Journal of Applied Analysis Vol. 9, No. 1 (2003), pp. 19–55

CROWDED AND SELECTIVE ULTRAFILTERS UNDER THE COVERING PROPERTY AXIOM

K. CIESIELSKI and J. PAWLIKOWSKI

Received May 17, 2002 and, in revised form, December 11, 2002

Abstract. In the paper we formulate an axiom CPA_{prism}^{game}, which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to ω_{1}, while 𝔠 = ω_{2}: the independence number 𝔦, the reaping number 𝔯, the almost disjoint number 𝔞, and the ultrafilter base number 𝔲. We will also show that CPA_{prism}^{game} implies the existence of crowded and selective ultrafilters as well as nonselective P -points. In addition we prove that under CPA_{prism}^{game} every selective ultrafilter is ω_{1}-generated. The paper finishes with the proof that CPA_{prism}^{game} holds in the iterated perfect set model.

1. Introduction and preliminaries

The Covering Property Axiom, CPA, constitutes an attempt to axiomatize the iterated perfect set (Sacks) model. In this paper we will consider its

2000 Mathematics Subject Classification. Primary 54D80, 03E17; Secondary 03E35. Key words and phrases. Ultrafilter, selective, crowded, P -point, rational numbers, in-

dependence, reaping, almost disjoint, covering.

The work of the first author was partially supported by NATO Grant PST.CLG.977652 and 2002/03 West Wirginia University Senate Research Grant. The second author wishes to thank West Virginia University for its hospitality during 1998–2001, where the results presented here were obtained.

ISSN 1425-6908 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag.

prominent version, CPA_{prism}^{game}, as well its three weaker variations: CPA_{prism}, CPA_{cube}^{game}, and CPA_{cube}. They are related to each other by the following implications.

[figure: CPA_{prism}^{game}; CPA_{cube}^{game}; CPA_{prism}; CPA_{cube}]

Chart 1.

Although in some cases the stronger versions of CPA are useful (see e.g. [8, Chapter 6]), it is known that the axiom CPA_{prism}^{game} captures the essence of the Sacks model concerning the standard cardinal characteristics of continuum. This follows from a resent result of J. Zapletal [27] who proved that for a “nice” cardinal invariant κ if κ < 𝔠 holds in any forcing extension then κ < 𝔠 follows already from CPA_{prism}^{game}.

The Covering Property Axiom is quite simple in formulation and use, nevertheless it requires some new concepts. To facilitate the absorption of these concepts we decided to introduce the axiom in three steps, beginning with its simplest form. (More on CPA can be found in [8].) Thus, in Section 2 we formulate the simplest version of the axiom, CPA_{cube}, and show that it implies that every selective ultrafilter is generated by ω_{1} sets and that the reaping number 𝔯 is equal to ω_{1}. In Section 3 we will formulate axiom CPA_{cube}^{game} and show that it implies CPA_{cube} as well as the existence of a family F ⊂ [ω]^{ω} of cardinality ω_{1} which is simultaneously maximal almost disjoint, MAD, and reaping. In particular, CPA_{cube}^{game} implies that 𝔞 = ω_{1} < 𝔠. In Section 4 we will formulate axioms CPA_{prism}^{game} and CPA_{prism} and show that CPA_{prism}^{game} implies all other versions of the axiom. We will also show there that CPA_{prism}^{game} implies the existence of selective and crowded ultrafilters as well as nonselective P -points. In addition we prove there that CPA_{prism}^{game} implies the existence of a family F ⊂ [ω]^{ω} of cardinality ω_{1} which is simultaneously independent and splitting. In particular, under CPA_{prism}^{game} we have 𝔰 = 𝔦 = 𝔲 = ω_{1} < 𝔠. In the last section of the paper we will prove the prism fusion lemma, which has been used in Section 4, and show that CPA_{prism}^{game} holds in the iterated perfect set model.

Our set theoretic terminology is standard and follows that of [6]. In particular, |X| stands for the cardinality of a set X and 𝔠 = |ℝ|. A Cantor set 2^{ω} will be denoted by a symbol ℭ. We use the term Polish space for a complete separable metric space without isolated points. For a Polish space X, the symbol Perf(X) will stand for the collection of all subsets of X homeomorphic to a Cantor set ℭ. For a function f: X → ℝ and A ⊂ X an image of A under f is denoted by f[A], that is, f[A] = {f(x): x ∈ A}.

For an ideal I on ω containing all finite subsets of ω we will use the following generalized selectivity terminology. We say (see Farah [14]) that an ideal I is selective provided for every sequence F_{0} ⊃ F_{1} ⊃ ⋯ of sets from I^{+} =^{def} P(ω) \ I there exists an F_{∞} ∈ I^{+} (called a diagonalization of this sequence) such that F_{∞} \ {0, …, n} ⊂ F_{n} for all n ∈ F_{∞}. Notice that this definition agrees with the definition of selectivity given by Grigorieff in [15, p. 365]. (The ideals selective in the above sense Grigorieff calls inductive but he also proves [15, Corollary 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 ∈ D such that A ⊆^{∗} B; set D is open in I^{+} if B ∈ D provided there is an A ∈ D such that B ⊆^{∗} A. For D̄ = ⟨D_{n} ⊂ I^{+} : n < ω⟩ we say that F_{∞} ∈ I^{+} is a diagonalization of D̄ provided F_{∞} \ {0, …, n} ∈ D_{n} for every n < ω. Following Farah [14] we say that an ideal I on ω is semiselective provided for every sequence D̄ = ⟨D_{n} ⊂ I^{+} : n < ω⟩ of dense and open subsets of I^{+} the family of all diagonalizations of D̄ is dense in I^{+}.

Following Grigorieff [15, 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 [14, Section 2] terms such ideals as having the Q^{+}-property. Note also that Baumgartner and Laver in [2] call such ideals selective, despite the fact that they claim to use Grigorieff’s terminology from [15].)

We have the following implications between these notions. (See Farah [14, Section 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 = {X \ F: F ∈ 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 [15]. Recall also that the existence of selective ultrafilters cannot be proved in ZFC. (Kunen [21] proved that there are no selective ultrafilters in the random real model. This also follows from the fact that every selective ultrafilter is a P -point, while Shelah proved that there are models with no P -points, see e.g. [1, Theorem 4.4.7].)

2. Axiom CPA_{cube} and its consequences

For a Polish space X we will consider Perf(X) 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.

Axiom CPA_{cube} will be 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.

Naive-CPA: If E is dense in Perf(X) then |X \ ⋃ E| < 𝔠. It is a very good candidate for our axiom in the 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)} .  (2.1)

However, it is well known (see e.g. [24, Theorem 5.10]) that there are s_{0}-sets of cardinality 𝔠. Thus, our Naïve-CPA “axiom” cannot be consistent with ZFC.

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 if C = ∏_{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.^{1} We say that P ∈ Perf(X) is a cube if it is determined by an (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 the witness function for P and C is a subcube of the domain of f.

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

∀f ∈ F_{cube} ∃g ∈ F_{cube} (g ⊂ f & range(g) ∈ E).  (2.2)

1In a language of forcing a coordinate function f is simply a nice name for an element from X.

