈ D1.

Note also that if f from Example 4.5.6 is replaced by its (n−1)-st antideriva-tive then we get also the following example.

Example 4.5.7 For any 0 < n < ω1 there exists an f ∈ Dn such that

cov (Dn, “Cn”) ≥ cov (f, “Cn”) ≥ cov(M) > ω.

Example 4.5.8 There exists an f ∈ C0 such that

cov(C0, “D1perf”) ≥ cov(f, “D1

perf”) ≥ cov(N ) > ω.

Moreover, for every n < ω if F ∈ Cn is such that F (n) = f then

cov(Cn, “C∞perf”

)≥ cov(Cn, “Dn+1

perf ”) ≥ cov(f, “Dn+1perf ”) ≥ cov(N ) > ω;

in particular

cov(Borel, “C∞perf”

)≥ cov

(Cn, “C∞perf”

)≥ cov(N ) > ω.

Proof. A function f justifying cov(f, “D1perf”) ≥ cov(N ) was pointed my

Morayne: just take any f ∈ C for which there is a set A ⊂ R of positive measurefor which |f−1(a)| = c for all a ∈ A. (See [85, thm. 6.1].) The remaining factsare obvious.

Chapter 5

Applications of the axiomCPA

gameprism

First notice that the proof identical to that for Theorem 2.1.1 gives also itsprism version that reads as follows.

Theorem 5.0.9 Assume that CPAgameprism holds and let X be a Polish space. If

D ⊂ Perf(X) is Fprism-dense and it is closed under perfect subsets then thereexists a partition of X into ω1 disjoint sets from D ∪ {{x}:x ∈ X}.

Notice that using Theorem 5.0.9 we can obtain the following generalizationsof Theorems 4.1.1 and 4.1.6.

Corollary 5.0.10 The graphs of covering functions in Theorems 4.1.1 and 4.1.6can be chosen as pairwise disjoint.

5.1 Nice Hamel basis

To prove the next theorem, which is a version of Theorem 5.0.9, we will needa variant of Mycielski’s theorem [71], in which we will use the following ter-minology. Let X be a Polish space, Fn be a countable family of closed n-aryrelations on X, and let F =

⋃1<n<ω Fn = {Ri ⊂ Xni : i < ω}. (In our main

applications relations Ri(z1, . . . , zn) will be the linear dependence relations:q1z1+· · ·+qnzn = 0.) We say that {x1, . . . , xn} ∈ [X]n is F-dependent providedthere exists an R ∈ Fn such that R(x1, . . . , xn) holds; otherwise we say that{x1, . . . , xn} is F-independent. We say that a subset S of X is F-independentprovided every finite subset of S is F-independent. The family F has finitecharacter provided for every F-independent set {x1, . . . , xn−1} ∈ [X]n−1 andevery R ∈ Fn the set

{x ∈ X:R(x1, . . . , xn−1, x)}is finite.

71


Note that if X = R and F is either the family of all linear dependencerelations or the family of all algebraic dependence relations then F has finitecharacter.

Proposition 5.1.1 Let X be a Polish space and F be a countable family ofclosed n-ary relations on X such that F has finite character. Then for everyprism P in X there is a subprism Q of P such that Q is F-independent.

Proof. We will prove it using Prism-Fusion Lemma 3.1.3. Let g ∈ Φprism(α)be such that P = g[Cα] and let {Rk: k < ω} be an enumeration of F with infiniterepetitions. We assume that each Rk is an nk-ary relation.

For k < ω let Dk be the collection of all families {Ei: i < 2k} of pairwisedisjoint sets from A = Bα, each of the diameter less than 2−k, and such that

• Rk(g(x1), . . . , g(xnk)) does not hold as long as each xj is chosen from a

different set from {Pi: i < 2k}.

Clearly condition (P1) is satisfied. We need to check condition (∗).So, fix k < ω, γ < α, and {Ei: i < 2k} ∈ Dk. Since family F has finite

character, we can find (by an easy induction of length 2k+1) a sequence S ={pji ∈ Ei: i < 2k & j < 2} such that {g(pji ): i < 2k & j < 2} is F-independentand πγ(p0i ) = πγ(p1i ) for every i < 2k. Thus, Rk(g(x1), . . . , g(xnk

)) does nothold for any different xj ’s from S. Now, since Rk is closed, we can find an ε > 0small enough that the clopen balls Eji = B(pji , ε) justify (∗).

Now, by Lemma 3.1.3, there exist a Pk ={P ki : i < 2k

}∈ Dk such that Q0 =⋂

k<ω

⋃i<2k P ki ∈ Sα. It is enough to show that Q = g[Q0] is F-independent.

So, let R ∈ F be n-ary and choose different points xi ∈ Q0 for i < n. Let k < ωbe such that R = Rk and 21−k is less than the minimum distance between xi’s.Then xi’s belong to different sets from {P ki : i < 2k} and soR(g(x0), . . . , g(xn−1))does not hold.

Remark 5.1.2 Note that Proposition 5.1.1 is false if we replace prisms withcubes. In particular, there is a cube P in R without linearly independent sub-cube.

Proof. Indeed, let P1 and P2 be disjoint perfect subsets of R such that P1∪P1

is linearly independent over Q. Let f :P1 × P2 → R be defined by a formulaf(x1, x2) = x1 + x2. Then P = f [P1 ×P2] is a cube in R. To see that P has nolinearly independent subcube let Q = Q1 × Q2 be a subcube of P and choosedifferent a1, b1 ∈ Q1 and a2, b2 ∈ Q2. Then {a1+a2, a1+b2, b1+a2, b1+a2} ⊂ Qand they are clearly linearly dependent.

Corollary 5.1.3 For every prism P in R there is a subprism Q of P such thatQ is algebraically (so linearly) independent.

From Theorem 5.0.9 and Corollary 5.1.3 we conclude immediately the fol-lowing fact.


Corollary 5.1.4 CPAgameprism implies that R is a union of ω1 disjoint closed alge-

braically independent sets.

The next theorem tells that we can, in fact, have considerable stronger result.Its proof is quite similar to that for Theorems 2.1.1 and 5.0.9.

Theorem 5.1.5 CPAgameprism implies that there is a family H of ω1 pairwise dis-

joint perfect subsets of R such that H =⋃H is a Hamel basis.

Proof. For a prism P in R let H(P ) be a subsubprism of P which is linearlyindependent. For a countable family P of perfect linearly independent subsetsof R and a prism P we define a linearly independent subprism Q = Q(P, P ) ofP as follows. Let L = LIN(

⋃P) be a linear subspace (of R over Q) generated

by⋃P. If H(P ) \ L is of second category in H(P ) then we use Claim 1.1.5 to

find Q ⊂ H(P ) \L. Otherwise we use Claim 1.1.5 to choose Q ⊂ H(P )∩L. Wealso put Q(P, {x}) = {x}.

Now consider the following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q({Qη: η < ξ}, Pξ).

By CPAgameprism strategy S is not a winning strategy for Player II. So there exists

a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which Player II loses, thatis, R =

⋃ξ<ω1

Qξ.Notice that for every ξ < ω1 prism Qξ is linearly independent and either is

disjoint with Lξ = LIN(⋃η<ξ Qη) or it is contained in Lξ. Let H0 be the family

of those Qξ, ξ < ω1, for which Qξ ∩ Lξ = ∅.It is easy to see that

⋃H0 is a Hamel basis. However, H0 may contain

singletons. To get H as in the theorem, first note that refining H0, if necessary,we may assume that for every ε > 0 family H0 contains ω1 sets of diameterless than ε. Moreover, by fixing an x ∈

⋃H0 we can replace each P ∈ H0 with

qx+P for some q ∈ Q and the resulting family will still be pairwise disjoint withunion being a Hamel basis. Thus, without loss of generality, we can assume thatevery open interval in R contains ω1 sets from H0. Now, for every singleton {x}in H0 we can choose a sequence P x1 > P x2 > P x3 > · · · from H0 converging tox, and replace a family {x} ∪ {P xn :n < ω} with its union. (We assume that wechoose different sets P xn for different singletons.) If H is such a modification ofH0 then H is as desired.

Note that essentially the same proof gives also the following theorem.

Theorem 5.1.6 CPAgameprism implies that there is a family B of ω1 disjoint perfect

subsets of R such that B =⋃B is a transcendental basis.

Let I be an additive invariant ideal on R. We say that a subset X of R

is I-rigid provided X /∈ I but X�(r + X) ∈ I for every r ∈ R. An easyinductive construction gives a non-measurable subset X of R without a Baireproperty which is [R]<c-rigid. (First such a construction, under CH, comes from


Sierpinski [80]. Compare also [47].) Thus, under CH or MA there are N ∩M-rigid sets. Recently Cichon, Jasinski, Kamburelis, and Szczepaniak [19] studiedthese sets and noticed that there are no N ∩M-rigid sets in the random andCohen models. The next corollary shows that the existence of such sets followsfrom CPAgame

prism.

Corollary 5.1.7 CPAgameprism implies there exists an N ∩M-rigid set X which is

neither measurable not has it the Baire property.

