Covering Property Axiom CPA

Transcription

1 Covering Property Axiom CPA Krzysztof Ciesielski Department of Mathematics, West Virginia University, Morgantown, WV , USA K Cies@math.wvu.edu web page: Janusz Pawlikowski 1 Department of Mathematics, University of Wroc law pl. Grunwaldzki 2/4, Wroc law, Poland pawlikow@math.uni.wroc.pl and Department of Mathematics, West Virginia University, Morgantown, WV , USA pawlikow@math.wvu.edu Draft of March 11, We still plan to make considerable changes to this manuscript. Comments andlist of typos are welcome! 1 The work of the second author was partially supported by KBN Grant number 2 P03A

2 2 Covering Property Axiom CPA March 12, 2001

3 Contents I CPA for Sacks forcing: a combinatorial core of the iterated perfect set model 5 Preface to Part I 7 Overview of Part I Preliminaries Axiom CPA cube and its consequences: properties (A) (E) Perfectly meager sets, universally null sets, and continuous images of sets of cardinality continuum cof(n )=ω 1 and total failure of Martin s Axiom Some consequences of cof(n )=ω 1 : Blumberg theorem, strong measure zero sets, magic set, and cofinality of Boolean algebras Selective ultrafilters and the reaping numbers r and r σ On a convergence of subsequences of real-valued functions Remarks on a form and consistency of the axiom CPA cube Games and axiom CPA game cube CPA game cube and disjoint coverings MAD families and numbers a and r Remark on a form of CPA game cube Prisms and axioms CPA game prism and CPA prism Fusion for prisms On F-independent prisms CPA prism, additivity of s 0, and more on (A) Multi-games and other remarks on CPA game prism and CPA prism CPA prism and coverings with smooth functions Chapter overview; properties (H )and(q) Proof of Proposition Proposition 4.2.1: a generalization of a theorem of Morayne Theorem 4.1.6: on cov (D n, C n ) < c Examples related to cov operator

4 4 Covering Property Axiom CPA March 12, Applications of CPA game prism Nice Hamel bases Some additive functions and more on Hamel bases Selective ultrafilters and number u Non-selective P -points and number i Crowded ultrafilters on Q CPA and properties (F ) and (G) cov(s 0 )=c, many ultrafilters, and ω 1 -intersections of open sets Surjections onto nice sets must be continuous on big sets Sums of Darboux and continuous functions Remark on a form of CPA sec cube CPA in the Sacks model Notations and basic forcing facts Consistency of CPA II CPA for other forcings 123 Notation 125 Bibliography 129 Index 135

5 Part I CPA for Sacks forcing: a combinatorial core of the iterated perfect set model 5

6

7 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 are false under the continuum hypothesis. However, the proofs that these facts are indeed true in this model are usually very technical and involve heavy forcing machinery. In this manuscript we extract a combinatorial principle, an axiom similar to the Martin axiom, that is true in the model and show that this axiom implies the above mentioned properties in a simple mathematical way. The proofs are essentially simpler than the original arguments. To follow all but the last chapter of this manuscript only a moderate knowledge 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 a model of the set theory ZFC in which continuum c = ω 2 and many of the consequences of the continuum hypothesis CH fail. In this paper we will describe a combinatorial axiom of the form similar to Martin axiom which holds in the iterated 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 with the perfect set forcing being a build-in parameter. A similar axioms hold also for several other forcings and their description will be a subject of Part II of this manuscript. In this text, however, we will concentrate only on the axiom associated 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 set model was the most studied from the class of forcing models we are interested in, we had a good supply of statements against which we could test the power of our axiom. In particular, we used for this purpose the statements listed below as (A) (H). The citations in the parentheses refer to the proofs that a given property holds in the iterated perfect set model. For the definitions see the end next section. 7

8 8 Covering Property Axiom CPA March 12, 2001 (A) (Miller [72]) For every subset S of R of cardinality c there exists a (uniformly) continuous function f: R [0, 1] such that f[s] =[0, 1]. (B) (Miller [72]) Every perfectly meager set S R has cardinality less than c. (C) (Laver [64]) Every universally null set S R has cardinality less than c. (D) (Folklore, see e.g. [74] or [4, p. 339]) The cofinality of the measure ideal N is less than c. (E) (Baumgartner, Laver [6]) There exist selective ultrafilters on ω and any such ultrafilter is generated by less than c many sets. (F) (Balcerzak, Ciesielski, Natkaniec [2]) There is no Darboux Sierpiński- 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) (Steprāns [93]) For every Darboux function g: R R there exists a continuous nowhere constant function f: R R such that f + g is Darboux. (H) (Steprāns [94]) The plane R 2 can be covered by less than c many sets each of which is a graph of a partial function defined and differentiable on a perfect subset of either horizontal or vertical axis. The counterexamples, under CH, for (B) and (C) are Luzin and Sierpiński sets. They have been constructed in [67] 1 and [86], respectively. The negation of (A) is witnessed by either Luzin or Sierpiński set, as noticed in [86, 87]. The counterexamples, under CH, for (F) and (G) can be found in [2] and [60], 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 we call the Covering Property Axiom and denote by CPA, requires some extra definitions which are unnecessary for most of applications, we will introduce the axiom in several approximations, from the easiest to state and use to the most powerful but more complicated. All the versions of the axiom will be formulated and discussed in the main body of the chapters. The sections that follow contain only the consequences of the axioms. In particular, most of the sections can be omitted in the reading without causing the difficulty in following the rest of the material. Thus, we start in Chapter 1 with a formulation of the simplest form of our axiom, CPA cube, which is based on a natural notion of a cube in a Polish space. In Section 1.1 we show that CPA cube implies the properties (A) (C), while Section 1.2 is devoted to the proofs that CPA cube implies the property (D) (i.e., cof(n )=ω 1 ) and the following fact, known as the total failure of Martin s Axiom (I) c >ω 1 and for every non-trivial ccc forcing P there exists ω 1 -many dense sets in P such that no filter intersects all of them. 1 Constructed also a year earlier by Mahlo [68].