It is easy to see that the notion of F_{cube}-density is a generalization of the notion of density as defined in the first paragraph of this section:

if E is F_{cube}-dense in Perf(X) then E is dense in Perf(X).

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

Example 2.1. Let X = ℭ × ℭ and let E be the family of all P ∈ Perf(X) such that either

• all vertical sections of P are countable, or

• all horizontal sections 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^{2} 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}.

It is also worth noticing that in order to check that E is F_{cube}-dense it is enough to consider in condition (2.2) only functions f defined on the entire space ℭ^{ω}, that is

Fact 2.2. 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). (2.3)

Proof. To see this, let Φ be the family of all bijections h = ⟨h_{n}⟩_{n<ω} between the perfect cubes ∏_{n∈ω} D_{n} and ∏_{n∈ω} C_{n} in 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).

Now take an arbitrary f: C → X from F_{cube} and choose an h ∈ Φ mapping ℭ^{ω} onto C. Then f̂ = f ∘h ∈ F_{cube} maps ℭ^{ω} into X and, using (2.3), we can find a ĝ ∈ F_{cube} such that ĝ ⊂ f̂ and range(ĝ) ∈ E. Then g = f ↾ h[dom(ĝ)] satisfies (2.2).

One of the most convenient tools for proving F_{cube}-density is the following fact.

2This version of the axiom, as well as its prism version CPA_{prism}, can be also formulated replacing the inequalities “≤ ω_{1}” with “< 𝔠” and removing the condition “𝔠 = ω_{2}.” Such a version of CPA_{cube} implies 𝔠 ≥ ω_{2}. Also all consequences of the axioms CPA_{cube} and CPA_{prism} presented in this paper follow also from the modified versions of these axioms.

However, we do not know if the modified axioms are consistent with 𝔠 > ω_{2}. We know only that the modified CPA_{prism} implies that 𝔠 is a successor cardinal. (See [7] or [8].)

Claim 2.3. Consider ℭ^{ω} with its usual topology and its usual product measure. If G ⊂ ℭ^{ω} is either comeager or of full measure in ℭ^{ω} then it contains a perfect cube ∏_{i<ω} P_{i}.

Proof. It follows easily, by induction on coordinates, from the following well known fact.

For every comeager (full measure) subset H of ℭ × ℭ there are a 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 [20, Exercise 19.3]. (Its version for ℝ^{2} is also proved, for example, in [9, condition (⋆), p. 416].) The measure version follows easily from the fact that

for every full measure subset H of [0, 1] × [0, 1] there are a perfect set P ⊂ ℭ and a positive inner measure subset Ĥ of [0, 1] such that P × Ĥ ⊂ H

which is proved by Eggleston [13] and, independently, by Brodskiǐ [4].

Next we will proceed to demonstrate some consequences of CPA_{cube}. The most important combinatorial fact for us concerning semiselective ideals is the following property. (See Theorem 2.1 and Remark 4.1 in [14].) This is a generalization of a theorem of Laver [22] who proved this fact for the ideal I = [ω]^{<ω}.

Proposition 2.4 (Farah [14]). 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 2.5. 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. [18, 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.  (2.4)

(If U ⊂ ℭ × [ω]^{ω} is a universal analytic set then S = ℭ × U ⊂ ℭ × ℭ × [ω]^{ω} satisfies (2.4), where we identify ℭ × ℭ with ℭ.) 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 from S.  (2.5)

Note that, by Proposition 2.4, the 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.

Clearly |W| ≤ ω_{1}. Also, by (2.4), for an analytic set A ⊂ [ω]^{ω} there exist a Q ∈ E_{0} and an x ∈ Q such that A = S_{x}. So, by (2.5), {x} × [W_{Q}]^{ω} is either contained in or disjoint from {x} × S_{x} = {x} × A.

Recall (see e.g. [1] or [26]) 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 the minimum cardinality of a reaping family. 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 ∈ W such that for every n < ω either W ⊆^{∗} A_{n} or W ⊆^{∗} ω \ A_{n}. (See [5] or [26].) Clearly 𝔯 ≤ 𝔯_{σ}.

Corollary 2.6. 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. The family W from Theorem 2.5 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} = ∅.

Corollary 2.7. 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 2.6. Then W generates F.

Indeed, if A ∈ F then there exists a W ∈ W such 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 4.8 we will prove that some version of our axiom implies that there exists a selective ultrafilter on ω. In particular, the assumptions of the next corollary are implied by such a version of our axiom.

Corollary 2.8. If CPA_{cube} holds and there exists a selective ultrafilter F on ω then 𝔯_{σ} = ω_{1} < 𝔠.

Proof. Let W ∈ [F]^{≤ω_{1}} 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}^{∗} belong 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 ∈ W be such that W ⊂ A. Then for every n < ω either W ⊆^{∗} A_{n} or W ⊆^{∗} ω \ A_{n}.

We are particularly interested in the number 𝔯_{σ} since it is related to different variants of sets of uniqueness coming from harmonic analysis, as described in the survey paper [5]. In particular, from [5, Theorem 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}.

3. CPA_{cube}^{game} and numbers 𝔞 and 𝔯

Before we get to the formulation of our next version of the axiom it is good to note that in many applications we would prefer to have a full covering of a Polish space X rather that the almost covering as claimed by CPA_{cube}. To get better access to the missing singletons we will extend the notion of a cube by allowing also the constant cubes: a family C_{cube}(X) of constant “cubes” is defined as the family of all constant functions from a perfect cube C ⊂ ℭ^{ω} into X. We define also F_{cube}^{∗}(X) as

F_{cube}^{∗} = F_{cube} ∪ C_{cube}.  (3.1)

Thus, F_{cube}^{∗} is the family of all continuous functions from a perfect cube C ⊂ ℭ^{ω} 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 P is a singleton. The terms “P ∈ Perf^{∗}(X) is a cube” and “Q is a subcube of a cube P ∈ Perf^{∗}(X)” are defined in a natural way.

Consider also the following game GAME_{cube}(X) of length ω_{1}. The game has two players, Player I and Player II. At each stage ξ < ω_{1} of the game Player I can play an arbitrary cube P_{ξ} ∈ Perf^{∗}(X) and Player II must respond with a subcube Q_{ξ} of P_{ξ}. The game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ is won by Player I provided

⋃_{ξ<ω_{1}} Q_{ξ} = 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(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) is a subcube of P_{ξ}, where ⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩ is any partial game. A game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ is played according to a strategy S for Player II provided Q_{ξ} = S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) for every ξ < ω_{1}. A strategy S for Player II is a winning strategy for Player II provided Player II wins any game played according to the strategy S.

Here is our new version of the axiom.^{3}

CPA_{cube}^{game}: 𝔠 = ω_{2} and for any Polish space X Player II has no winning

strategy in the game GAME_{cube}(X). Notice that

Proposition 3.1. Axiom CPA_{cube}^{game} implies CPA_{cube}.

Proof. Let E ⊂ Perf(X) be F_{cube}-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 S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = s(P_{ξ}).