Proof. Let H = {Qξ: ξ < ω1} be from Theorem 5.1.5 and for every ξ < ω1

let Lξ = LIN(⋃

η<ξ Qη

). Then R is an increasing union of Lξ’s and each Lξ

belongs to N ∩M, since it is a proper Borel subgroup of R.Let X0 = {xξ: ξ < ω1} ⊂ R be a non-measurable set without the Baire

property and defineX =

⋃ξ<ω1

(xξ + Lξ).

To see that X is as desired it is enough to notice that for every r ∈ Lζ

X�(r +X) ⊂⋃ξ<ζ

[(xξ + Lξ) ∪ (r + xξ + Lξ)] ∈ N ∩M.

This finishes the proof.

Our next application of Theorem 5.1.5 is the following

Corollary 5.1.8 CPAgameprism implies there exists a function f : R → R such that

for every h ∈ R the difference function ∆h(x) = f(x+h)−f(x) is Borel; however,for every α < ω1 there is an h ∈ R such that ∆h is not of Borel class α.

Note that answering a question of Laczkovich [58] Filipow and RecRlaw [40]gave an example of such an f under CH. RecRlaw also asked (private commu-nication) whether such a function can be constructed in absence of CH. Corol-lary 5.1.7 gives an affirmative answer to this question. It is an open questionwhether such a function exists in ZFC.

Proof. The proof is quite similar to that for Corollary 5.1.7.Let H = {Qξ: ξ < ω1} be from Theorem 5.1.5. For every ξ < ω1 define

Lξ = LIN(⋃

η<ξ Qη

)and choose a Borel subset Bξ of Qξ of Borel class greater

than ξ. DefineX =

⋃ξ<ω1

(Bξ + Lξ)

and let f be the characteristic function χX of X.To see that h is as required note that

∆h(x) = χ(h+X)\X − χX\(h+X).


So, it is enough to show that each of the sets (h +X) \X and X \ (h +X) isBorel, though they can be of arbitrary high class. For this, notice that for everyh ∈ Lα+1 \ Lα we have

h+X = h+⋃ξ<ω1

(Bξ + Lξ) =⋃ξ≤α

(h+Bξ + Lξ) ∪⋃

α<ξ<ω1

(Bξ + Lξ)

and that the sets⋃ξ≤α(h+Bξ+Lξ) ⊂ Lα+1 and

⋃α<ξ<ω1

(Bξ+Lξ) are disjoint.So

(h+X) \X =⋃ξ≤α

(h+Bξ + Lξ) \X =⋃ξ≤α

(h+Bξ + Lξ) \⋃ξ≤α

(Bξ + Lξ)

is Borel, since each set Bξ + Lξ is Borel. (It is a subset of Qξ + Lξ, which ishomeomorphic to Qξ × Lξ via addition function.) Similarly, set X \ (h+X) isBorel.

Finally notice that for h ∈ Qα \Bα the set

(h+X) \X =⋃ξ≤α

(h+Bξ + Lξ)

is of Borel class greater than α, since so is (h+Qα) ∩ [(h+X) \X] = h+Bα.Thus, ∆h(x) can be of an arbitrary hight Borel class.

5.2 Selective and crowded ultrafilters; numbersi and u

This section is based on our paper [29]. We will use here the terminologyintroduced in Preliminaries Section. Recall, in particular, that every weaklyselective ultrafilter is selective and that the ideal I = [ω]<ω is selective. We willalso need the following fact.

Fact 5.2.1 The ideal I of nowhere dense subset of rationals Q is weakly selec-tive.

Proof. Let A ∈ I+ and take an f :A → ω. If there is a B ∈ I+ ∩ P(A) suchthat f � B constant then we are done. So, assume that it is not the case andlet A0 ⊂ S be dense on some interval. By induction on n < ω define a sequence{bn ∈ A0:n < ω} dense in A0 such that f restricted to B = {bn:n < ω} isone-to-one. Then B is as desired.

In what follows we will also need the following fact on weakly selective ideals,which can be found in Grigorieff [44, prop. 14].

Proposition 5.2.2 Let I be a weakly selective ideal on ω and A ∈ I+. IfT ⊂ A<ω is a tree such that

A \ {j < ω: s j ∈ T} ∈ I for every s ∈ Tthen there exists a branch b of T such that b[ω] ∈ I+.


Theorem 5.2.3 CPAgameprism implies that for every selective ideal I on ω there

exists a selective ultrafilter F on ω such that F ⊂ I+.In particular if CPAgame

prism holds then there is a selective ultrafilter on ω.

The proof is based on the following lemma.

Lemma 5.2.4 Let I be a weakly selective ideal on ω.

(a) For every A ∈ I+ and a prism P in ωω there exist B ∈ I+, B ⊂ A, and asubprism Q of P such that either

(i) g � B is one-to-one for every g ∈ Q, or else

(ii) there exists a k < ω such that g � B is constant equal to k for everyg ∈ Q.

(b) For every A ∈ I+ and a prism P in [ω]ω there exist B ∈ I+, B ⊂ A, anda subprism Q of P such that either

– x ∩B = ∅ for every x ∈ Q, or else

– B ⊂ x for every x ∈ Q.

Proof. (a) Fix an A ∈ I+, an f ∈ Fprism(ωω) from Cα onto P , and assumethat for no subprism Q of P and B ∈ I+ ∩ P(A) condition (ii) holds. We willfind Q satisfying (i). This will be done in four steps.

Step 1. For every E ∈ Sα and n < ω the set

ZnE = {i ∈ A: (∀E′ ∈ Sα, E′ ⊂ E)(∃g ∈ E′) f(g)(i) = n}

belongs to I.

By way of contradiction assume that there are E ∈ Sα and n < ω for whichthe set B = ZnE belongs to I+. Notice that without loss of generality we canassume that E = Cα. This is the case, since we can replace f with f ◦ h, whereh ∈ Φprism(α) is such that E = h[Cα]. We will use Simple Fusion Lemma 3.1.4with A = Sα to find Q satisfying (ii), contradicting our assumption.

Let B = {ik: k < ω} and for k < ω let Dk be the collection of all pairwisedisjoint families {Ej ∈ Sα: j < m} such that f(g)(ik) = n for all g ∈

⋃j<mEj .

It is easy to see that sets Dk satisfy conditions (P1)–(P3) from Lemma 3.1.4,where condition (P3) follows from the fact that ik ∈ ZnE . Now, if Pk ∈ Dk aresuch that E0 =

⋂k<ω

⋃Pk ∈ Sα then subprism Q = f [E0] of P satisfies (ii)

with B ⊂ A.

Step 2. There is no n < ω for which there exist A ∈ I+ and E ∈ Sα such thatA ⊂ A and

f(g)(i) ≤ n for all g ∈ E and i ∈ A. (5.1)

By way of contradiction assume that there exist n < ω, E ∈ Sα, and A ∈I+ ∩P(A) for which (5.1) holds. We can assume that n is the smallest number


with this property. As in Step 1 we can also assume that E = Cα. Also byStep 1 we have that n > 0. We will obtain a contradiction with the minimalityof n.

We will use Game-Fusion Lemma 3.1.1 with A = Bα to construct a treeT ⊂ A<ω and the mappings T 5 s (→ Es ∈ [A]<ω and T 5 s (→ Ds ⊂ [A]<ω

such that T satisfies the assumptions of Proposition 5.2.2 and for every branchb of T the sequence 〈〈Eb�k,Db�(k+1)〉: k < ω〉 is a game as in Lemma 3.1.1.

The construction is done by induction on levels. Thus, we start with E0 ={{Cα}}. Next, if Es = {E0, . . . , Ens

} is already defined for some s ∈ T notethat, by Step 1,

⋃i≤ns

ZnEi∈ I. We will define T -successors of s by

s m ∈ T if and only if m ∈ A \⋃i≤ns

ZnEi.

This will take care of the assumptions of Proposition 5.2.2. Next pick an mfrom A \

⋃i≤ns

ZnEiand define Dsˆm as a family of all pairwise disjoint sets

{Eji ∈ A: i ≤ ns & j < 2} such that for every i ≤ ns and j < 2

Eji ⊂ Ei and f(g)(m) %= n for every g ∈ Eji . (5.2)

Then Dsˆm clearly satisfies (P1) from Lemma 3.1.1. To see that it also satisfies(∗) note that for every i ≤ ns there exists an E′

i ∈ Sα, E′i ⊂ Ei, such that

f(g)(m) %= n for all g ∈ E′i. This is the case since m ∈ A \ ZnEi

. Now forevery 0 < γ < α it is easy to find disjoint E0

i , E1i ∈ A subsets of Ei for which

πγ [E0i ] = πγ [E1

i ]. Then {Eji ∈ A: i ≤ ns & j < 2} belongs to Dsˆm and soDsˆm satisfies (∗). The construction is finished by picking up Esˆm from Dsˆmaccording to the strategy S from Lemma 3.1.1.

Now, by Proposition 5.2.2, there is a branch b of T such that Ab = b[ω]belongs to I+. Clearly Ab ⊂ A. Let Eb =

⋂k<ω

⋃Eb�k. Then Eb ∈ Sα.

To get a contradiction with the minimality of n it is enough to notice thatf(g)(m) < n for all g ∈ Eb and m ∈ Ab. This, however, follows immediatelyfrom (5.1) and (5.2).

Step 3. For every E ∈ Sα and n < ω the set

Z≤nE = {i ∈ A: (∀E′ ∈ Sα, E

′ ⊂ E)(∃g ∈ E′) f(g)(i) ≤ n}

belongs to I.