9 Covering Property Axiom CPA March 12, The consistency of (I) was first proved by Baumgartner [5] in a model obtained by adding Sacks reals side-by-side. In Section 1.3 we will present some consequences of cof(n )=ω 1 that seems to be related to the iterated perfect set model. In particular we prove that cof(n )=ω 1 implies that (J) c >ω 1 and there exists a Boolean algebra B of cardinality ω 1 which is not a union of strictly increasing ω-sequence of subalgebras of B. The consistency of (J) was first proved by Just and Koszmider [54] in a model obtained by adding Sacks reals side-by-side, while Koppelberg [61] showed that Martin s Axiom contradicts (J). In Section 1.4 we show that CPA cube implies that every selective ultrafilter is generated by ω 1 sets (i.e., the second part of the property (E)) and that (K) r = ω 1, 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 CPA cube implies the following version of a theorem of Mazurkiewicz [70]. (L) For each Polish space X and uniformly bounded sequence f n : X R n<ω of Borel measurable functions there are the sequences: P ξ : ξ < ω 1 of compact subsets of X and W ξ [ω] ω : ξ<ω 1 such that X = ξ<ω 1 P ξ and for every ξ<ω 1 : f n P ξ n Wξ is a monotone uniformly convergent sequence of uniformly continuous functions. We will also show that CPA cube + selective ultrafilter on ω implies the following variant of (L): (M) Let X an arbitrary set and f n : X R be a sequence of functions such that the set {f n (x): n<ω} is bounded for every x X. Then there are the sequences: P ξ : ξ<ω 1 of subsets of X and W ξ F: ξ<ω 1 such that X = ξ<ω 1 P ξ and for every ξ<ω 1 : f n P ξ n Wξ is monotone and uniformly convergent. It should be noted here that a result essentially due to Sierpiński (see Example 1.5.2) implies that (M) is false under the Martin s Axiom. The last section of this chapter consists of the remarks on a form and consistency of CPA cube. In particular, we note there that CPA cube is false in a model obtained by adding Sacks reals side-by-side. In Chapter 2 we revise slightly the notion of a cube and introduce a cubegame GAME cube, a covering game of length ω 1 which is a foundation for our next (stronger) variant of the axiom, CPA game cube. In Section 2.1, as its application, we will show that CPA game cube implies that

10 10 Covering Property Axiom CPA March 12, 2001 (N) c >ω 1 and for every Polish space there exists a partition of X into ω 1 disjoint closed nowhere dense measure zero sets. In Section 2.2 we will show that CPA game cube implies that (O) c > ω 1 and there exists a family F [ω] ω of cardinality ω 1 which is simultaneously maximal almost disjoint and reaping. Chapter 3 begins with a definition of a prism, which is a generalization of a notion of cube in a Polish space. This notion, perhaps the most important notion of this text, is then used to our next generation of the axioms, CPA game prism and CPA prism, which are prism (stronger) counterparts of axioms CPA game cube and CPA cube. Since the notion of a prism is rather unknown, in the first two sections of 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 form cubes (Section 3.2). In the last section of this chapter we discuss some application of CPA prism. In particular, we will prove that CPA prism implies the following generalization of the property (A) (A ) there exists a family G of uniformly continuous functions from R to [0, 1] such that G = ω 1 and for every S [R] c there exists a g G with g[s] =[0, 1]. We will also show that CPA prism implies that (P) add(s 0 ), the additivity of the ideal s 0, is equal to ω 1 < c. We finish Chapter 3 with several remarks on CPA game prism. In particular, we prove that CPA game prism implies axiom CPAgame prism (X ) in which the game is played simultaneously over ω 1 Polish spaces. Chapters 4 and 5 deal with the applications of the axioms CPA prism and CPA game prism, respectively. Chapter 4 contains a deep discussion of a problem of covering R 2 and Borel functions from R to R by continuous functions of different smooth levels. In particular, we show that CPA prism implies the following strengthening of the property (H) (H ) there exists a family F of less then continuum many C 1 functions from R to R (i.e., differentiable functions with continuous derivatives) such that R 2 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 than continuum many C 1 functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f. We will also examine which functions can be covered by less than c many C n functions for n>1 and give the examples showing that all covering theorems discussed are the best possible. Chapter 5 concentrates on several specific applications of CPA game prism. Thus in Section 5.1 we will show that CPA game prism implies

11 Covering Property Axiom CPA March 12, (R) there is a family H of ω 1 pairwise disjoint perfect subsets of R such that H = 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 property which is N M-rigid, that is, such that X (r + X) N Mfor every r R; (T) 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<ω 1 there is an h R such that h is not of Borel class α. Implication CPA game prism = (S) answers a question related to the work of Cichoń, Jasiński, Kamburelis, and Szczepaniak [19]. The implication CPA game prism = (T) shows that a recent construction of such a function from CH due to Filipów and Rec law [42] (and answering a question of Laczkovich [63]) can be also repeated with a help of our axiom. In Section 5.2 we show that CPA game prism implies that (U) there is a discontinuous, almost continuous, additive function f: R R whose graph is of measure zero. A first construction of such a function, under Martin s Axiom, was given by Ciesielski in [23]. It is unknown whether it can be constructed in ZFC. We also prove there that, under CPA game prism (V) there exists a Hamel basis H such that E + (H) has measure zero, where E + (A) is a linear combination of A R with non-negative rational coefficients. This relates to work of Erdős [39], H. Miller [75], and Muthuvel [77] who constructed such and similar Hamel bases under different set theoretical assumptions. It is unknown whether (V) holds in ZFC. In Section 5.3 we deduce from CPA game prism that every selective ideal on ω can be extended to a maximal selective ideal. In particular, the first part of the condition (E) holds and u = r σ = ω 1, where u is the smallest cardinality of the base for a non-principal ultrafilter on ω. In Section 5.4 prove that CPA game prism implies that (W) there exist many non-selective P -points as well as a family F [ω] ω of cardinality ω 1 which is simultaneously independent and splitting. In particular i = ω 1, where i is the smallest cardinality of an infinite maximal independent family. We will finish this chapter with the proof that CPA game prism implies that (X) there exists a non-principal ultrafilter on Q which is crowded. In Chapter 6 we formulate the most general form of our axiom, CPA, and show that it implies all the other versions of the axiom. In Section 6.1 we will conclude from CPA that