By CPA_{cube}^{game} it is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ in which Q_{ξ} = s(P_{ξ}) for every ξ < ω_{1} and Player II loses, that is, X = ⋃_{ξ<ω_{1}} Q_{ξ}. Now, let E_{0} = {Q_{ξ}: ξ < ω_{1} & Q_{ξ} ∈ Perf(X)}. Then |X \ ⋃ E_{0}| ≤ ω_{1}, so CPA_{cube} is justified.

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

𝔞 = min{|A|: A is infinite and MAD}.

The fact that 𝔞 = ω_{1} holds in the iterated perfect set model was apparently first noticed by Spinas (see Andreas Blass [3, Section 11.5]) though it seems that the proof of this result was never provided.

Theorem 3.2. CPA_{cube}^{game} implies that 𝔞 = ω_{1}.

Our proof of Theorem 3.2 is based on the following lemma.

3Note that if we remove the assumption 𝔠 = ω_{2} from the axiom then the remaining part follows from the continuum hypothesis. Thus, for most of our applications the assumption that 𝔠 ≥ ω_{2} is essential. On the other hand, the main body of the axiom and 𝔠 ≥ ω_{2} imply that 𝔠 = ω_{2}.

Lemma 3.3. For every countably infinite almost disjoint family W ⊂ [ω]^{ω} and a cube P ∈ Perf([ω]^{ω}) there exist a W ∈ [ω]^{ω} and a subcube Q of P such that W ∪ {W } is almost disjoint but W ∪ {W, x} is not almost disjoint for every x ∈ Q.

Proof. Let W = {W_{i}: i < ω}. For every i < ω choose sets V_{i} ⊂ W_{i} such that: the V_{i}’s are pairwise disjoint, each W_{i}\V_{i} is finite, but V_{ω} = ω\⋃_{i<ω} V_{i} is infinite. Let

B = {x ∈ P: (∀i ≤ ω) |x ∩ V_{i}| < ω}

and notice that B is a Borel subset of P. (In fact, B is an F_{σδ}-set.) So, by Claim 2.3, there is a subcube P^{∗} of P such that either P^{∗} ⊂ B or P^{∗}∩B = ∅.

If P^{∗} ∩ B = ∅ then W = V_{ω} and Q = P^{∗} satisfy the conclusion of the lemma. So, suppose that P^{∗} ⊂ B. Let h: ℭ^{ω} → P^{∗}, h ∈ F_{cube}, be a coordinate function making P^{∗} a cube, let λ be the standard product probability measure on ℭ^{ω}, and define a Borel measure μ on P^{∗} by the formula μ(B) = λ(h^{−1}(B)).

For i, n < ω let

P_{i}^{n} = {x ∈ P^{∗}: x ∩ V_{i} ⊂ n}.

Then all the sets P_{i}^{n} are Borel (in fact, they are closed) and P^{∗} = ⋃_{n<ω} P_{i}^{n} for every i < ω. Thus for each i < ω there exists an n(i) < ω such that

μ (P_{i}^{n(i)}) > 1 − 2^{−i}.

Then the set T = ⋃_{j<ω} ⋂_{j<i<ω} P_{i}^{n(i)} has a μ-measure 1 so, by Claim 2.3, there is a subcube Q of P^{∗} which is a subset of T. Let

W = ⋃_{i<ω} [V_{i} ∩ n(i)] .

We claim that W and Q satisfy the lemma.

It is obvious that W is almost disjoint with each W_{i}. 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_{i}^{n(i)}. Thus, there exists a j < ω such that x ∈

⋂_{j<i<ω} P_{i}^{n(i)}. So, x ∩ ⋃_{j<i<ω} V_{i} = ⋃_{j<i<ω}(x ∩ V_{i}) ⊂ ⋃_{j<i<ω}(V_{i} ∩ n(i)) ⊂ W and the set

x \ W ⊂ x ∩ (V_{ω} ∪ ⋃_{i≤j} V_{i}) = (x ∩ V_{ω}) ∪ ⋃_{i≤j}(x ∩ V_{i})

is finite.

Proof of Theorem 3.2. For a countably infinite almost disjoint family W ⊂ [ω]^{ω} and a cube P ∈ Perf([ω]^{ω}) let W (W, P) ∈ [ω]^{ω} and a subcube Q(W, P) of P be as in Lemma 3.3. For P = {x} ∈ Perf^{∗}([ω]^{ω}) we put Q(W, P) = P and define W (W, P) as some arbitrary W almost disjoint with each set from W and such that A∩x is infinite for some A ∈ W ∪{W }.

Let A_{0} ⊂ [ω]^{ω} be an arbitrary infinite almost disjoint family and consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q(A_{0} ∪ {W_{η}: η < ξ}, P_{ξ}),

where W_{η}’s are defined inductively by W_{η} = W (A_{0} ∪ {W_{ζ}: ζ < η}, P_{η}). In other words, Player II remembers (recovers) the sets W_{η} associated with the sets P_{η} played so far, and he uses them (and Lemma 3.3) to get the next answer Q_{ξ} = Q(A_{0} ∪ {W_{η}: η < ξ}, P_{ξ}), while remembering (or recovering each time) the set W_{ξ} = W (A_{0} ∪ {W_{η}: η < ξ}, P_{ξ}).

By CPA_{cube}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which Player II loses, that is, [ω]^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}.

Now, notice that the family A = A_{0} ∪ {W_{ξ}: ξ < ω_{1}} is a MAD family. It is clear that A is almost disjoint, since every set W_{ξ} was chosen as almost disjoint with every set from A_{0} ∪ {W_{ζ}: ζ < ξ}. To see that A is maximal it is enough to note that every x ∈ [ω]^{ω} belongs to a Q_{ξ} for some ξ < ω_{1}, and so there is an A ∈ A_{0} ∪ {W_{η}: η ≤ ξ} such that A ∩ x is infinite.

By Theorem 3.2 we see that CPA_{cube}^{game} implies the existence of MAD family of size ω_{1}. Next we will show that such a family can be simultaneously a reaping family. This result is similar in flavor to that from Theorem 4.20.

Theorem 3.4. CPA_{cube}^{game} implies that there exists a family F ⊂ [ω]^{ω} of cardinality ω_{1} which is simultaneously MAD and reaping.

Proof. The proof is just a slight modification of that for Theorem 3.2.

For a countably infinite almost disjoint family W ⊂ [ω]^{ω} and a cube P ∈ Perf([ω]^{ω}) let W_{0} ∈ [ω]^{ω} and a subcube Q_{0} of P be as in Lemma 3.3. Let A ∈ [ω]^{ω} be almost disjoint with every set from W ∪ {W_{0}}. By Laver’s theorem [22] we can also find a subcube Q_{1} of Q_{0} and a W_{1} ∈ [A]^{ω} such that

• either W_{1} ∩ x = ∅ for every x ∈ Q_{1},

• or else W_{1} ⊂ x for every x ∈ Q_{1}. Let Q(W, P) = Q_{1} and W(W, P) = {W_{0}, W_{1}}. If P ∈ Perf^{∗}([ω]^{ω}) is a singleton then we put Q(W, P) = P and we can easy find W_{0} and W_{1} satisfying the above conditions.