The argument is almost identical to that used in Step 1. By way of contra-diction assume that there are E ∈ Sα and n < ω for which the set A = Z≤n

E

belongs to I+. As before we note that without loss of generality we can assumethat E = Cα. We will use Simple Fusion Lemma 3.1.4 with A = Sα to find anE ∈ Sα such that (5.1) is satisfied. This clearly contradicts our Step 2.

Let A = {ik: k < ω} and for k < ω let Dk be the family of all pairwisedisjoint families {Ej ∈ Sα: j < m} such that f(g)(ik) ≤ n for all g ∈

⋃j<mEj .

It is easy to see that sets Dk satisfy conditions (P1)–(P3) from Lemma 3.1.4,where condition (P3) follows from the fact that ik ∈ Z≤n

E . Now, if Pk ∈ Dk aresuch that E =

⋂k<ω

⋃Pk ∈ Sα then (5.1) is satisfied.


Step 4. There exist B ∈ I+ and E ∈ Sα such that B ⊂ A and

f(g) � B is one-to-one for every g ∈ E. (5.3)

Certainly proving this will finish the proof of part (a) of the lemma sincethen Q = f [E] is as required.

The proof that follows is very similar to the argument used in Step 2. Asbefore, we will use Game-Fusion Lemma 3.1.1 with A = Bα to construct a treeT ⊂ A<ω and the mappings T 5 s (→ Es ∈ [A]<ω and T 5 s (→ Ds ⊂ [A]<ω

such that T satisfies the assumptions of Proposition 5.2.2 and for every branchb of T the sequence 〈〈Eb�k,Db�(k+1)〉: k < ω〉 is a game as in Lemma 3.1.1.

The construction is done by induction on levels. Thus, we start with E0 ={{Cα}}. Next, if Es = {E0, . . . , En} is already defined for some s ∈ T let

ns = max

g(k): g ∈

⋃i≤n

Ei & k ∈ range(s)

.

Notice that ns is finite, since the sets Ei are compact. Also by Step 3 we have⋃i≤n Z

≤ns

Ei∈ I. We will define T -successors of s by

s m ∈ T if and only if m ∈ A \⋃i≤n Z

≤ns

Ei.

Therefore the assumptions of Proposition 5.2.2 will be satisfied. Next pick anm ∈ A \

⋃i≤n Z

≤ns

Eiand define Dsˆm as a family of all pairwise disjoint sets

{Eji ∈ A: i ≤ n & j < 2} such that for every i ≤ n and j < 2

Eji ⊂ Ei and f(g)(m) > ns for every g ∈ Eji . (5.4)

Then Dsˆm clearly satisfies (P1) from Lemma 3.1.1. To see that it also satisfies(∗) note that for every i ≤ n there exists an E′

i ∈ Sα, E′i ⊂ Ei, such that

f(g)(m) > ns for all g ∈ E′i. This is the case since m ∈ A \ Z≤ns

Ei. Now for

every 0 < γ < α it is easy to find disjoint E0i , E

1i ∈ A subsets of Ei for which

πγ [E0i ] = πγ [E1

i ]. Then {Eji ∈ A: i ≤ ns & j < 2} belongs to Dsˆm and soDsˆm satisfies (∗). The construction is finished by picking up Esˆm from Dsˆmaccording to the strategy S from Lemma 3.1.1.

Now, by Proposition 5.2.2, there is a branch b of T such that B = b[ω]belongs to I+. Clearly B ⊂ A. Let E =

⋂k<ω

⋃Eb�k ∈ Sα. Then, by (5.4) and

the definition of ns, E satisfies (5.3).This finishes the proof of (a).

(b) Since the characteristic function χ gives an embedding from [ω]ω into2ω ⊂ ωω prism P can be identified with χ[P ] = {χx:x ∈ P}. Applying part (a)to χ[P ] we can find a subprosm Q of P , k < 2, and B ∈ I+, B ⊂ A, such thatχx � B ≡ k for every x ∈ Q. If k = 0 this gives x ∩ B = ∅ for every x ∈ Q. Ifk = 1 we have B ⊂ x for every x ∈ Q.


Proof of Theorem 5.2.3. Let I be a selective ideal on ω. For a countablefamily A ⊂ I+ linearly ordered by ⊂∗ let C(A) ∈ I+ be such that C(A) ⊂∗ Afor every A ∈ A.

For A ∈ I+ and f ∈ Fprism(ωω) put P = range(f) and let B(A,P ) ∈ [A]ω

and a subprism Q(A,P ) of P be as in Lemma 5.2.4(a). If f ∈ Cprism(ωω)and P = range(f) = {x} then we put Q(A,P ) = P and take B(A,P ) ∈ [A]ω

satisfying the conclusion of Lemma 5.2.4(a).Consider the following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q(C({Bη: η < ξ}), Pξ),

where sets Bη are defined inductively by Bη = B(C({Bζ : ζ < η}), Pη).By CPAgame

prism strategy S is not a winning strategy for Player II. So thereexists a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which and Player IIloses, that is, ωω =

⋃ξ<ω1

Qξ.Now, let F be a filter generated by {Bξ: ξ < ω1} and notice that F is a

selective ultrafilter. It is a filter, since {Bξ: ξ < ω1} is decreasing with respectto ⊂∗. It also easy to see that

for every f ∈ ωω there exists a B ∈ F such that f � B is eitherone-to-one or constant.

Indeed, if f ∈ ωω then there exists a ξ < ω1 such that f ∈ Qξ. Then B = Bξ isas desired.

Now, to see that F is an ultrafilter take an A ⊂ ω and let f ∈ ωω be acharacteristic function of A. Then B ∈ F as above is a subset of either A or itscomplement.

It is easy to see that the above two properties imply that F is a selectiveultrafilter.

Notice that CPAgameprism implies also that we have many different selective

ultrafilters. This fact was first noticed (in a model obtained by adding manyside-by-side Sacks reals) by Hart in [45].

Remark 5.2.5 CPAgameprism implies that there are ω2 different selective ultrafil-

ters.

Proof. This can be easily deduced by a simple transfinite induction from

(∗) for every family U = {Fξ: ξ < ω1} of ultrafilters on ω there is a selectiveultrafilter F /∈ U ,.

Property (∗) is proved as above, however operator C({Bη: η < ξ}) is replacedwith Cξ({Bη: η < ξ}) /∈ Fξ.

Now, from the above we obtain that “2ω1 = ω2”+CPAgameprism (which is con-

sistent) implies that there are 2ω1 different selective ultrafilters. Since CPA isalso consistent with 2ω1 > ω2 it is worth to notice that the existence of 2ω1 dif-ferent selective ultrafilters can be also deduced from a slightly stronger versionof CPAgame

prism (which follows from CPA) also in this case.


Recall that the number u is defined as the smallest cardinality of the basefor a non-principal ultrafilter on ω. Thus Theorem 5.2.3 and Corollaries 1.4.4and 1.4.5 imply that

Corollary 5.2.6 CPAgameprism implies that u = rσ = ω1.

Recall also that a family J ⊂ [ω]ω is an independent family provided the set⋂A∈A

A ∩⋂B∈B

(ω \B)

is infinite for every disjoint finite subsets A and B of J . It is often convenientto express this definition in a slightly different notation. Thus, for W ⊂ ω letW 0 = W and W 1 = ω \W . A family J ⊂ [ω]ω is independent provided the set⋂

W∈J0

W τ(W )

is infinite for every finite subset J0 of J and τ :J0 → {0, 1}.The independence cardinal i is defined as follows:

i = min{|J |:J is infinite maximal independent family}.

According to Andreas Blass [8, sec. 11.5] the fact that the equation i = ω1

holds in the iterated perfect set model was first proved by Eisworth and Shelah(unpublished).

Theorem 5.2.7 CPAgameprism implies that i = ω1.

Proof is based on the following lemma. We say that a family W ⊂ [ω]ω

separates points provided for every k < ω there are U, V ∈ W such that k ∈U \ V .

Lemma 5.2.8 For every countable independent family W ⊂ [ω]ω separatingpoints and a perfect prism P in [ω]ω there existW ∈ [ω]ω and a perfect subprismQ of P such that W ∪ {W} is independent but W ∪ {W,x} is not independentfor every x ∈ Q.

Proof. Let W = {Wi: i < ω} and let ϕ:ω → 2ω be a Marczewski’s functionfor W, that is, for i, k < ω

ϕ(k)(i) ={

1 for k ∈Wi

0 for k /∈Wi.

Note that ϕ is one-to-one, since W separates points. Notice also that for everyk, n < ω and τ ∈ 2n

k ∈⋂i<n

Wτ(i)i ⇔ (∀i < n) k ∈W τ(i)

i ⇔ (∀i < n) ϕ(k)(i) = τ(i)⇔ τ ⊂ ϕ(k).