12 12 Covering Property Axiom CPA March 12, 2001 (Y) cov(s 0 )=c and (Z) if U = {U ξ : ξ<ω 1 } 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 of the property (F): (F ) for an arbitrary function h from a subset S of a Polish space X onto a Polish space Y there exists a uniformly continuous function f from a subset 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 there exists a function f Cperf such that f h = c and f can be extended to a function f C 1 (R) such that either either f C 1 or f is an autohomeomorphism of R with f 1 C 1. 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 [62]. 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. A Cantor set 2 ω will be denoted by a symbol C. In this paper we use term Polish space for a complete separable metric space without isolated points. For a Polish space X symbol Perf(X) will stand for a collection of all subsets of X homeomorphic to a Cantor set C; as usual C(X) will stand for the family of all continuous functions from X into R. A function f: R R is Darboux if a conclusion of the intermediate value theorem holds for f or, equivalently, when f maps every interval onto an interval; f is a Sierpiński-Zygmund function if its restriction f Y is discontinuous for every subset Y of R of cardinality c; f is nowhere constant if it is not constant on any non-trivial interval. A set S R is perfectly meager if S P is meager in P for every perfect set P R, and S is universally null provided for every perfect set P R the set S 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 and its covering as cof(i) = min{ B : B generates I} cov(i) = min{ B : B I& B = X}.

13 Covering Property Axiom CPA March 12, Symbol N will stand for the ideal of Lebesgue measure zero subsets of R. For a fixed Polish space X the ideal of its meager subsets will be denoted by M, and we will use the symbol s 0 (or s 0 (X)) to denote the σ-ideal of Marczewski s s 0 -sets, that is, s 0 = {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 following generalized selectivity terminology. We say (see Farah [40]) that an ideal I is selective provided for every sequence F 0 F 1 of sets from I + there exists an F I + (called a diagonalization of this sequence) with the property that F \{0,...,n} F n for all n F. Notice that this definition agrees with the definition of selectivity given by Grigorieff in [47, p. 365]. (The ideals selective in the above sense Grigorieff calls inductive but he also proves [47, cor. 1.15] that the 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 dense in I + provided for every B I + there exists an A Dsuch that A B; set D is open in I + if B Dprovided there is an A Dsuch that B A. For D = D n I + : n<ω we say that F I + is a diagonalization of D provided F \{0,...,n} D n for every n<ω. Following Farah [40] we say that an ideal I on ω is semiselective provided for every sequence D = D n I + : n<ω of dense and open subsets of I + the family of all diagonalizations of D is dense in I +. Following Grigorieff [47, p. 390] we say that I is weakly selective (or weak selective) provided for every A I + and f: A ω there exists a B I + such that f B is either one-to-one or constant. (Farah in [40, sec. 2] terms such ideals 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 s terminology from [47].) We have the following implications between these notions. (See Farah [40, 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 if and only if it is weakly selective. This follows, for example, directly from the definitions of these notions as in Grigorieff [47]. Recall also that the existence of selective ultrafilters cannot be proved in ZFC. (This is the case since every selective ultrafilter is a P -point, while Shelah proved that there are models with no P -points, see e.g. [4, thm ].)

14 14 Covering Property Axiom CPA March 12, 2001

15 Chapter 1 Axiom CPA cube and its consequences: properties (A) (E) For a Polish space X we will consider Perf(X), the family of all subsets of X homeomorphic to a Cantor set C, as ordered by inclusion. Thus, a family E Perf(X) is dense in Perf(X) provided for every P Perf(X) there exists a Q 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 portion E 0 of E covers almost all of X in a sense that X \ E 0 < c. If the word appropriately in the above is ignored, then it implies the following statement. Naïve-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 the properties 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 s 0 of Marczewski s sets, since G is clearly hereditary. Thus we have { s 0 = X \ } E: E is dense in Perf(X). (1.1) However, it is well known (see e.g. [73, thm. 5.10]) that there are s 0 -sets of cardinality c. Thus, our Naïve-CPA axiom cannot be consistent with ZFC. 15

16 16 Covering Property Axiom CPA March 12, 2001 In order to formulate the real axiom CPA cube we need the following terminology and notation. A subset C of a product C η of the Cantor set is said to be a perfect cube (or just cube) ifc = n η C n, where C n Perf(C) for each n. For a fixed Polish space X let F cube stand for the family of all continuous injections from a perfect cube C C ω onto a set P from Perf(X). We consider each function f F cube 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 F cube onto P, and Q is a subcube of a cube P Perf(X) provided Q = f[c], where f F cube is a witness function for P and C is a subcube of the domain of f. We say that a family E Perf(X) isf cube -dense (or cube-dense) inperf(x) provided every cube P Perf(X) contains a subcube Q E. More formally, E Perf(X) isf cube -dense provided f F cube g F cube (g f & range(g) E). (1.2) It is easy to see that the notion of F cube -density is a generalization of a notion of density as defined in the first paragraph of this chapter: if E is F cube -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 following simple example. Example Let X = C C and let E be the family of all P Perf(X) such that either all vertical sections P x = {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 F cube -dense in Perf(X). With these notions in hand we are ready to formulate our axiom CPA cube. CPA cube : c = ω 2 and for every Polish space X and every F cube -dense family E Perf(X) there is an E 0 E such that E 0 ω 1 and X \ E 0 ω 1. The proof that CPA cube is consistent with ZFC (it holds in the iterated perfect set model) will be presented in the next chapters. In the reminder of this chapter we will take a closer look at CPA cube and its consequences. In what follows we consider F cube as ordered by the inclusion. Therefore F F cube is dense in F cube provided for every f F cube there exists a g F with g f. Proposition CPA cube is equivalent to the following. CPA 0 cube : c = ω 2 and for every Polish space X and every dense subfamily F of F cube there is an F 0 [F] ω1 such that X \ g F 0 range(g) ω 1.