Let A_{0} ⊂ [ω]^{ω} be an arbitrary infinite almost disjoint family and consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q (A_{0} ∪ ⋃{W_{η}: η < ξ}, P_{ξ}) ,

where W_{η}’s are defined inductively by W_{η} = W(A_{0} ∪ ⋃{W_{η}: η < ξ}, P_{η}).

By CPA_{cube}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which and Player II loses, that is, [ω]^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}. Then the family F = A_{0} ∪ ⋃{W_{ξ}: ξ < ω_{1}} is MAD and reaping.

4. On CPA_{prism}^{game}, selective and crowded ultrafilters, nonselective P -points, and numbers 𝔦 and 𝔲

The axioms CPA_{cube}^{game} and CPA_{cube} dealt with the notion of F_{cube}-density, where F_{cube} is the family of all injections f: C → X with C being a perfect cube in ℭ^{ω}. In the applications of these axioms we were using the facts that different subfamilies of Perf(X) were F_{cube}-dense. Unfortunately, in many cases the notion of F_{cube}-density is too weak to do the job — in the applications that follow the families E ⊂ Perf(X) will not be F_{cube}-dense, but they will be dense in a weaker sense defined below. Luckily, this weaker notion of density still leads to consistent axioms.

4.1. Prisms and CPA_{prism}^{game}.

To define this weaker notion of density, let us first take another look at the notion of cube. Let A be a non-empty countable set of ordinal numbers. The notion of a perfect cube in ℭ^{A} can be defined the same way as it was done for ℭ^{ω}. However, it will be more convenient for us to define it as follows. Let Φ_{cube} be the family of all continuous injection f: ℭ^{A} → ℭ^{A} such that

f(x)(α) = f(y)(α) for all α ∈ A and x, y ∈ ℭ^{A} with x(α) = y(α).

In other words Φ_{cube} is the family of all functions of the form f = ⟨f_{α}⟩_{α∈A}, where each f_{α} is an injection from ℭ into ℭ. Then the family of all perfect cubes in ℭ^{A} for an appropriate A is equal to

CUBE = {range(f): f ∈ Φ_{cube}}

and F_{cube} is the family all continuous injections f: C → X with C ⊂ ℭ^{ω} and C ∈ CUBE.

In the definitions that follow the notion of “cube” will be replaced by that of a “prism.” So, let Φ_{prism}(A) be the family of all continuous injection f: ℭ^{A} → ℭ^{A} with the property that

f(x) ↾ α = f(y) ↾ α ⇔ x ↾ α = y ↾ α for all α ∈ A and x, y ∈ ℭ^{A} (4.1)

or, equivalently, such that for every α ∈ A

f ↾↾ α =^{def} {⟨x ↾ α, y ↾ α⟩: ⟨x, y⟩ ∈ f}

is a one-to-one function from ℭ^{A∩α} into ℭ^{A∩α}. For example, if A = {0, 1, 2} then f ∈ Φ_{prism}(A) provided there exist continuous functions f_{0} : ℭ → ℭ, f_{1} : ℭ^{2} → ℭ, and f_{2} : ℭ^{3} → ℭ such that

f(x_{0}, x_{1}, x_{2}) = ⟨f_{0}(x_{0}), f_{1}(x_{0}, x_{1}), f_{2}(x_{0}, x_{1}, x_{2})⟩

for all x_{0}, x_{1}, x_{2} ∈ ℭ and maps f_{0}, ⟨f_{0}, f_{1}⟩, and f are one-to-one. Functions f from Φ_{prism}(A) were first introduced, in more general setting, in [19] where they are called projection-keeping homeomorphisms. Note that

Φ_{prism}(A) is closed under compositions  (4.2)

and that for every ordinal number α > 0

if f ∈ Φ_{prism}(A) then f ↾↾ α ∈ Φ_{prism}(A ∩ α).  (4.3)

Let

ℙ_{A} = {range(f): f ∈ Φ_{prism}(A)}.

We will write Φ_{prism} for ⋃_{0<α<ω_{1}} Φ_{prism}(α) and define

ℙ_{ω_{1}} =^{def} ⋃_{0<α<ω_{1}} ℙ_{α} = {range(f): f ∈ Φ_{prism}}.

Following [19] we will refer to elements of ℙ_{ω_{1}} as iterated perfect sets. Also let F_{prism}(X) (or just F_{prism}, if X is clear from the context) be the family of all continuous injections f: P → X where P ∈ ℙ_{ω_{1}} and X is a fixed Polish space.

We say that a family E ⊂ Perf(X) is F_{prism}-dense provided

∀f ∈ F_{prism} ∃g ∈ F_{prism} (g ⊂ f & range(g) ∈ E).

Similarly as in Fact 2.2, using (4.2) we can also prove that

Fact 4.1. E ⊂ Perf(X) is F_{prism}-dense if and only if

∀α < ω_{1} ∀f ∈ F_{prism}, dom(f) = ℭ^{α} ∃g ∈ F_{prism} (g ⊂ f & range(g) ∈ E) (4.4)

Notice also that Φ_{cube} ⊂ Φ_{prism}, so every cube is also a prism. From this and Fact 4.1 it also easy to see that

if E ⊂ Perf(X) is F_{cube}-dense then E is also F_{prism}-dense. (4.5)

The converse of (4.5), however, is false. (See [8, Remark 3.3.2].)

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) witness function f ∈ F_{prism} onto P. By Fact 4.1 to establish F_{prism}-density we can always assume that the witness function f is in a standard form, that is, defined on the entire set ℭ^{α} for an appropriate α < ω_{1}. Then Q is a subprism of a prism P ∈ Perf(X) provided Q = f[E], where f ∈ F_{prism} is as above and E ∈ ℙ_{α}. Also singletons {x} in X will be identified with constant functions from E ∈ ℙ_{ω_{1}} to {x}, and these functions will be considered as elements of F_{prism}^{∗}, similarly as in (3.1).

Now we are ready to state the next version of our axiom, in which the game GAME_{prism}(X) is an obvious generalization of GAME_{cube}(X).

CPA_{prism}^{game}: 𝔠 = ω_{2} and for any Polish space X Player II has no winning

strategy in the game GAME_{prism}(X). Notice that if a prism P ∈ Perf(X) is considered with a witness function

f ∈ F_{prism} in a standard form (i.e., f is from ℭ^{α} onto P) then P is also a cube and any subcube of P is also a subprism of P. Thus, any Player II strategy in a game GAME_{cube}(X) can be translated to a strategy in a game GAME_{prism}(X). (You need to identify appropriately ℭ^{α} with ℭ^{ω}: first you identify ℭ^{α} with ℭ^{ω} × ℭ^{α\{0}}, which is important for a finite α, and then this second space identify with ℭ^{ω} coordinatewise.) In particular, CPA_{prism}^{game} implies CPA_{cube}^{game}. Also, essentially the same argument as used for Proposition 3.1 gives also the following.