Now, if [τ ] = {t ∈ 2ω: τ ⊂ t} then sets {[τ ]: τ ∈ 2<ω} form a base for 2ω and

k ∈⋂i<n

Wτ(i)i ⇔ ϕ(k) ∈ [τ ]. (5.5)

Thus, independence of W implies that ϕ[ω] is dense in 2ω. We will denote ϕ[ω]by Q, since it is homeomorphic with the set of rational numbers. Note also thatfrom (5.5) it follows immediately that

(a) if W ⊂ ω is such that ϕ[W ] and Q \ ϕ[W ] are dense then W ∪ {W} isindependent;

(b) if W,x ⊂ ω are such that for some τ ∈ 2<ω either ϕ[x ∩W ] ∩ [τ ] = ∅ orϕ[W ] ∩ [τ ] ⊂ ϕ[x] then W ∪ {W,x} is not independent.

Now identify ω with Q = ϕ[ω] and let I be the ideal of nowhere dense subsets ofϕ[ω]. Since I is weakly selective we can apply Lemma 5.2.4(b) to find subprismQ of P and a V ∈ [ω]ω \ I such that either

• x ∩ V = ∅ for every x ∈ Q, or else

• V ⊂ x for every x ∈ Q.

Since V /∈ I, there exists a τ ∈ 2<ω such that ϕ[V ] is dense in [τ ]. TrimmingV , if necessary, we can assume that ϕ[V ] ⊂ [τ ] and that ϕ[ω \ V ] is also densein [τ ]. Now let W ⊃ V be such that ϕ[W ] ∩ [τ ] = ϕ[V ] and both ϕ[W ] andϕ[ω \W ] are dense in Q. Then, by (a) and (b), W ∪ {W} is independent whileW ∪ {W,x} is not independent for every x ∈ Q.

Proof of Theorem 5.2.7. For a countable independent family W ⊂ [ω]ω

separating points and an f ∈ Fprism([ω]ω) put P = range(f) and letW (W, P ) ∈[ω]ω and a subprism Q(W, P ) of P be as in Lemma 5.2.8. If f ∈ Cprism([ω]ω)and P = range(f) = {x} then we put Q(W, P ) = P and define W (W, P ) as anarbitrary W such that W ∪ {W} is independent while W ∪ {W,x} is not.

Let A0 ⊂ [ω]ω be an arbitrary countable independent family separatingpoints and consider the following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q(A0 ∪ {Wη: η < ξ}, Pξ),

where sets Wη are defined inductively by Wη = W (A0 ∪ {Wζ : ζ < η}, Pη).By CPAgame

cube strategy S is not a winning strategy for Player II. So thereexists a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which and Player IIloses, that is, [ω]ω =

⋃ξ<ω1

Qξ.Now, notice that family J = A0 ∪ {Wξ: ξ < ω1} is a maximal independent

family. It is clear that J is independent, since every set Wξ was chosen sothat A0 ∪ {Wζ : ζ ≤ ξ} is independent. To see that family J is maximal it isenough to note that every x ∈ [ω]ω belongs to a Qξ for some ξ < ω1, and soA0 ∪ {Wζ : ζ ≤ ξ} ∪ {x} is not independent.

By Theorem 5.2.7 we see that CPAgameprism implies the existence of an indepen-

dent family of size ω1. Next, answering a question of Hrusak [46] we show that


such a family can be simultaneously a splitting family. This is similar in flavorto Theorem 2.2.3. In the proof we will use the following lemma.

Lemma 5.2.9 For every countable family V ⊂ [ω]ω and a perfect set P in [ω]ω

there exists a W1 ∈ [ω]ω such that V ∪{W1} is independent and W1 splits everyA ∈ P .

Proof. We follow the argument from [34, p. 121] that s ≤ d.For every A ∈ [ω]ω let bA be a strictly increasing bijection from ω onto

A. Then b: [ω]ω → ωω defined by b(A) = bA is continuous. In particularb[P ] = {bA:A ∈ P} is compact, so there exists a strictly increasing f ∈ ωω suchthat bA(n) < f(n) for every A ∈ P and n < ω. For n < ω let fn denote then-fold composition of f and let Sn = {m < ω: fn(0) ≤ m < fn+1(0)}. Thenfn(0) ≤ bA(fn(0)) < f(fn(0)) = fn+1(0) for every A ∈ P and n < ω. Inparticular, for every A ∈ P

Sn ∩A %= ∅.So, if T ⊂ ω be infinite and co-infinite and W1 =

⋃n∈T Sn then W1 splits every

A ∈ P . Thus, it is enough to take infinite and co-infinite T ⊂ ω such thatV ∪ {W1} is independent.

Theorem 5.2.10 CPAgameprism implies that there exists a family F ⊂ [ω]ω of car-

dinality ω1 which is simultaneously independent and splitting.

Proof. The proof is just a slight modification of that for Theorem 5.2.7.(Compare also Theorem 2.2.3.)

For a countable independent family W ⊂ [ω]ω separating points and anf ∈ Fprism([ω]ω) put P = range(f) and let W0 ∈ [ω]ω and a subprism Q of Pbe as in Lemma 5.2.8. Let W1 be as in Lemma 5.2.9 used with P = Q andV =W ∪ {W0}. We put Q(W, P ) = Q1 and W(W, P ) = {W0,W1}.

If f ∈ Ccube([ω]ω) and P = range(f) = {x} then we put Q(W, P ) = Pand W(W, P ) = {W0,W1}, where W0 and W1 are such that W ∪ {W0,W1} isindependent and W1 splits P = {x}.

Let A0 ⊂ [ω]ω be an arbitrary countable independent family separatingpoints and consider the following strategy S for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q(A0 ∪⋃{Wη: η < ξ}, Pξ),

where Wη’s are defined inductively by Wη =W(A0 ∪⋃{Wη: η < ξ}, Pη).

By CPAgameprism strategy S is not a winning strategy for Player II. So there


⋃ξ<ω1

Qξ. Then the family F = A0 ∪⋃{Wξ: ξ < ω1} is

independent and splitting.

Let Perf(Q) stand for the family of all closed subsets A of Q without isolatedpoints, that is, such that their closures clR(A) in R are perfect sets. Recallthat an ideal I on Q of is crowded provided I+ = P(Q) \ I is generated by


the sets from Perf(Q). Crowded ultrafilters were studied by several authors(see e.g. [35, 33]) in connection with the reminder βQ \ Q of the Cech-Stonecompactification βQ of Q.

In what follows we will also use the following simple fact.

Fact 5.2.11 Every non-scattered set B ⊂ Q contains a subset from Perf(Q).

Proof. Since B is non-scattered, decreasing it, if necessary, we can assumethat B is dense in itself. Let {kn:n < ω} be an enumeration of ω with infinitelymany repetitions and let Q \ B = {an:n < ω}. By induction construct clopensets 〈Un ⊂ Q:n < ω〉 and 〈pn ∈ B \

⋃i<n Ui:n < ω〉 such that an ∈ Un and

|pn − pkn| < 2−n. Then Q \

⋃n<ω Un ⊂ B is as desired.

The following theorem answers in positive a question of Hrusak [46] onwhether there exists a crowded ultrafilter in the iterated perfect set model.

Theorem 5.2.12 CPAgameprism implies there exists a non-principal ultrafilter on

Q which is crowded.

Fix a p ∈ R \ Q and for a family D ⊂ P(Q) let F (D) denote a filter on Q

generated by the family D ∪ {In ∩ Q:n < ω}, where In = [p − 2−n, p + 2−n].The proof of the theorem is based on the following lemma, in which [Q]ω isconsidered with the same topology as [ω]ω upon natural identification.

Lemma 5.2.13 Let D ⊂ Perf(Q) be a countable family such that p ∈ clR(D)for every D ∈ D. Then for every prism P in [Q]ω there exist a subprism Q ofP and a Z ∈ Perf(Q) such that p ∈ clR(Z) and either

(i) Z ∩ x = ∅ for every x ∈ Q, or else

(ii) Z ⊂ x for every x ∈ Q.

Proof. In what follows we will identify [Q]ω with 2Q, the identification mappinggiven by the characteristic function. Thus, we will consider P as a prism in 2Q.

Let {Dn ∈ Perf(Q):n < ω} be a cofinal sequence in F (D) with a propertythat Dn+1 ⊂ Dn ⊂ In for every n < ω. Choosing a subsequence, if necessary,we can find disjoint intervals Jn such that Kn = Dn ∩ Jn ∈ Perf(Q).

First we will show that there exist a sequence 〈Bn ⊂ Kn:n < ω〉 of non-scattered sets and a subprism P0 of P such that

• g � Bn is constant for every g ∈ P0 and n < ω.

Let f ∈ Fprism(2Q) be from Cα onto P . We will find P0 and Bn’s with thehelp of Prism-Fusion Lemma 3.1.3 used with A = Sα. Thus, for k < ω let Dk bethe collection of all pairwise disjoint families E = {Ej ∈ Sα: j < m} for whichthere exists a non-scattered set BEn ⊂ Kn with the property that

g � Bn is constant for every g ∈ f[⋃E

].


Clearly sets Dk satisfy condition (P1). To see that condition (') is also satisfiedtake E = {Ej ∈ Sα: j < m} ∈ Dk−1 and γ < α. It is easy to see that itis enough to find a non-scattered set B ⊂ Kn and prisms E′

j ⊂ Ej such that

g � B is constant for every g ∈ f[⋃

j<mE′j

]. So, let B0 = Kn, take Q0 ⊂ B0

homeomorphic to Q, and apply Lemma 5.2.4(a) to the ideal of nowhere densesubsets of Q0 and a prism f [E0] to find E′

0 ⊂ E0 and a somewhere dense subsetB1 of Q0 such that g � B1 is constant for every g ∈ f [E′

0]. Repeating this trickm-many times we find B = Bm and the sets E′

j as desired. Thus, condition (')is satisfied.