17 Covering Property Axiom CPA March 12, Proof. First notice that E Perf(X) isf cube -dense if and only if F cube (E) is dense in F cube, (1.4) where F cube (E) ={g F cube : range(g) E}. Now, to see that CPA 0 cube implies CPA cube take an F cube -dense family E Perf(X). Then, using CPA 0 cube with F = F cube(e), we can find an appropriate F 0 [F] ω1. Clearly E 0 = {range(g): g F 0 } satisfies the conclusion of CPA cube. To see the other implication, take a dense subfamily F of F cube and let E = {range(g): g F}. Then, by (1.4), E is F cube -dense in Perf(X), since F F cube (E). By CPA cube we can find an E 0 [E] ω1 such that X \ E 0 < c. For each E E 0 take an f E Fsuch that range(f E )=E. Then F 0 = {f E : E E 0 } satisfies the conclusion of CPA 0 cube. Condition CPA 0 cube has a form which closer to our main axiom CPA than CPA cube. Also CPA 0 cube does not require a new notion of F cube-density. So why bother with CPA cube at all? The reason is that the consequences presented in this chapter follows more naturally from CPA cube than from CPA 0 cube. Moreover, from CPA cube it is also clearer that the axiom describes, in fact, a property of perfect subsets of X, rather than of some coordinate functions. Note that if F all stands for the family of all injections from perfect subsets of C ω into X then we can use schema (1.2) to define a notion of F-density of E Perf(X) for any family F F all : E Perf(X) isf-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) if and only if E is F all -dense in Perf(X). It is also worth to notice that in order to check that E is F cube -dense it is enough to consider in condition (1.2) only functions f defined on the entire space C ω, that is Fact E Perf(X) is F cube -dense if and only if f F cube, dom(f) =C ω, g F cube (g f & range(g) E). (1.6) Proof. To see this, let Φ be the family of all bijections h = h n n<ω between perfect subcubes n ω D n and n ω C n of C ω such that each h n is a homeomorphism between D n and C n. Then f h F cube for every f F cube and h Φ with range(h) dom(f). (1.7) Now take an arbitrary f: C X from F cube and choose an h Φ mapping C ω onto C. Then ˆf = f h F cube maps C ω into X and, using (1.6), we can find ĝ F cube such that ĝ ˆf and range(ĝ) E. Then g = f h[dom(ĝ)] satisfies (1.2).

18 18 Covering Property Axiom CPA March 12, 2001 Next, let us consider s cube 0 = { X \ } E: E is F cube -dense in Perf(X) = {S X: cube P Perf(X) subcube Q P \ S}. (1.8) It can be easily shown, in ZFC, that s cube 0 forms a σ-ideal. However, we will not use this fact in this text in that general form. This is the case, since we will usually assume that CPA cube holds while CPA cube implies the following stronger fact. Proposition If CPA cube holds then s cube 0 =[X] ω1. Proof. It is obvious that CPA cube implies s cube 0 [X] <c. The other inclusion is always true and it follows from the following simple fact. Fact [X] <c s cube 0 s 0 for every Polish space X. Proof. Choose S [X] <c. In order to see that S s cube 0 note that the family E = {P Perf(X): P S = } is F cube -dense in Perf(X). Indeed, if function f: C ω X is from F cube then there is a perfect subset P 0 of C which is disjoint with the projection π 0 (f 1 (S)) of f 1 (S) into the first coordinate. Then f [ i<ω P i] S =, where Pi = C for all 0 <i<ω. Therefore, f [ i<ω P i] E. Thus, S X \E s cube 0. The inclusion s cube 0 s 0 follows immediately from (1.1), (1.8), and (1.3). 1.1 Perfectly meager sets, universally null sets, and continuous images of sets of cardinality continuum An important quality of the ideal s cube 0, and so the power of the assumption s cube 0 =[X] <c, is well depicted by the following fact. Proposition If X is a Polish space and S X does not belong to s cube 0 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: C ω X be a continuous injection such that f[c] S for every cube C. Let g: C C be a continuous function such that g 1 (y) is perfect for every y C. Then h 0 = g π 0 f 1 : f[c ω ] C is uniformly continuous. Moreover, if T = S f[c ω ] then h 0 [T ]=C since T h 1 0 (y) =T f[π 1 0 (g 1 (y))] = S f[g 1 (y) C C ] for every y C.