Proposition 4.2. Axiom CPA_{prism}^{game} implies the following prism version of the axiom CPA_{cube}:

CPA_{prism}: 𝔠 = ω_{2} and for every Polish space X and every F_{prism}-dense family E ⊂ Perf(X) there exists an E_{0} ⊂ E such that |E_{0}| ≤ ω_{1} and |X \ ⋃ E_{0}| ≤ ω_{1}.

By (4.5) it is also obvious that CPA_{prism} implies CPA_{cube}. All these implications are summarized in Chart 1.

In what follows for a fixed 0 < α < ω_{1} and 0 < β ≤ α the symbol π_{β} will stand for the projection from ℭ^{α} onto ℭ^{β}. We will always consider ℭ^{α} with the following standard metric ρ: fix an enumeration {⟨β_{k}, n_{k}⟩: k < ω} of α × ω and for distinct x, y ∈ ℭ^{α} define

ρ(x, y) = 2^{− min{k<ω: x(β_{k})(n_{k})≠y(β_{k})(n_{k})}}.  (4.6)

The open ball in ℭ^{α} with a center at z ∈ ℭ^{α} and radius ε > 0 will be denoted by B_{α}(z, ε). Notice that in this metric any two open balls are either disjoint or one is a subset of another. Also for every γ < α and ε > 0

π_{γ}[B_{α}(s, ε)] = π_{γ}[B_{α}(t, ε)] for every s, t ∈ ℭ^{α} with s ↾ γ = t ↾ γ. (4.7)

It is also easy to see that any B_{α}(z, ε) is a clopen set and, in fact, it is a perfect cube in ℭ^{α}, so it belongs to ℙ_{α}. In fact, more can be said:

if B_{α} =^{def} {B ⊂ ℭ^{α}: B is clopen in ℭ^{α}} then B_{α} ⊂ ℙ_{α}.  (4.8)

This is the case, since any clopen E in ℭ^{α} is a finite union of disjoint open balls, each of which belongs to ℙ_{α}, and it is easy to see that ℙ_{α} is closed under finite unions of disjoint sets.

From this we conclude immediately that

a clopen subset of E ∈ ℙ_{α} belongs to ℙ_{α}  (4.9)

and

a clopen subset of a prism is its subprism.  (4.10)

Notice also that if P ∈ ℙ_{α} then

P ∩ π_{β}^{−1}(P′) ∈ ℙ_{α} for every P′ ∈ ℙ_{β} with P′ ⊂ π_{β}[P]. (4.11)

Indeed, let f ∈ Φ_{prism}(β) and g ∈ Φ_{prism}(α) be such that f[ℭ^{β}] = P′ and g[ℭ^{α}] = P. Let Q = (g ↾ β)^{−1}[P′] = (g ↾ β)^{−1} ∘ f[ℭ^{β}] ∈ ℙ_{β}. Then π_{β}^{−1}(Q) belongs to ℙ_{α} and P ∩ π_{β}^{−1}(P′) = g[π_{β}^{−1}(Q)] ∈ ℙ_{α}.

4.2. Fusion Lemmas.

One of the main technical tools used to prove that a family of perfect sets is dense is the so called fusion lemma. It says that for an appropriately chosen decreasing sequence {P_{n}: n < ω} of perfect sets its intersection P = ⋂_{n<ω} P_{n}, called the fusion, is still a perfect set. The simple structure of perfect cubes makes it quite easy to formulate a “cube fusion lemma” in which the fusion set P is also a cube. However, so far we did not have any need for such a lemma (at least in an explicit form), since its use was always hidden in the proofs of the results we quoted, like Claim 2.3 or Proposition 2.4. On the other hand, the new and more complicated structure of prisms does not leave us the option of avoiding fusion arguments any longer — we have to face it up front.

For a fixed 0 < α < ω_{1} let {⟨β_{k}, n_{k}⟩: k < ω} be an enumeration of α × ω used in the definition (4.6) of the metric ρ and let

A_{k} = {⟨β_{i}, n_{i}⟩: i < k} for every k < ω.  (4.12)

Lemma 4.3 (Fusion Sequence). Let 0 < α < ω_{1} and for every k < ω let E_{k} = {E_{s} ∈ ℙ_{α}: s ∈ 2^{A_{k}}}. Assume that for every k < ω, s, t ∈ 2^{A_{k}}, r ∈ ⋃_{i<ω} 2^{A_{i}}, and β < α we have:

(i) the diameter of E_{s} is less than or equal to 2^{−k}, (ii) if r ⊂ s then E_{s} ⊂ E_{r}, (ag) (agreement) if s ↾ (β × ω) = t ↾ (β × ω) then π_{β}[E_{s}] = π_{β}[E_{t}], (sp) (split) if s ↾ (β × ω) ≠ t ↾ (β × ω) then π_{β}[E_{s}] ∩ π_{β}[E_{t}] = ∅. Then Q = ⋂_{k<ω} ⋃ E_{k} belongs to ℙ_{α}.

Proof. For x ∈ ℭ^{α} let x̄ ∈ 2^{α×ω} be defined by x̄(β, n) = x(β)(n).

First note that, by conditions (i) and (sp), for every k < ω the sets in E_{k} are pairwise disjoint and each of the diameter at most 2^{−k}. Thus, taking into account (ii), the function h: ℭ^{α} → ℭ^{α} defined by

h(x) = r ⟺ {r} = ⋂_{k<ω} E_{x̄↾A_{k}}

is well defined and is one-to-one. It is also easy to see that h is continuous and that Q = h [ℭ^{α}]. Thus, we need to prove only that h ∈ Φ_{prism}(α), that is, that h is projection-keeping.

To show this fix β < α, put S = ⋃_{i<ω} 2^{A_{i}}, and notice that, by (i) and (ag), for every x ∈ ℭ^{α} we have

{h(x) ↾ β} = π_{β} [⋂{E_{x̄↾A_{k}}: k < ω}]
= ⋂{π_{β}[E_{x̄↾A_{k}}]: k < ω}
= ⋂{π_{β}[E_{s}]: s ∈ S & s ⊂ x̄}
= ⋂{π_{β}[E_{s}]: s ∈ S & s ↾ (β × ω) ⊂ x̄}.

Now, if x ↾ β = y ↾ β then for every s ∈ S

s ↾ (β × ω) ⊂ x̄ ⇔ s ↾ (β × ω) ⊂ ȳ

so h(x) ↾ β = h(y) ↾ β.

On the other hand, if x ↾ β ≠ y ↾ β then there exists a k < ω big enough such that for s = x̄ ↾ A_{k} and t = ȳ ↾ A_{k} we have s ↾ (β × ω) ≠ t ↾ (β × ω). But then {h(x) ↾ β} and {h(y) ↾ β} are subsets of π_{β}[E_{s}] and π_{β}[E_{t}], respectively, which, by (sp), are disjoint. So, h(x) ↾ β ≠ h(y) ↾ β.