So, by Lemma 3.1.3 there are sets Pk ∈ Dk such that E0 =⋂k<ω

⋃Pk ∈ Sα.

It is easy to see that the sets Bn = BPnn and Q0 = f [E0] satisfy (•). Notice also

that by Fact 5.2.11 we can assume that Bn ∈ Perf(Q) for every n < ω.Now let A be a selector from the family {Bn:n < ω}. Then A ∈ I+, where

I is the ideal of finite subsets of Q. Applying Lemma 5.2.4(a) to A and P0 wecan find i < 2, S ∈ [A]ω, and a subprism Q of P0 such that g � S is constantfor every g ∈ Q. Put Z =

⋃n∈S Bn. Then, by (•), g � Z is constant for every

g ∈ Q. Finally, note that Z ∈ Perf(Q) since Z is closed, as Bn → p /∈ Q.

Proof of Theorem 5.2.12. For a prism P in [Q]ω and a countable familyD ⊂ Perf(Q) for which p ∈ clR(D) for every D ∈ D let Z(D, P ) ∈ Perf(Q) and asubprism Q(D, P ) of P be as in Lemma 5.2.13. Consider the following strategyS for Player II:

S(〈〈Pη, Qη〉: η < ξ〉, Pξ) = Q({Zη: η < ξ}, Pξ),

where sets Zη are defined inductively by Zη = Z({Bζ : ζ < η}, Pη).By CPAgame

prism strategy S is not a winning strategy for Player II. So thereexists a game 〈〈Pξ, Qξ〉: ξ < ω1〉 played according to S in which Player II loses,that is, [Q]ω =

⋃ξ<ω1

Qξ.Now, let F = F ({Zξ: ξ < ω1}). Then clearly F is a crowded non-principal

filter. To see that it is maximal, take an x ∈ [Q]ω. Then there is a ξ < ω1

such that x ∈ Qξ. Then either Zξ ∩ x = ∅ or Zξ ⊂ x. Thus, either x or itscomplement belong to F .

The construction of crowded ultrafilter is quite similar to that of selectiveultrafilter. This similarity suggest a possibility of construction of a crowdedultrafilter which is also selective. This, however, cannot be done:

Proposition 5.2.14 There is no non-principal crowded ultrafilter on Q whichis also selective.

Proof. Let F be a non-principal crowded ultrafilter on Q and let {x} =⋂F∈F clR(F ) ∈ R \Q. Then In = (x− 2−n, x+2−n)∩Q belongs to F for every

n < ω. Define f : I0 → ω by putting f(q) = n for every q ∈ In \ In+1. Thenclearly f is not constant on any B ∈ F ∩ P(I0). But it is also not one-to-oneon any such B, since f is one-to-one only on scattered sets.


It is worth to mentioned that CPAgameprism implies also the existence of many

other kinds of ultrafilters, like these constructed in [45]. In fact, many construc-tions that are done under CH can be carried also under CPAgame

prism. However, thisalways needs some combinatorial lemma, such as Lemma 5.2.4, which allows toreplace points with the prisms.


Chapter 6

CPA and properties (F∗)and (G)

To express the most general version of our axiom we need yet another versionof game which we denote by GAMEsection

prism (X). As before, the game is of lengthω1, has two players, Player I and Player II, and at each stage ξ < ω1 of thegame Player I can play an arbitrary prism, now represented explicitly by anfξ ∈ F∗

prism(X), and Player II responds with a subprism gξ ∈ F∗prism(X) of fξ.

The game 〈〈fξ, gξ〉: ξ < ω1〉 is won by Player I provided there exists a c ∈ C

such thatX =

⋃ξ<ω1

gξ [{p ∈ dom(gξ):π0(p) = c}] ,

where π0 is a projection onto the first coordinate.Now, if Γ = {α < ω2: cof(α) = ω1} then CPA reads as follows.

CPA: c = ω2 and for any Polish space X Player II has no winning strategy inthe game GAMEsection

prism (X).

Moreover, there exist a family {〈fαξ ∈ F∗prism(X): ξ < ω1〉:α < ω2} of

(very simple) tactics for Player I, and the enumerations {cα:α < ω2} and{xα:α < ω2} of C and X, respectively, such that for every strategy S forPlayer II there is a closed unbounded subset ΓS of Γ such that for everyα ∈ ΓS and a game 〈〈fαξ , gξ〉: ξ < ω1〉 played according to S we have

X =⋃ξ<ω1

gξ [{p ∈ dom(gξ):π0(p) = cα}]

and

X \⋃

gξ∈Fprism

gξ [{p ∈ dom(gξ):π0(p) = cα}] ⊂ {xξ: ξ < α}.

87


In what follows we will not give any application for the additional part ofthe axiom. However, it is a powerful tool which we used in some earlier versionof this text and which, we believe, will find some applications in the future.

It is easy to see that CPA implies CPAgameprism. We have also the following

implication.

Proposition 6.0.15 Axiom CPA implies the following section version of theaxiom CPAprism:

CPAsecprism: c = ω2 and for every Polish space X, P ∈ Perf(C), and an F∗

prism-dense family F ⊂ F∗

prism(X) there exist a c ∈ P and an F0 ⊂ F such that|F0| ≤ ω1 and

X =⋃f∈F0

f [{p ∈ dom(f):π0(p) = c}] .

Proof. The argument is essentially identical to that used in the proof ofProposition 2.0.4. First note that the implication is true if P = C.

So, let F ⊂ F∗prism(X) be F∗

prism-dense. Thus for prism f ∈ F∗prism(X) there

exists a subprism s(f) ∈ F of f . Consider the following strategy S for Player II:

S(〈〈fη, gη〉: η < ξ〉, fξ) = s(fξ).

By CPA it is not a winning strategy for Player II. So there exists a game〈〈fξ, gξ〉: ξ < ω1〉 in which gξ = s(fξ) for every ξ < ω1 and Player II loses, thatis, there exists a c ∈ C such that

X =⋃ξ<ω1

gξ [{p ∈ dom(gξ):π0(p) = c}] .

Then F0 = {gξ: ξ < ω1} is as desired.To get the general version of CPAsec

prism from the version with P = C it isenough to identify in every prism the first coordinate Cantor set C with the setP and apply the above case when P = C to the family F ′ of all restrictionsf � {x ∈ dom(f):π0(x) ∈ P} of f ∈ F .

To state a cube version of CPAsecprism we need the following additional nota-

tion: for a fixed Polish space X the symbol F seccube(X) will stand for the family

of all continuous functions f from a perfect cube C ⊂ Cη, where 1 ≤ η ≤ ω,into X such that f is either one-to-one or constant.

It should be clear that CPAsecprism implies the following:

CPAseccube: c = ω2 and for every Polish space X and every F sec

cube-dense familyF ⊂ F sec

cube(X) there exist a c ∈ C and an F0 ⊂ F such that |F0| ≤ ω1 and

X =⋃f∈F0

f [{p ∈ dom(f):π0(p) = c}] .

Note that the axiom CPAseccube would be false if we use in it F∗

cube in place ofF sec

cube, as noticed in Remark 6.4.1. Thus, this complication is essential.For the reminder of this chapter we will concentrate on the consequences of

CPAseccube and CPAsec

prism.


6.1 cov(s0) = c and ω1-intersections of open sets

To appreciate the section part in CPA let us note that it implies Martin’s Axiomfor the Sacks forcing S = 〈Perf(C),⊂〉 in a sense that for every family D of lessthan c many dense subsets of S there exists a D-generic filter in S. This followsimmediately from the following fact, since for a dense subset D of S we haveC \

⋃D ∈ s0. (Compare also Corollary 3.3.7.)

Proposition 6.1.1 CPA implies that cov(s0) = c.

Proof. Let T = {Tξ: ξ < ω1} ⊂ s0. We will show that⋃T %= C. Since s0 is a

sigma ideal, we can assume that Tη ⊂ Tξ for every η < ξ < ω1.Consider the following Player II strategy S in a game GAMEsection

prism (R): ifPlayer I plays fξ ∈ F∗

prism(R), Player II looks at P = π0(dom(fξ)), choosesQ ∈ Perf(P ) disjoint with Tξ, and replies with a subprism gξ of fξ such thatπ0(dom(gξ)) = Q.

Since, by CPA, S is not winning, there exist a game 〈〈fξ, gξ〉: ξ < ω1〉 playedaccording to S and a c ∈ C such that

R =⋃ξ<ω1

gξ [{p ∈ dom(gξ):π0(p) = c}] .

We claim that c ∈ C \⋃T , so that

⋃T %= C.

Indeed, if we had c ∈ Tζ for some ζ < ω1 then we would have also c /∈π0(dom(gξ)) for all ζ ≤ ξ < ω1 and so

R =⋃η<ζ

gη [{p ∈ dom(gη):π0(p) = c}] .

But this would mean that R is a union of countable many nowhere dense setsrange(gη), η < ζ. This contradiction finishes the proof.

We have been asked by Joerg Brendle (private communication) whether thefollowing property (X), which he proved to be true in the iterated perfect setmodel, follows also from our axiom. The next theorem gives an affirmativeanswer to this question.