19 Covering Property Axiom CPA March 12, Corollary Assume s cube 0 =[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, CPA cube implies property (A). Proof. If S is as above then, by CPA cube, S / s cube 0. Thus, by Proposition there exists a uniformly continuous function h from a subset of S onto C. Let ĥ: X [0, 1] be an uniformly continuous extension of h and let g maps continuously C onto [0, 1]. Then f = ĥ g is as desired. For more on the property (A) see also Corollary It is worth to note here that the function f in Corollary cannot be required to be either monotone or in the class D 1 of all functions having finite or infinite derivative at every point. This follows immediately from the following proposition, since each function which is either monotone or D 1 belongs to the Banach class (See [43] or [84, p. 278].) (T 2 )= { f C(R): {y R: f 1 (y) >ω} N }. Proposition There exists, in ZFC, an S [R] c such that [0, 1] f[s] for every f (T 2 ). Proof. Let {f ξ : ξ<c} be an enumeration of all functions from (T 2 ) which range contains [0, 1]. Construct by induction a sequence s ξ,y ξ : ξ<c such that for every ξ<c (i) y ξ [0, 1] \ f ξ [{s ζ : ζ<ξ}] and f 1 ξ (y ξ ) ω. ( (ii) s ξ R \ {s ζ : ζ<ξ} ) ζ ξ f 1 ζ (y ζ ). Then the set S = {s ξ : ξ<c} is as required since y ξ [0, 1] \ f ξ [S] for every ξ<c. Theorem If S R is either perfectly meager or universally null then S s cube 0. In particular, CPA cube = s cube 0 =[R] <c = (B) & (C). Proof. Take an S R which is either perfectly meager or universally null and let f: C ω R be a continuous injection. Then S f[c ω ] is either meager or null in f[c ω ]. Thus G = C ω \ f 1 (S) is either comeager or of full measure in C ω. Hence the theorem follows immediately from the following fact. Claim If G C ω is either comeager or of full measure in C ω then it contains a perfect cube i<ω P i. Claim follows easily, by induction on coordinates, from the following well known fact.

20 20 Covering Property Axiom CPA March 12, 2001 For every comeager (full measure) subset H of C C there are perfect set P C and a comeager (full measure) subset Ĥ of C such that P Ĥ H. The category version is easy and can be found in [56, Exercise 19.3]. (Its version for R 2 is also proved, for example, in [33, condition ( ), p. 416].) The measure version follows easily form the fact that for every full measure subset H of [0, 1] [0, 1] there are perfect set P C and a positive inner measure subset Ĥ of [0, 1] such that P Ĥ H which is proved by Eggleston [38] and, independently, by Brodskiǐ [10]. 1.2 cof(n )=ω 1 and total failure of Martin s Axiom We will start with the proof that CPA cube implies that for every non-trivial ccc forcing P there exists ω 1 -many dense sets in P such that no filter intersects all of them. This follows immediately from the theorem below. The consistency of this fact with c >ω 1 was first proved by Baumgartner [5] in a model obtained by adding Sacks reals side-by-side. Theorem If CPA cube holds then every compact ccc topological space is a union of ω 1 nowhere dense sets. Proof. Let cba stand for complete Boolean algebra. First notice that in order to prove the theorem it is enough to show that for every ccc cba B there is a family of ω 1 -many maximal antichains in B such that no filter in B intersects all of them. Next notice that it is enough to prove this fact only for countable generated ccc cba s. Indeed, every ccc cba B contains a complete countably generated Boolean subalgebra B 0, which is still ccc. Now, every maximal antichain in B 0 is also a maximal antichain in B. So, if A = {A ξ : ξ<ω 1 } is a family of maximal antichains in B 0 such that no filter in B 0 intersects all of them, then A witness the same fact for B. Next let B be the algebra of Borel subsets of C =2 ω and note that it is a free countably generated cba, with free generators A i = {s C: s(i) =0}. Thus, by Loomis-Sikorski theorem (see [91, p. 117] or [66]), every countably generated ccc cba B 0 (which is just a ccc sigma complete Boolean algebra) is isomorphic to 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 is some σ-ideal of Borel sets containing all singletons. To prove that such algebra has ω 1 maximal antichains as desired, it is enough to prove that C is a union of ω 1 sets from I.

21 Covering Property Axiom CPA March 12, But this follows easily from CPA cube, since any cube P in C contains a subcube Q I: any cube P can be partitioned into c-many disjoint subcubes and, by the ccc property of I, only countable many of them can be outside I. Next we show that CPA cube implies that cof(n )=ω 1. So, under CPA cube, all cardinals from Cichoń s diagram (see e.g. [4]) are equal to ω 1. Let C H be the family of all subsets n<ω T n of ω ω such that T n [ω] n+1 for all n<ω. We will use the following characterization. Proposition (Bartoszyński [4, thm ]) { cof(n ) = min F : F C H & F = ω ω}. Lemma Family C H is F cube -dense in Perf(ω ω ). Proof. Let f: C ω ω ω be a continuous injection. By (1.6) it is enough to find a cube C in C ω such that f[c] C H. So, let { n i,m i : i<ω} be an enumeration of ω ω and for i<ωlet S i = {s C ω : s(n j )(m j ) = 0 for all i j<ω}. Notice that S i =2 i for every i<ω. By induction on i<ωwe will define sets X i = {x s C ω : s S i }. For the unique s S 0 we choose x s C ω arbitrarily. Now, if X i 1 is constructed for some 0 <i<ωwe extend it to X i as follows. For s S i \ S i 1 let s S i 1 be such that s(n j )(m j )=s (n j )(m j ) for all j i. We choose x s x s such that (i) x s (n) =x s (n) for all n n i, and (ii) f(x s ) 2 i = f(x s ) 2 i. The choice satisfying (ii) can be made since f is continuous. Note that we have defined a uniformly continuous mapping g, s x s, from S = i<ω S i onto X = {x s : s S}. So g can be extended continuously to a function ḡ from C ω =cl(s) ontoc = cl(x). Note that, by (i), C is a cube since s(n) =s (n) implies ḡ(s)(n) =ḡ(s )(n) for all s, s C ω and n<ω. Also, by (ii), f(x)(k) T k = {f(x s )(k): s S i } for all x C and 2 i k<2 i+1. So, f[c] n<ω T n C H. Corollary If CPA cube holds then cof(n )=ω 1. Proof. By CPA cube and Lemma there exists an F [C H ] ω1 such that ω ω \ F ω 1. This and Proposition imply cof(n )=ω 1. As a last application in this section let us consider the following covering number connected to a Blumberg s theorem (see next section) and studied by F. Jordan in [52]. Here B 1 stands for the class of all Baire class 1 functions f: R R.