In all of our applications the task of constructing sequences ⟨E_{k}: k < ω⟩ satisfying specific conditions (ag) and (sp) can be reduced to checking some simple density properties listed in our next lemma. In its statement we consider ℙ_{α} as ordered by inclusion and use the standard terminology from the theory of partially ordered sets: D ⊂ ℙ_{α} is dense provided for every E ∈ ℙ_{α} there is an E′ ∈ D with E′ ⊂ E; it is open provided for every E ∈ D if E′ ∈ ℙ_{α} and E′ ⊂ E then E′ ∈ D. Moreover, for a family E of pairwise disjoint subsets of ℙ_{α} we say that E′ ⊂ ℙ_{α} is a refinement of E provided E′ = {P_{E}: E ∈ E} where P_{E} ⊂ E for all E ∈ E.

Lemma 4.4. Let 0 < α < ω_{1} and k < ω. If E_{k} = {E_{s} ∈ ℙ_{α}: s ∈ 2^{A_{k}}} satisfies (ag) and (sp) then (A) there exists an E_{k+1} = {E_{s} ∈ ℙ_{α}: s ∈ 2^{A_{k+1}}} such that (i), (ii), (ag),

and (sp) hold for all s, t ∈ 2^{A_{k+1}} and r ∈ 2^{A_{k}}. Moreover, if D ⊂ [ℙ_{α}]^{<ω} is a family of pairwise disjoint sets such that ∅ ∈ D, D is closed under refinements, and (†) for every E ∈ D and E ∈ ℙ_{α} which is disjoint with ⋃ E there exists an

E′ ∈ ℙ_{α} ∩ P(E) such that {E′} ∪ E ∈ D then (B) there exists a refinement E′_{k} ∈ D of E_{k} satisfying (ag) and (sp), (C) there exists an E_{k+1} as in (A) such that E_{k+1} ∈ D.

The proof of this lemma will be postponed to the last section of this paper. One of the most important consequences of Lemma 4.4 is the following.

Corollary 4.5. Let 0 < α < ω_{1} and let {D_{k}: k < ω} be a collection of dense open subsets of ℙ_{α}. If for every k < ω

D_{k}^{∗} = {⋃ D: D ∈ [D_{k}]^{<ω} and the sets in D are pairwise disjoint}

then D̄ = ⋂_{k<ω} D_{k}^{∗} is open and dense in ℙ_{α}.

Proof. It is clear that D̄ is open. To see its density notice that the families

D_{k} = {D ∈ [D_{k}]^{<ω} and sets in D are pairwise disjoint}

satisfy condition (†). Let E ∈ ℙ_{α}, choose an E_{∅} ∈ D_{0} ⊂ D_{0}^{∗} below E, and put E_{0} = {E_{∅}}. Applying (C) from Lemma 4.4 by induction we can define families E_{k} ∈ D_{k}, k < ω, such that conditions (i), (ii), (ag), and (sp) from Lemma 4.3 are satisfied. But then Q = ⋂_{k<ω} ⋃ E_{k} ⊂ E belongs to D̄.

4.3. Selective ultrafilters and number 𝔲.

Recall that every weakly selective ultrafilter is selective and that the ideal I = [ω]^{<ω} is selective. Another example of a weakly selective ideal which we will use in what follows is given below.

Fact 4.6. The ideal I of nowhere dense subset of rationals ℚ is weakly selective.

Proof. Let A ∈ I^{+} and take an f: A → ω. If there is a B ∈ I^{+} ∩ P(A) such that f ↾ B constant then we are done. So, assume that it is not the case and let A_{0} ⊂ A be dense on some interval. By induction on n < ω define a sequence {b_{n} ∈ A_{0} : n < ω} dense in A_{0} such that f restricted to B = {b_{n}: n < ω} is one-to-one. Then B is as desired.

In what follows we will also need the following fact about weakly selective ideals, which can be found in Grigorieff [15, Proposition 14].

Proposition 4.7. Let I be a weakly selective ideal on ω and A ∈ I^{+}. If T ⊂ A^{<ω} is a tree such that

A \ {j < ω: sˆ j ∈ T } ∈ I for every s ∈ T

then there exists a branch b of T such that b[ω] ∈ I^{+}.

Theorem 4.8. CPA_{prism}^{game} implies that for every selective ideal I on ω there exists a selective ultrafilter F on ω such that F ⊂ I^{+}.

In particular if CPA_{prism}^{game} holds then there is a selective ultrafilter on ω.

The proof is based on the following lemma.

Lemma 4.9. Let I be a weakly selective ideal on ω.

(a) For every A ∈ I^{+} and every prism P in ω^{ω} there exist a B ∈ I^{+}, B ⊂ A, and a subprism Q of P such that either

(i) g ↾ B is one-to-one for every g ∈ Q, or else (ii) there exists an n < ω such that g ↾ B is constant equal to n for

every g ∈ Q.

(b) For every A ∈ I^{+} and every prism P in [ω]^{ω} there exist a B ∈ I^{+}, B ⊂ A, and a 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 ∈ F_{prism}(ω^{ω}) from ℭ^{α} onto P, and assume that for no subprism Q of P and B ∈ I^{+} ∩ P(A) condition (ii) holds. We will find Q satisfying (i).

For i, n < ω let D(i, n) = {E_{0} ∈ ℙ_{α}: (∀g ∈ E_{0}) f(g)(i) ≠ n} and for γ ≤ α, E ∈ ℙ_{α}, and A′ ⊂ ω put D_{γ}(E, i, n) = {π_{γ}[E_{0}]: E_{0} ∈ D(i, n)∩P(E)} and

D_{γ}(E, A′, n) = ⋂_{i∈A′} D_{γ}(E, i, n).

Notice that the sets D_{γ}(E, i, n) and D_{γ}(E, A′, n) are open in ℙ_{γ}. By induction on 0 < β ≤ α we are going to prove the following property. ψ_{β}: For all 0 < γ ≤ β, E ∈ ℙ_{α}, n < ω, and Â ∈ I^{+} ∩ P(A) there exists an

A′ ∈ I^{+} ∩ P(Â) such that D_{γ}(E, A′, n) ≠ ∅. In what follows for k < ω and E_{k} = {E_{s} ∈ ℙ_{β}: s ∈ 2^{A_{k}}} satisfying (i), (ag), and (sp) from Lemma 4.3 and E_{k+1} = {E_{s} ∈ ℙ_{β}: s ∈ 2^{A_{k+1}}} we will write

E_{k+1} ≺ E_{k}

provided (i), (ii), (ag), and (sp) hold for all s, t ∈ 2^{A_{k+1}} and r ∈ 2^{A_{k}}. One of the main facts used in the proof of ψ_{β} is the following property. (∗) If ψ_{γ} holds for all γ < β, Ā ∈ I^{+} ∩ P(A), n < ω, E ∈ ℙ_{α}, E_{k} is as

above and such that ⋃ E_{k} ⊂ π_{β}[E], and

Z(Ā, E_{k}, n) = {i ∈ Ā: (∃E_{k+1} ≺ E_{k}) ⋃ E_{k+1} ∈ D_{β}(E, i, n)}

then Ā \ Z(Ā, E_{k}, n) ∈ I. In order to prove (∗) fix an Â ∈ P(Ā) ∩ I^{+} and note that it is enough

to show that Â ∩ Z(Ā, E_{k}, n) ≠ ∅. Fix an Ē_{k+1} = {Ē_{s} ∈ ℙ_{β}: s ∈ 2^{A_{k+1}}} such that Ē_{k+1} ≺ E_{k}. We can find such an Ē_{k+1} by Lemma 4.4(A). Let γ = max{δ: ⟨δ, m⟩ ∈ A_{k+1}} < β.