Theorem 6.1.2 CPA implies the following property

(X) if U = {Uξ: ξ < ω1} is a family of open sets in C and |⋂U| = c then

⋂U

contains a perfect set.

Proof. Will be added.

6.2 Surjections onto nice sets must be continu-ous on big sets

In this section we will prove that CPAseccube implies


(F∗) For an arbitrary function h from a subset S of a Polish space X ontoa Polish space Y there exists a uniformly continuous function f from asubset of X into Y such that |f ∩ h| = c.

Note that (F∗) implies (F): if f : R → R is Darboux non-constant then its range,Y = f [R], is a non-trivial interval, so f is onto a Polish space Y .

We will start the proof of (F∗) with the following fact.

Fact 6.2.1 Assume that CPAseccube holds and that X is a Polish space. Then

for every function h from a subset S of X onto C there exists an f :E → X inF sec

cube(X) such that the set

Zf = {f(p) ∈ S: p ∈ E & h(f(p)) = π0(p)}

intersects f [C] for every subcube C of E. Moreover, function f is one-to-one.

Proof. By way of contradiction, assume that for every f :E → X from F seccube

there exists a subcube Cf of E such that f [Cf ] is disjoint with Zf . ThenF = {f � Cf : f ∈ F sec

cube} is dense in F seccube. So, by CPAsec

cube, there exist a c ∈ C

and an F0 ⊂ F such that |F0| ≤ ω1 and

X =⋃f∈F0

f [{p ∈ dom(f):π0(p) = c}] .

Let r ∈ S be such that h(r) = c. So, there exists an f ∈ F0, f :Cf → X, suchthat r ∈ f [{p ∈ Cf :π0(p) = c}]. Thus, there is a p ∈ Cf with π0(p) = c forwhich f(p) = r. But this means that h(f(p)) = h(r) = c = π0(p). Thereforef(p) belong to Zf and f [Cf ], contradicting the fact that these sets are disjoint.

Thus, there is an f ∈ F seccube having the main property. To see that it is one-

to-one it is enough to show that it is not constant. So, by way of contradictionassume that f is constant and find subcubes C and C ′ of dom(f) such thatπ0[C] ∩ π0[C ′] = ∅. So, there are p ∈ C and p′ ∈ C ′ such that f(p), f(p′) ∈Zf . Thus, since f is constant, we have π0(p) = h(f(p)) = h(f(p′)) = π0(p′)contradicting the fact that π0[C] and π0[C ′] are disjoint.

Corollary 6.2.2 CPAseccube implies (F∗).

Proof. Since Y contains a subset homeomorphic to C, by restricting h we canassume that Y = C. Let f :E → X and Zf be as in Fact 6.2.1. Clearly there iscontinuum many pairwise disjoint subcubes C of E, and Zf intersects f [C] forall these C. So, since f is one-to-one, Zf has cardinality c. Note also that forevery x = f(p) ∈ Zf we have h(x) = π0(f−1(x)). Thus, on a set Zf , h is equalto a continuous function π0 ◦ f−1.

If in (F∗) we have X,Y ⊂ R then we can also get the following strongerresult.

Corollary 6.2.3 CPAsecprism implies the following property:


(F′) for any function h from a subset S of R onto a perfect subset of R thereexists a function f ∈ “C∞perf” such that |f ∩ h| = c and f can be extended

to a function f ∈ “C1(R)” such that either either f ∈ C1 or f is anautohomeomorphism of R with f−1 ∈ C1.

Proof. Let f∗ be a continuous function as in condition (F∗), that is such that|f∗ ∩ h| = c, and let g: R → R be a Borel extension of f∗. Let {fξ: ξ < ω1} bethe functions from Theorem 4.1.1(a). Then f∗ ∩ h ⊂ g ⊂

⋃ξ<ω1

fξ. So thereexists a ξ < ω1 such that |fξ ∩ h| = c and f = fξ is as desired.

Notice also that function f in (F′) cannot chosen from D1perf . This is evident

for h = f � P , where f and P are from Example 4.5.2.

6.3 Sums of Darboux and continuous functions

In this section we show that the conjunction of properties (A) and (F∗) impliesthe property

(G) for every Darboux function g: R → R there exists a continuous nowhereconstant function f : R → R such that f + g is Darboux.

In particular, (G) follows from CPAseccube.

The proof presented below follows the argument given in [26]. Althoughit is more involved that many of the other proofs presented so far it is stillconsiderably simpler that the original proof of the consistency of (G) givenin [84].

Our proof will be based on the following three lemmas.

Lemma 6.3.1 Let d: R → R be Darboux and D ⊂ R be such that d � D isdense in d. If g: R → R is continuous and such that

[(d+ g)(α), (d+ g)(β)] ⊂ (d+ g)[[α, β]] for all α, β ∈ D (6.1)

then d+ g is Darboux.

Proof. An easy proof can be found in [84, lemma 4.1].

Lemma 6.3.2 Assume that (A) holds and for every n < ω let An ∈ [R]c. Thenfor every n < ω there exists a Bn ∈ [An]c such that the closures of Bn’s arepairwise disjoint.

Proof. First note that (A) implies that

∀〈Cn ∈ [R]c:n < ω〉 ∃C ∈ [C0]c ∀n < ω |Cn \ cl(C)| = c. (6.2)

Indeed, by condition (A), we can find a continuous function f : R → C suchthat |C0 ∩ f−1(c)| = c for every c ∈ C. By induction on n < ω choose an


increasing sequence sn ∈ 2n+1 such that |Cn \ f−1({c ∈ C: sn ⊂ c})| = c. Puts =

⋃n<ω sn. Then C = C0 ∩ f−1(s) satisfies (6.2).

Next note that there exists a sequence 〈Dn:n < ω〉 of closed subsets of R

such that∣∣An ∩ (

Dn \⋃i<nDi

)∣∣ = c. This sequence is constructed by inductionon n < ω with each set Dn chosen by applying condition (6.2) to the sequence〈Ak \

⋃i<nDi:n ≤ k < ω〉.

Finally, since each set Dn\⋃i<nDi is an Fσ-set, we can find its closed subset

En with Bn = En ∩ An having cardinality c. It is easy to see that the sets Bnare as required.

Remark 6.3.3 Note that Lemma 6.3.2 cannot be proved in ZFC. Indeed, if weassume that there exists a c-Luzin set1 and 〈An ∈ [R]c:n < ω〉 is a sequence ofc-Luzin sets such that every non-empty open interval contains one of An’s, thatfor such a sequence there are no Bn’s as in the lemma.

Lemma 6.3.4 If (A) and (F∗) hold then for every d: R → R and every count-able subset D of R there exists a countable family A ⊂ [R]c such that

(a) different elements of A have disjoint closures,

(b) d � A is uniformly continuous for every A ∈ A, and

(c) for every α, β ∈ D there is an A ∈ A such that d � A ⊂ [α, β]×[d(α), d(β)].

Proof. First for every α < β from D we choose Aβα ∈ [R]c such that the familyA0 = {Aβα:α, β ∈ D & α < β} satisfies (b) and (c). For this fix α < β from D.

If d is constant on [α, β] put Aβα = [α, β]. Otherwise, since d is Darboux,Y = d[[α, β]] is a non-trivial interval, so it is Polish. Let S = [α, β] ∩ d−1(Y )and notice that h = d � S maps S onto Y . So, by (F∗), we can find an Aβα ∈ [S]c

such that d � Aβα is uniformly continuous.It is easy to see that A0 satisfies (b) and (c) so it is enough to decrease its

elements to get condition (a), while assuring that they still have cardinality c.This can be done by applying Lemma 6.3.2.

Theorem 6.3.5 Assume that (A) and (F∗) hold. Then for every Darbouxfunction d: R → R there exist a complete metric ρ on C(R) and a dense Gδsubset G of 〈C(R), ρ〉 of nowhere constant functions such that f + g is Darbouxfor every g ∈ G.

In particular (A)&(F∗) implies (G).

Proof. Take a Darboux function d: R → R and choose a countable denseset D ⊂ R such that d � D is dense in d. Let A ⊂ [R]c be the family from

1A set L ⊂ R is a c-Luzin set if |L| = c but |L ∩ N | < c for every nowhere dense subset Nof R. It is well known (see e.g. [67, sec. 2]) and easy to see that no c-Luzin set can be mappedcontinuously onto [0, 1]. Thus (A) implies that there is no c-Luzin set.


Lemma 6.3.4 and let {An:n < ω} be an enumeration of A∪{{d}: d ∈ D \⋃A}.

Let ρ0 be the uniform convergence metric on C(R), that is,

ρ0(f, g) = min{1, sup{|f(x)− g(x)|:x ∈ R}}

and, for f, g ∈ C(R), let

ρ1(f, g) = 2−min{n<ω:f�An �=g�An}.

(If {n < ω: f � An %= g � An} = ∅ we assume that ρ1(f, g) = 0.) Then ρ1 is apseudometric on C(R). Consider C(R) with the following metric ρ:

ρ(f, g) = max{ρ0(f, g), ρ1(f, g)}

and notice that 〈C(R), ρ〉 forms a complete metric space. We will prove that ifG is a set of all nowhere constant continuous functions g: R → R for which d+gsatisfies (6.1) of Lemma 6.3.1 then G contains a dense Gδ subset of 〈C(R), ρ〉.For this we will show that for every α < β from D the following two kinds ofsets contain dense open subsets of 〈C(R), ρ〉:

Hβα = {g ∈ C(R): g is not constant on (α, β)}

andGβα = {g ∈ C(R): [(d+ g)(α), (d+ g)(β)] ⊂ (d+ g)[[α, β]]}.