22 22 Covering Property Axiom CPA March 12, 2001 cov(b 1, Perf(R)) is the smallest cardinality of F B 1 such that for each P Perf(R) there is an f F with f P not continuous. Jordan proves also [52, thm. 7(a)] that cov(b 1, Perf(R)) is equal to the covering number of the space Perf(R) (considered with the Hausdorff metric) by the elements of some σ-ideal Z p and notices [52, thm. 17(a)] that every compact set C Perf(R) contains a dense G δ subset which belongs to Z p. So, by Claim 1.1.5, the elements of Perf(R) Z p are F cube -dense in Perf(R). Thus Corollary CPA cube implies that cov(b 1, Perf(R)) = cov(z p )=ω Some consequences of cof(n )=ω 1 : Blumberg theorem, strong measure zero sets, magic set, and cofinality of Boolean algebras In this section we will show some consequences of cof(n ) = ω 1 (so also of CPA cube ) which are related to the Sacks model. In 1922 Blumberg [9] proved that for every f: R R there exists a dense subset D of R such that f D is continuous. This theorem sparked a lot of discussion and generalizations, see e.g. [22, pp ]. In particular, Shelah [85] showed that there is a model of ZFC in which for every f: R R there is a nowhere meager subset D of R such that f D is continuous. The dual measure result, that is the consistency of a statement for every f: R R there is a subset D of R of positive outer Lebesgue measure such that f D is continuous, has been also recently established by Ros lanowski and Shelah [82]. Below we note that each of these properties contradicts CPA cube. (See also in [29].) Theorem Let I {N, M}. Ifcof(I) =ω 1 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 essentially identical. Let {N ξ R 2 : ξ<ω 1 } be a family cofinal in the ideal of null subsets of R 2 and for each ξ<ω 1 let N ξ = {x R:(N ξ ) x / N}, where (N ξ ) x = {y R: x, y N ξ }. By Fubini theorem each Nξ each x Nξ \ ζ<ξ N ζ we choose f(x) so that f(x) / ζ<ξ(n ζ ) x. is null. For Then function f is as desired. Indeed, if f D is continuous for some D R then f D is null in R 2. In particular, there exists a ξ<ω 1 such that f D N ξ. But this means that D ζ ξ N ζ.

23 Covering Property Axiom CPA March 12, Note that essentially the same proof works if we assume only that cof(i) is equal to the additivity number add(i) of I. Corollary Assume CPA cube. Then there exists an f: R R such that if f D is continuous then D N M. Proof. By CPA cube we have cof(n ) = cof(m) =ω 1. Let f N and f M be from Theorem constructed for the ideals N and M, respectively. Let G R be a dense G δ of measure zero and put f =[f M G] [f N (R \ G)]. Then this f is as desired. Although CPA cube implies that all cardinals from Cichoń s diagram are as small as possible, this does not extend to all possible cardinal invariants. For example we note that CPA cube refutes Borel conjecture. Corollary If CPA cube holds then there exists an uncountable strong measure zero set. Proof. This follows from the fact (see e.g. [4, thm ]) that cof(n )=ω 1 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) iffor every different nowhere constant functions f,g C(R) wehavef[m] g[m]. It has been proved by Berarducci and Dikranjan [7, thm. 8.5] that a magic set exists under CH, while Ciesielski and Shelah [32] constructed a model with no magic set. It is relatively easy to see that if M is magic then M is not meager and f[m] [0, 1] for every f C(R). (See [17, cor and thm. 5.6(5)].) So CPA cube implies there is no magic set of cardinality c. On the other hand it was noticed by Todorcevic (see [17, p. 1097]) that in the iterated perfect set model there is a magic set (clearly of cardinality ω 1 ). We like to note here that the same is implied by CPA cube. (See also in [29].) Proposition If the cofinality cof(m) of the ideal M of meager sets is equal ω 1 then there exists a magic set. In particular, CPA cube implies that there exists a magic set. Proof. An uncountable set L R is a 2-Lusin set provided for every disjoint subsets {x ξ : ξ<ω 1 } and {y ξ : ξ<ω 1 } of L, where the enumerations are one-toone, the set of pairs { x ξ,y ξ : ξ<ω 1 } is not a meager subset of R 2. In [17, prop. 4.8] it was noticed that every ω 1 -dense 2-Lusin set is a magic set. It is also a standard and easy diagonal argument that cof(m) =ω 1 implies the existence of a ω 1 -dense 2-Lusin set. (The proof presented in [95, prop. 6.0] works also under the assumption cof(m) =ω 1.) So, CPA cube = cof(n )=ω 1 = cof(m) =ω 1 = there is a magic set.