First assume that γ = 0. Then for every s ∈ 2^{A_{k+1}} the set

Z_{s} = {i ∈ Â: D_{β}(E, i, n) ∩ P(Ē_{s}) = ∅}

belongs to I, since otherwise Q = f [π_{β}^{−1}[Ē_{s}] ∩ E] and B = Z_{s} ∈ I^{+} ∩P(A) would satisfy the condition (ii), contradicting our assumption. Let us define A′ = Â \ ⋃{Z_{s}: s ∈ 2^{A_{k+1}}} ∈ I^{+} and notice that A′ ⊂ Z(Ā, E_{k}, n). Indeed, take an i ∈ A′ and for every s ∈ 2^{A_{k+1}} choose E_{s} ∈ D_{β}(E, i, n) ∩ P(Ē_{s}).

Then E_{k+1} =^{def} {E_{s}: s ∈ 2^{A_{k+1}}} ≺ E_{k}, since (i), (ii), and (sp) hold for E_{k+1} as they were true for Ē_{k+1}, and (ag) is satisfied trivially, by the maximality of γ. Condition ⋃ E_{k+1} ∈ D_{β}(E, i, n) is guaranteed by the choice of E_{s}’s, so indeed i ∈ Z(Ā, E_{k}, n).

Next assume that γ > 0. Let B = {⟨δ, m⟩ ∈ A_{k+1} : δ < γ}, and define E_{k+1}^{∗} = {E_{t}^{∗} ∈ ℙ_{γ}: t ∈ 2^{B}}, where E_{t}^{∗} = π_{γ}[Ē_{s}] for any s ∈ 2^{A_{k+1}} with t ⊂ s. Note that, by (ag), the definition of E_{t}^{∗} is independent of the choice of s. It is easy to see that E_{k+1}^{∗} satisfies (ag) and (sp), where α is replaced by γ. For Â_{0} ∈ P(Â) ∩ I^{+} and t ∈ 2^{B} define D(Â_{0}, t) as the collection of all E_{0} ∈ ℙ_{γ} for which there exists an A′ ∈ P(Â_{0}) ∩ I^{+} such that

E_{0} ∈ ⋂ {D_{γ} (π_{β}^{−1}[Ē_{s}] ∩ E, A′, n) : t ⊂ s ∈ 2^{A_{k+1}}} .

It is clear that each D(Â_{0}, t) is open, since so is each D_{γ} (π_{β}^{−1}[Ē_{s}] ∩ E, A′, n).

It is also important to notice that D(Â_{0}, t) is dense below E_{t}^{∗}. To see it, fix an E_{0} ∈ ℙ_{γ} ∩ P(E_{t}^{∗}) and let {s_{1}, …, s_{m}} be an enumeration of {s ∈ 2^{A_{k+1}} : t ⊂ s}. By induction on i ≤ m define two decreasing sequences {E_{i} ∈ ℙ_{γ}: i ≤ m} and {Â_{i} ∈ I^{+} : i ≤ m} such that E_{i} ∈ D_{γ} (π_{γ}^{−1}(E_{i−1}) ∩ (π_{β}^{−1}[Ē_{s_{i}}] ∩ E), Â_{i}, n) provided 0 < i ≤ m. The inductive step can be made since condition ψ_{γ} holds. Then

E_{m} ∈ ⋂{D_{γ}(π_{β}^{−1}[Ē_{s}] ∩ E, Â_{m}, n): t ⊂ s ∈ 2^{A_{k+1}}}

and we have E_{m} ∈ D(Â_{0}, t) ∩ P(E_{0}).

Let D be the collection of all pairwise disjoint families E ∈ [ℙ_{γ}]^{<ω} for which there exists an A′ ∈ P(Â)∩I^{+} working simultaneously for all E_{0} ∈ E, that is, such that for all t ∈ 2^{B} and E_{0} ∈ E if E_{0} ⊂ E_{t}^{∗} then

E_{0} ∈ ⋂ {D_{γ} (π_{β}^{−1}[Ē_{s}] ∩ E, A′, n) : t ⊂ s ∈ 2^{A_{k+1}}} .

Notice that D satisfies condition (†) from Lemma 4.4 used with α replaced by γ. Indeed, if E ∈ D is witnessed by A′ ∈ P(Â)∩I^{+} and E ∈ ℙ_{γ} is disjoint with ⋃ E choose E′ ∈ ℙ_{γ} below E which is either disjoint with ⋃ E_{k+1}^{∗} or contained in some E_{t}^{∗} ∈ E_{k+1}^{∗}. If E′ ∩ ⋃ E_{k+1}^{∗} = ∅ then {E′} ∪ E ∈ D is witnessed by A′. If E′ ⊂ E_{t}^{∗} ∈ E_{k+1}^{∗} by the density of D(A′, t) below E_{t}^{∗} we can find an A′′ ∈ P(A′) ∩ I^{+} and

E′′ ∈ P(E′) ∩ ⋂ {D_{γ} (π_{β}^{−1}[Ē_{s}] ∩ E, A′′, n) : t ⊂ s ∈ 2^{A_{k+1}}} .

Then {E′′} ∪ E ∈ D is witnessed by A′′.

Now, by Lemma 4.4(B), there exists an Ê_{k+1} = {Ê_{t} ∈ D: t ∈ 2^{B}} ∈ E

satisfying (ag) and (sp) such that Ê_{t} ⊂ E_{t}^{∗} for all t ∈ 2^{B}. Let A′ ∈ P(Â)∩I^{+} witness Ê_{k+1} ∈ E. We will show that A′ ⊂ Z(Ā, E_{k}, n). So fix an i ∈ A′. Since for every t ∈ 2^{B} and t ⊂ s ∈ 2^{A_{k+1}} we have Ê_{t} ∈ D_{γ}(π_{β}^{−1}[Ē_{s}]∩E, A′, n) there exists an Ê_{s} ∈ D(π_{β}^{−1}[Ē_{s}] ∩ E, i, n) with π_{γ}[Ê_{s}] = Ê_{t}. Let E_{s} =

π_{β}[Ê_{s}] ⊂ Ē_{s} and notice that E_{k+1} =^{def} {E_{s}: s ∈ 2^{A_{k+1}}} ≺ E_{k}. Indeed, E_{k+1} satisfies (i), (ii), and (sp) since they were true for Ē_{k+1} and E_{k+1} is a refinement of Ē_{k+1}. Condition (ag) is satisfied by E_{k+1} since, by the maximality of γ, it is non-trivial for β̂ ≤ γ and for such β̂ it guaranteed by (ag) for Ê_{k+1}. Finally, ⋃ E_{k+1} ∈ D_{β}(E, i, n) is guaranteed by our definition, so indeed i ∈ Z(Ā, E_{k}, n). This finishes the proof of (∗).