Then ⋂ {Hβα ∩Gβα:α, β ∈ D & α < β

}⊂ G

will contain a dense Gδ subset of 〈C(R), ρ〉.To see that Hβ

α contains a dense open subset first note that, by (a) and (c)of Lemma 6.3.4, elements of A are nowhere dense. Next, take an f ∈ C(R),and fix an ε > 0. Let B(f, ε) be the open ρ-ball centered at f and of radiusε. We will find g ∈ B(f, ε) and δ > 0 such that B(g, δ) ⊂ B(f, ε) ∩ Hβ

α .So take an n < ω such that 2−n < ε/4, choose a non-empty open intervalJ ⊂ (α, β) \

⋃ni=1Ai, and pick different x, y ∈ J . It is easy to find g ∈ C(R)

such that g(x) %= g(y), g � (R \ J) = f � (R \ J), and ρ0(f, g) < 2−n. Then,by the choice of J , g ∈ B(f, ε/4). Now, if δ = min{|g(x) − g(y)|/4, ε/4} thenB(g, δ) ⊂ B(f, ε) ∩Hβ

α .To see that each Gβα contains a dense open subset take f ∈ C(R) and ε > 0.

As previously we will find g ∈ B(f, ε) and δ > 0 such that B(g, δ) ⊂ B(f, ε)∩Gβα.Find α = x0 < x1 < · · · < xm = β, xi ∈ D, such that the variation of f on eachinterval [xi, xi+1] is less than ε/8. Also, since d � [xi, xi+1] is Darboux, we canpartition each [xi, xi+1] even further to get also that |d(xi)− d(xi+1)| < ε/8 forall i < m.

Pick an n < ω such that 2−n < ε/8 and {xi: i ≤ m} ⊂⋃ni=1Ai. For every

i < m choose an index ki > n such that d � Aki ⊂ (xi, xi+1)×[d(xi), d(xi+1)]. By(A) for every i < m we can also pick a uniformly continuous function hi from Aki


onto [(d+f)(xi), (d+f)(xi+1)]. For each i < m define h � Aki= (hi−d−f) � Aki

and note that

h[Aki] ⊂ hi[Aki

]− d[Aki]− f [Aki

] ⊂ [−ε/2, ε/2]

since [(d + f)(xi), (d + f)(xi+1)] ⊂ [d(xi) + f(xi) − ε/4, d(xi) + f(xi) + ε/4],d[Aki ] ⊂ [d(xi), d(xi+1)] ⊂ [d(xi) − ε/8, d(xi) + ε/8], and f [Aki ] is a subset of(f(xi) − ε/8, f(xi) + ε/8). Define also h as 0 on

⋃ni=1Ai and extend it to a

uniformly continuous function from R into [−ε/2, ε/2]. Put g = f + h and notethat g ∈ B(f, ε). Also let k = max{n, k0, . . . , km−1} and δ ∈ (0, 2−k). We claimthat B(g, δ) ⊂ B(f, ε) ∩Gβα.

Indeed, it is easy to see that B(g, δ) ⊂ B(f, ε). To see that B(g, δ) ⊂ Gβαtake a g0 ∈ B(g, δ). By the choice of δ, h, and g for every i ≤ m we have

(d+ g0)(xi) = (d+ g)(xi) = (d+ f + h)(xi) = (d+ f)(xi)

and, for A =⋃j<mAkj ,

g0 � A = g � A = (f + h) � A = (f + (hi − d− f)) � A = (hi − d) � A.

So,

[(d+ g0)(α), (d+ g0)(β)] = [(d+ f)(x0), (d+ f)(xm)]

⊂⋃j<m

[(d+ f)(xj), (d+ f)(xj+1)]

=⋃j<m

hj [Akj ]

=⋃j<m

(d+ (hj − d))[Akj ]

=⋃j<m

(d+ g0)[Akj]

⊂ (d+ g0)[[α, β]]

proving that g0 ∈ Gβα. This finishes the proof of the theorem.

Corollary 6.3.6 CPAseccube implies (G).

Let us notice also that in (G) we cannot require that function g is “C1”.This follows from the following fact which, for the functions from the class C1,was first noticed by Steprans [84, p. 118]. Since Steprans leaves his statementwithout any comments concerning its proof, we include here a missing argument.

Proposition 6.3.7 There exists, in ZFC, a Darboux function d: R → R suchthat d+ f is Darboux for every non-constant “C1” function f .


Proof. Let R = {xξ: ξ < c} and let {fξ: ξ < c} be an enumeration of all non-constant “C1” functions. Notice that for every ξ < c there exists a non-emptyopen interval Iξ such that fξ � Iξ is strictly monotone. (Just take an x ∈ R

such that f ′ξ(x) %= 0, which exists since fξ is non-constant. Then Iξ is chosen asa neighborhood of x on which f ′ξ is non-zero.)

Function d we construct will be strongly Darboux in a sense that d−1(r) isdense in R for every r ∈ R. For such d in order to show that d + fξ is notDarboux it is enough to show that (d + fξ)−1(yξ) is not dense in R for someyξ ∈ R. (See e.g. [21, prop. 7.2.4].)

By induction we construct a sequence {〈Qξ, dξ, yξ〉: ξ < c} such that for everyξ < c we have

(i) Sets {Qη: η ≤ ξ} are countable and pairwise disjoint, and xξ ∈⋃η≤ξ Qη.

(ii) dξ:Qξ → R and d−1ξ (xξ) is dense in R.

(iii) yζ /∈ (dη + fζ)[Qη ∩ Iζ ] for every ζ, η ≤ ξ.Notice, that if we have such a sequence then, by (i), R =

⋃ξ<c

Qξ so d =⋃ξ<µ dξ: R → R. It is strongly Darboux by condition (ii), while, by (iii), for

every ζ < c the set (d+ fζ)−1(zζ) is not dense in R, since it misses Iζ . Thus, dis as desired.

To make an inductive ξ-th step first choose a countable dense subset Q0ξ of

R disjoint with ⋃η<ξ

Qη ∪⋃ζ<ξ

(Iζ ∩ f−1

ζ (yζ − xξ)).

This can be done since, by the choice of of the intervals Iζ ’s, each of the setsIζ ∩ f−1

ζ (yζ − xξ) has at most one element.Define dξ � Q0

ξ as constantly equal to xξ. This guarantees (ii), while condi-tion (iii) is satisfied for the part defined so far: for every ζ < ξ and x ∈ Iζ ∩Q0

ξ

we have dξ(x) + fζ(x) = fζ(x) + xξ %= yζ , since otherwise we would havex ∈ Iζ ∩ f−1

ζ (yζ − xξ) contradicting the choice of Q0ξ .

If xξ ∈⋃η<ξ Qη we define Qξ = Q0

ξ . Otherwise we put Qξ = Q0ξ ∪ {xξ}

and if dξ(xξ) is not defined yet (i.e., if xξ /∈ Q0ξ) we define dξ(xξ) by choosing

dξ(xξ) /∈ {yζ − fζ(xξ): ζ < ξ}. This guarantee that (i) is satisfied, while (iii) ispreserved.

Finally, we choose yξ to have

yξ ∈ R \⋃η≤ξ

(dη + fξ)[Qη].

This will guarantee that (iii) holds also for ζ = ξ.

Let us also note that the following question, for the case of D1 due toSteprans [84, Question 5.1], remains open.

Problem 6.3.8 Does there exist, in ZFC, a Darboux function d: R → R suchthat d+ f is Darboux for every nowhere constant “D1” function f? What if werestrict the choice of f to D1?


6.4 Remark on a form of CPAseccube.

Remark 6.4.1 Note that if X = C and h is any autohomeomorphism of C thenthe conclusion of Fact 6.2.1 may hold only for one dimensional cube E ⊂ C1.

Proof. Indeed, for every cube E ⊂ Cη = C1 × Cη\1 with η > 1 the set{p ∈ E:h(f(p)) = π0(p)} is closed with vertical sections having at most onepoint. So, it is meager in E and it cannot meet all subcubes of E. Since in theproof of Fact 6.2.1 was independent of the dimension of cubes, this implies thatthe axiom CPAsec

cube would have been false if we did not include one dimensionalcubes C1 in the definition of F sec

cube.

Notice also, that if CPA[X] stands for CPA for a fixed Polish space X then,similarly as in Remark 1.6.2 we can also prove

Remark 6.4.2 For any Polish space X axiom CPA[X] implies CPA.

Chapter 7

CPA in the Sacks model

In what follows we will show that CPA holds in the generic extension V [G] ofa model V of ZFC+CH, where G is a V -generic filter over S∗

ω2, ω2 countable

support iteration of the Sacks forcing. We will use here terminology from [4].Let S = 〈Perf(C),⊂〉. Recall that the perfect set (Sacks) forcing S∗ is usually

defined as the set of all trees T (P ) = {x � n ∈ 2<ω:x ∈ P & n < ω}, whereP ∈ S, and is ordered by the inclusion, that is, s ∈ S is stronger than t ∈ S,s ≤ t, if s ⊂ t. It is important to realize that

P ⊂ Q if and only if T (P ) ⊂ T (Q),

so T : S → S∗ establishes isomorphism between forcings S and S∗. Also if fors ∈ S∗ we define lim(s) = {x ∈ 2ω:∀n < ω (x � n ∈ s)} then lim: S∗ → S is theinverse of T .