24 24 Covering Property Axiom CPA March 12, 2001 Recall also that the existence of a magic set for the class D 1 can be proved in ZFC. This follows from [18, thm. 3.1], since every function from D 1 belongs to the class (T 2 ). (Compare also [18, cor. 3.3 and 3.4].) For an infinite Boolean algebra B its cofinality cof(b) is defined as the least infinite cardinal number κ such that B is a union of strictly increasing sequence of type κ of subalgebras of B. Koppelberg [61] proved that Martin s Axiom implies that cof(b) =ω for every Boolean algebra B with B < c while Just and Koszmider [54] proved that in the model obtained by adding at least ω 2 Sacks reals side-by-side there exists a Boolean algebra B of cardinality ω 1 such that cof(b) =ω 1 < c. (See also [37].) We like to notice here that this follows also form CPA cube, since we have: Theorem cof(n )=ω 1 implies that there exists a Boolean algebra B of cardinality ω 1 such that cof(b) =ω 1. The proof of Theorem presented below follows the argument given in [28]. It will be based on the following lemma. Lemma If cof(n )=ω 1 then for every infinite countable Boolean algebra A there exists a family {a ξ n A: n<ω& ξ<ω 1 } with the following properties. (i) a ξ n a ξ m = 0 for every n<m<ωand ξ<ω 1. (ii) For every increasing sequence A n : n<ω of proper subalgebras of A with A = n<ω A n there exists a ξ<ω 1 such that a ξ n / A n for all n<ω. Proof. In the argument that follows every sequence Ā = A n: n<ω as in (ii) will be identified with a function f Ā = f ω A for which f 1 (n) =A n \ i<n A i. We will denote the set of all such functions by X. Also, let {b n : n<ω} be an enumeration of A and for each n<ωlet B n be a finite algebra generated by {b i : i<n}. Thus, A = i<n B i. Since cof(n )=ω 1, the dominating number d = min { K : K ω ω &( f ω ω )( g K)( n <ω) f(n) <g(n)} is equal to ω 1. (See e.g. [4].) So, there exists a dominating family K ω ω of cardinality ω 1. We can also assume that the sequences in K are strictly increasing and that for every g Kfunction ḡ defined by ḡ(n) = i n g(i) also belongs to K. Next notice that for every f X there exist d = d k : k < ω Kand r = r k : k<ω Ksuch that for every k<ω (a) f(b) <r k for all b B k ; and (b) there are disjoint b 0,...,b 2k B dk with r dk 1 <f(b 0 ) < <f(b 2k ). Indeed, the existence of r satisfying (a) follows directly from the definition of a dominating family. Moreover, since all algebras A n = f 1 ({0,...,n}) are

25 Covering Property Axiom CPA March 12, proper, for every number d<ωthere exist disjoint b 0,...,b 2d Asuch that r d <f(b 0 ) < <f(b 2d ). Let h ω ω be such that b 0,...,b 2d B d+h(d) for every d<ωand let g Kbe a function dominating h. Then d =ḡ is as required. The above implies, in particular, that for every f X there are d, r Ksuch that f satisfies (b) and the sequence f r = f (B dk \ B dk 1 ): k<ω belongs to X( d, r) = k<ω(r dk ) B d k \B dk 1. Now, since cof(n )=ω 1, by Proposition (applied to k<ω ωb d k \B dk 1 in place of ω ω ) we can find an ω 1 -covering of X( d, r) by sets T of the form [ ] k+1 k<ω T k, where T k ω B d k \B dk 1 for all k<ω. Since the total number of these sets T (for different d, r K) is equal to ω 1, to finish the proof it is enough to show that for any such T there is one sequence a n : n<ω satisfying (i) and such that (ii) holds for every for every Ā = A n: n<ω for which f r Ā belongs to T and f Ā satisfies (b). So, let T be as above and let T be the set of all functions f Ā satisfying (b) for which f r T. By induction on k<ωwe will construct a sequence Ā c k B dk \ B dk 1 : k<ω such that ( ) f(c k ) >r dk k for every k<ωand f T. So fix a k<ωand let {f i : i<k} be such that {f i B dk \ B dk 1 : i<k} = {f B dk \ B dk 1 : f T } T k. We show inductively that for every m<k there is a c B dk such that f j (c) >r dk for all j m. (1.9) So, fix an m<kand let a B dk such that f j (a c )=f j (a) >r dk for all j < m. If f m (a) > r dk then c = a satisfies property (1.9). Thus, assume that f m (a c )=f m (a) r dk. By (b) we can find b 0,...,b 2k B dk such that r dk 1 <f m (b 0 ) < <f m (b 2k ). By Pigeon Hole Principle we can find an I [{0,...,2k}] k+1 and a b {a, a c } such that f m (b b i )=f m (b i ) for all i I. Without loss of generality we can assume that I = {0,...,k} and b b i = b i for all i k. Then f m (b c b i ) >r dk for all i k. Moreover, for every j<mthere is at most one i j k for which f j (b c b ij ) r dk since for different i, i k we have f j ((b c b i ) (b c b i )) = f j (b c ) >r dk. Thus, by Pigeon Hole Principle, there is an i k such that c = b c b i satisfies (1.9). This finishes the proof of ( ). Clearly the sequence c k B dk \ B dk 1 : k<ω satisfies (ii) for every Ā with f Ā T. Thus, we need only to modify it to get also the condition (i). To do it, use the fact that

26 26 Covering Property Axiom CPA March 12, 2001 r dk <f(c k ) <r dk+1 for every k<ωand f T to construct the sequences: ω = I 0 I 1 of infinite subsets of ω, increasing k j I j : j<ω, and c k j {c kj,c c k j }: j<ω such that for every j<ω f(ā kj c l ) >r dl for every f T and l>k j with l I j, where ā kj = c k 0 c k j. Then the sequence ā kj : j<ω is a strictly decreasing sequence satisfying (ii) and it is now easy to see that by putting a j =ā kj ā c k j+1 we obtain the desired sequence. Proof of Theorem Algebra B we construct will be a subalgebra of the algebra P(ω) of all subsets of ω. First, let K ω ω be a dominating family with K = ω 1 and fix a partition {D k : k<ω} of ω into infinite subsets. For every sequence ā = a n : n<ω of pairwise disjoint subsets of ω and k<ωput a k = {a n : n D k }. In addition, for every h Kwe put a h = {a nh (k): k<ω} where n h (k) = min{n D k : n>max{h(k),k}}. We also put F (ā) ={a k: k<ω} {a h : h K} [P(ω)] ω1. Next, we will construct an increasing sequence B ξ [P(ω)] ω1 : ξ ω 1 of subalgebras of P(ω) aiming for B = B ω1. Thus, we choose B 0 as an arbitrary subalgebra of P(ω) with B 0 = ω 1 and for limit ordinal numbers λ ω 1 we put B λ = ξ<λ B ξ. The algebra B ξ+1 is formed from B ξ in the following way. Let {b η : η < ω 1 } be an enumeration of B ξ and for η < ω let A ξ η be a subalgebra of B ξ generated by {b ζ : ζ < η}. For each such algebra we apply Lemma to find the sequences ā γ = a γ n: n<ω, γ<ω 1, satisfying (i) and (ii) and let G(A ξ η)= F (ā γ ). γ<ω 1 B ξ+1 is defined as the algebra generated by B ξ η<ω 1 G(A ξ η). This finishes the construction of B. Clearly, B = ω 1. To prove that cof(b) =ω 1 it is enough to show that B is not a union of an increasing sequence B = B n : n<ω of proper subalgebras. So, by way of contradiction, assume that such a sequence B exists. For every n<ωchoose b n B \ B n and find ξ,η < ω 1 such that {b n : n<ω} A ξ η. Then the algebras A n = B n A ξ η form an increasing sequence of proper subalgebras of A = A ξ η. Thus, one of the sequences ā γ satisfies (ii) for Ā. So, if we put ā γ =ā = a n : n<ω we conclude that {a k : k<ω} {ah : h K} B. Let f(k) = min{n <ω: a k B n} and let h Kbe such that f(k) <h(k) for all k<ω. The final contradiction is obtained by noticing that a h cannot belong to any B k. Indeed, if a h B k for some k, then a h a k = a n h (k) belongs to

27 Covering Property Axiom CPA March 12, B max{f(k),k}, since a k B f(k). But max{f(k),k} max{h(k),k} <n h (k) so we get a nh (k) B nh (k) contradicting the fact that a nh (k) belongs to Ā\A n h (k), which is disjoint with B nh (k). 1.4 Selective ultrafilters and the reaping numbers r and r σ In this section, which in part is based on [30], we will show that CPA cube implies that every selective ultrafilter is generated by ω 1 sets and that the reaping number r is equal to ω 1. The actual construction of a selective ultrafilter will require a stronger version of the axiom and will be done in Theorem We will use here the terminology introduced in the Preliminaries Section. In particular, recall that the ideal [ω] <ω of finite subsets of ω is semiselective. The most important combinatorial fact for us concerning semiselective ideals is the following property. (See theorem 2.1 and remark 4.1 in [40].) This is a generalization of a theorem of Laver [65] who proved this fact for the ideal I =[ω] <ω. Proposition (Farah [40]) Let I be a semiselective ideal on ω. For every analytic set S C ω [ω] ω and every A I + there exist a B I + P(A) and a perfect cube C in C ω such that C [B] ω is either contained in or disjoint with S. With this fact in hand we can prove the following theorem. Theorem Assume that CPA cube holds. If I is a semiselective ideal then there is a family W I +, W ω 1, 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 the family {S x : x C} (where S x = {y [ω] ω : x, y S}) contains all analytic subsets of [ω] ω. (See e.g. [51, lemma 39.4].) In fact, we will take S such that for any analytic set A in [ω] ω {x C: S x = A} = c. (1.10) For this particular set S consider the family E of all Q Perf(C) for which there exists a W Q I + such that Q [W Q ] ω is either contained in or disjoint with S. (1.11) Note that, by Proposition 1.4.1, family E is F cube -dense in Perf(C). So, by CPA cube, there exists an E 0 E, E 0 ω 1, such that C \ E 0 < c. Let W = {W Q : Q E 0 }. It is enough to see that this W is as required.

28 28 Covering Property Axiom CPA March 12, 2001 Clearly W ω 1. Also, by (1.10), for an analytic set A [ω] ω there exist a Q E 0 and an x Q such that A = S x. So, by (1.11), {x} [W Q ] ω is either contained in or disjoint with {x} S x = {x} A. Recall (see e.g. [4] or [98]) that a family W [ω] ω is a reaping family provided 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 r σ is defined as the smallest cardinality of a family W [ω] ω such that for every sequence A n [ω] ω : n<ω there exists a W Wsuch that for every n<ωeither W A n or W ω \ A n. (See [15] or [98].) Clearly r r σ. Corollary If CPA cube holds then for every semiselective ideal I there exists a family W I +, W ω 1, such that for every A [ω] ω there is a W W for which either W A or W ω \ A. In particular, CPA cube implies that r = ω 1 < c. Proof. Family W from Theorem 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 CPA cube implies the second of the property (E). Corollary If CPA cube holds then every selective ultrafilter F on ω is generated by a family of size ω 1 < c. Proof. If F is a selective ultrafilter on ω then I = P(ω) \F is a selective ideal and I + = F. Let W I + = F be as in Corollary Then W generates F. Indeed, if A Fthen there exists a W Wsuch that either W A or W ω \ A. But it is impossible that W ω \ A since then we would have = A W F. As mentioned above, in Theorem we will prove that some version of our 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 If CPA cube holds and there exists a selective ultrafilter F on ω then r σ = ω 1 < c. Proof. Let W [F] ω1 be a generating family for F. We will show that it justifies r σ = ω 1. Indeed, take a sequence A n [ω] ω : n<ω. For every n<ω let A n belongs to F {A n,ω\ A n }. Since F is selective, there exists an A F such that A A n for every n<ω. Let W Wbe such that W A. Then for every n<ωeither W A n or W ω \ A n. We got some special interest in number r σ since it is related to different variants of sets of uniqueness coming from harmonic analysis, as described in a survey paper [15]. In particular, from [15, thm. 12.6] it follows that an appropriate version of our axiom implies that all covering numbers described in the paper are equal to ω 1.