Perfect set forcing is usually represented as S∗ rather that in its more naturalform S since the conditions in S∗ are absolute, unlike those in S. However, inlight of our axiom, we prefer think of this forcing in terms of S.

Recall also that for a countable A ⊂ ω2 we defined Φprism(A) as the familyof all projection-keeping homeomorphisms f :CA → CA. We also defined SA as{range(f): f ∈ Φprism(A)}. However, for the purpose of this chapter we like toredefine SA to

SA = {[range(f)]: f ∈ Φprism(A)},where [E] = {g ∈ Cω2 : g � A ∈ E} for every E ⊂ CA. Also, we put

Sω2 =⋃ {

S(A):A ∈ [ω2]≤ω}

and order it by inclusion. For 0 < α < ω2 we define

Sα = {{πα(f): f ∈ E}:E ∈ Sω2}.

(Thus, for a countable α we have two different definitions of Sα. However, boththese representations lead to obviously isomorphic forcings and it will be alwaysclear from the contect which definition we have in mind.)

97


It is known that forcing Sγ is equivalent to the countable support iterationS∗γ of S of length 0 < γ ≤ ω2 in a sense that

Fact 7.0.3 S∗γ contains a dense subset order isomorphic to Sγ .

This fact is stated, for instance, by Kanovei in [52], but it is left there withouta proof.1 However, elements of S∗

γ which for set isomorphic to Sγ are alreadyconsidered in [67] and [84], where they are determined conditions.

Theorem 7.0.4 CPA holds in the iterated perfect set model. In particular, itis consistent with ZFC set theory.

Proof. Start with a model V of ZFC+CH and let V [G] be a generic extensionof V with respect to forcing Sω2 . By Fact 7.0.3 forcing Sω2 is equivalent to S∗

ω2,

so it preserves cardinals and c = ω2 in V [G]. For α ≤ ω2 let Gα = G � α. ThenGα is V -generic over Sα.

By Remark 6.4.2 it is enough to prove only CPA[X] for X = C. In this casethe sequence {xα:α < ω2} will be equal to {cα:α < ω2}.

Let {cα:α < ω2} be an enumeration, in V [G], of C such that for every α-thelement ω1 α of Γ, α > 0, we have:

• {cξ: ξ < ω1 α} = C ∩ V [Gα];

• xω1 α is the Sacks generic real in V [Gα+1] over V [Gα].

Moreover, for each α ∈ Γ let {fαξ : ξ < ω1} be an enumeration of F∗prism(X)∩ V .

We will show that these sequences satisfy CPA in V [G].So let S be a strategy for Player II. Thus, S is a function from a subset of

D =⋃ξ<ω1

(F∗

prism(X)×F∗prism(X)

)ξ×F∗prism(X) into F∗

prism(X). Since Sω2 isω2-cc and satisfies axiom A, there is a closed unbounded subset ΓS of Γ suchthat for every α ∈ Γ

ω1 α = α & S ∩ V [Gα] = S ∩ [(D ×F∗prism(X)) ∩ V [Gα]] ∈ V [Gα]. (7.1)

Since the quotient forcing Sω2/Sα is equivalent to Sω2 we can assume thatα = 0, that is, that V [Gα] is our ground model V .

Let 〈〈fξ, gξ〉: ξ < ω1〉 be a game played according to S, wher fξ − f0ξ for

every ξ < ω1. Then G = {gξ: ξ < ω1} ∈ V is F∗prism-dense. It is enough to show

that for c = c0 we have

X \ V ⊂⋃

gξ∈Fprism

gξ [{p ∈ dom(gξ):π0(p) = c0}] .

1In [52] the author considers forcings SA for different partial ordered sets which are notnecessary well ordered, so they have no standard iteration equivalences S∗

A. Thus, he provesbasic properties of forcings SA, like preservation of ω1, directly their definition. It couldbe interesting to follow this path and prove the basic facts on Sγ we need straight from thedefinition, instead of deducing them from the known properties of S∗

γ . This certainly would belabor taking, but would allow to avoid all together the complicated matter of iterated forcing,with conditions being the names with respect to initial segments of the iteration.


So, take an r ∈ X \ V . Then there exists an Sω2-name τ for r such that

Sω2 ‖− τ ∈ X \ V.

We can also choose τ such that it is an S(A)-name for some A ∈ [ω2]≤ω with0 ∈ A. Then all the information on r is coded by GA = G � A. Increase Ato a subset A of ω2 of order type ω1. Then r ∈ V [{cω1 ξ: ξ ∈ A}] and S(A) isisomorphic to Sω1 . Applying this isomorphism we can assume that A = ω1, τis a Sω1-name for r, and

Sω1 ‖− τ ∈ X \ V.Now, for any such a name τ and any R ∈ Sω1 there exist P ∈ Sω1 , P ⊂ R, anda continuous injection function f :P → X (so f ∈ Fprism(X) ∩ V ) which “readsτ continuously” in a sense that

Q ‖− τ ∈ f [Q] (7.2)

for every Q ⊂ P , Q ∈ Sω1 . (See [84, Lemma 3.1] or [67, Lemma 6, p. 580]. Thisalso can be deduced from our Lemma 3.2.1.) So, the set

D = {Q ∈ Sω1 : (∃ξ < ω1) Q = dom(gξ) & Q ‖− τ ∈ gξ[Q]} ∈ V

is dense in Sω1 . (For R ∈ Sω1 take f as in (7.2), find ξ < ω1 with f = fξ, andnotice that Q = dom(gξ) justifies the density of D.)

Take Q ∈ D ∩Gω1 and ξ < ω1 such that Q = dom(gξ). Then there is z ∈ Qsuch that π0(z) = c and gξ(z) = r. This finishes the proof.


Part II

CPA for other forcings

101

Notation

• r – the reaping number, 9.

• u – 11.

• i – the independence number, 11.

• C – the Cantor set 2ω, 12.

• Perf(X) – all subsets of a Polish space X homeomorphic to C, 12.

• C(X) – the family of all continuous functions from X into R, 12.

• cof(I) – the cofinality of an ideal I, 12.

• cov(I) – the covering number of an ideal I, 12.

• N – the ideal of Lebesgue measure zero subsets of R, 12.

• M – the ideal of meager subsets of a Polish space, 12.

• s0 – the ideal of Marczewski’s s0-subsets of a fixed Polish space, 12.

• I+ – coideal of the ideal I on X, I+ = P(X) \ I, 12.

• Fcube – all continuous injections from a perfect cube C ⊂ Cω onto a fixedPolish space X, 16.

• CPAcube – 16.

• scube0 – 18.

• “D1” – the class of all f : R → R having finite or infinite derivative at everypoint, 19.

• scube0 – the class of all f : R → R satisfying Banach condition T2, 19.

• CH – the family of all∏n<ω Tn such that Tn ∈ [ω]≤n for all n < ω, 21.

• B1 – the class of all Baire class 1 functions f : R → R, 21.

• d – the dominating number, 24.

• rσ – 28.

103


• Ccube(X) – 35.

• F∗cube(X) – 35.

• Perf∗(X) – 35.

• GAMEcube(X) – 35.

• CPAgamecube – 35.

• a – minimal cardinality of a maximal almost disjoint family A ⊂ [ω]ω,37.

• Φprism(A) – 41.

• SA – prisms in CA, 42.

• Fprism(X) – 42.

• F∗prism – 42.

• GAMEprism(X) – 42.

• CPAgameprism – 42.

• CPAprism – 43.

• πβ – projection from Cα onto Cβ , 43.

• ρ – a nice metric on Cα, 43.

• Bα(z, ε) – an open ball in Cα with center at z and radius ε, 44.

• Bα – clopen subsets of Cα, 44.

• FusionGameα(A) – 45.

• sprism0 – 50.

• add(s0) – the additivity of the ideal s0, 52.

• Dn(X) – n-times differentiable functions from X to R, 55.

• Cn(X) – all functions f ∈ Dn(X) with continuous n-th derivative, 55.

• C∞(X) – all infinitely many times differentiable functions from X into R, 55.

• “Dn(X)”, “Cn(X)”, “C∞(X)” – 55.

• Cnperf , Dnperf , C∞perf , “C∞perf” – 55.

• cov(A,F) – 57.

• dec(A,F) – 59.

• IntTh(A,F) – A and F satisfy Intersection Theorem, 60.

• GAMEsectionprism (X) – 87.


• Γ – {α < ω2: cof(α) = ω1}, 87.

• CPA – Covering Property Axiom, 87.

• CPAsecprism – 88.

• CPAseccube – 88.


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Index

Will be added.

113

Documents

Covering Property Axiom CPA - Department of Mathematicskcies/prepF/BookCPA/00DecCPA/00DecCPA.pdf · Covering Property Axiom CPA KrzysztofCiesielski DepartmentofMathematics, WestVirginiaUniversity,