To prove ψ_{β} assume that ψ_{γ} holds for all γ < β. Fix E ∈ ℙ_{α}, n < ω, and Â ∈ I^{+}∩P(A). We need to find an A′ ∈ I^{+}∩P(Â) such that D_{β}(E, A′, n) ≠ ∅, that is, ⋂_{i∈A′} D_{β}(E, i, n) ≠ ∅. We will construct a tree T ⊂ Â^{<ω} and the mapping T ∋ s ↦ E_{s} ∈ [ℙ_{β}]^{<ω} such that E_{∅} = {E} and for every r ∈ T and s = rˆ i ∈ T we have E_{s} ≺ E_{r} and ⋃ E_{s} ∈ D_{β}(E, i, n). Notice that, by (∗), for every r ∈ T we can define E_{rˆ i} for all i ∈ Z(Â, E_{r}, n). So we can ensure that T satisfies the assumptions of Proposition 4.7. Let b be a branch of T with A′ = b[ω] ∈ I^{+}. By Lemma 4.3 E_{0} = ⋂_{k<ω} ⋃ E_{b↾k} belongs to ℙ_{β} and E_{0} ∈ ⋂_{i∈A′} D_{β}(E, i, n). This concludes the proof of ψ_{β}.

For the conclusion of the proof we first need to refine the prism P. For every i < ω let h_{i}: ℭ^{α} → ω ⊂ ℝ be defined by h_{i}(g) = f(g)(i). Clearly each h_{i} is continuous. Hence each set h_{i}^{−1}(n) is open in ℭ^{α}, so

D_{i} = {E ∈ ℙ_{α}: h_{i} is constant on E}

is dense and open in ℙ_{α}. Thus, by Corollary 4.5, there exists an E ∈ ⋂_{i<ω} D_{i}^{∗}, where

D_{i}^{∗} = {⋃ D: D ∈ [D_{i}]^{2^{i}} and the sets in D are pairwise disjoint} .

Let P_{0} = f[E]. Then P_{0} is a subprism of P. We will find a subprism Q of P_{0}. Notice also that, by our construction, for every i < ω there is a set V_{i} ∈ [ω]^{≤2^{i}} such that

f(g)(i) ∈ V_{i} for all g ∈ E and i < ω.

Notice also that since ψ_{α} holds, so is the conclusion of (∗) for β = α. In particular, for every n < ω and E_{k} = {E_{s} ∈ ℙ_{α}: s ∈ 2^{A_{k}}} satisfying (i), (ag), and (sp) from Lemma 4.3 and such that ⋃ E_{k} ⊂ E we have

Z(A, E_{k}, n) = {i ∈ A: (∃E_{k+1} ≺ E_{k}) ⋃ E_{k+1} ∈ D(i, n)}

and A \ Z(A, E_{k}, n) ∈ I.

We will construct a tree T ⊂ A^{<ω} as in Proposition 4.7 and the mapping T ∋ s ↦ E_{s} ∈ [ℙ_{α}]^{<ω}. The construction is done by induction on the levels of T. We start with E_{∅} = {E} and, for every r ∈ T and s = rˆ i ∈ T, we ensure that E_{s} ≺ E_{r} and

⋃ E_{s} ∈ ⋂ {D(i, n): n ∈ V_{j} for some j ∈ range(r)} .  (4.13)

Notice that for every r ∈ T if Z_{r} = ⋂ {Z(A, E_{r}, n): n ∈ ⋃_{j∈range(r)} V_{j}} then A \ Z_{r} ∈ I. Moreover, for all i ∈ Z_{r} we can find E_{rˆ i} as in (4.13). So, T as above can be constructed. Take a branch b of T with B = b[ω] ∈ I^{+} and E_{0} = ⋂_{k<ω} ⋃ E_{b↾k} ∈ ℙ_{β}. Then B and Q = f[E_{0}] satisfy (i). This finishes the proof of (a).

(b) Since the characteristic function χ gives an embedding from [ω]^{ω} into 2^{ω} ⊂ ω^{ω} the prism P can be identified with χ[P] = {χ_{x}: x ∈ P }. Applying part (a) to χ[P] we can find a subprism Q of P, k < 2, and B ∈ I^{+}, B ⊂ A, such that χ_{x} ↾ B ≡ k for every x ∈ Q. If k = 0 this gives x ∩ B = ∅ for every x ∈ Q. If k = 1 we have B ⊂ x for every x ∈ Q.

Proof of Theorem 4.8. Let I be a selective ideal on ω. For a countable family A ⊂ I^{+} linearly ordered by ⊂^{∗} let C(A) ∈ I^{+} be such that C(A) ⊂^{∗} A for every A ∈ A.

For A ∈ I^{+} and f ∈ F_{prism}(ω^{ω}) put P = range(f) and let B(A, P) ∈ [A]^{ω} and a subprism Q(A, P) of P be as in Lemma 4.9(a). If f ∈ C_{prism}(ω^{ω}) and P = range(f) = {x} then we put Q(A, P) = P and take B(A, P) ∈ [A]^{ω} satisfying the conclusion of Lemma 4.9(a).

Consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q(C({B_{η}: η < ξ}), P_{ξ}),

where the sets B_{η} are defined inductively by B_{η} = B(C({B_{ζ}: ζ < η}), P_{η}).

By CPA_{prism}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which and Player II loses, that is, ω^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}.

Now, let F be a filter generated by {B_{ξ}: ξ < ω_{1}} and notice that F is a selective ultrafilter. It is a filter, since {B_{ξ}: ξ < ω_{1}} is decreasing with respect to ⊂^{∗}. It also easy to see that

for every f ∈ ω^{ω} there exists a B ∈ F such that f ↾ B is either one-to-one or constant.

Indeed, if f ∈ ω^{ω} then there exists a ξ < ω_{1} such that f ∈ Q_{ξ}. Then B = B_{ξ} is as desired.

Now, to see that F is an ultrafilter take an A ⊂ ω and let f ∈ ω^{ω} be a characteristic function of A. Then B ∈ F as above is a subset of either A or its complement.

It is easy to see that the above two properties imply that F is a selective ultrafilter.

Notice that CPA_{prism}^{game} implies also that we have many different selective ultrafilters. The consistency of this fact, in a model obtained by adding many side-by-side Sacks reals, was first noticed by Hart in [16].

Remark 4.10. CPA_{prism}^{game} implies that there are ω_{2} different selective ultrafilters.

Proof. This can be easily deduced by a simple transfinite induction from (∗) for every family U = {F_{ξ}: ξ < ω_{1}} of ultrafilters on ω there is a

selective ultrafilter F ∉ U. Property (∗) is proved as above, where we use I = [ω]^{<ω} and the operator C({B_{η}: η < ξ}) is replaced with C_{ξ}({B_{η}: η < ξ}) ∉ F_{ξ}.

Now, from the above we obtain that “2^{ω_{1}} = ω_{2}”+CPA_{prism}^{game} (which is consistent) implies that there are 2^{ω_{1}} different selective ultrafilters. Since CPA is also consistent with 2^{ω_{1}} > ω_{2} it is worth noticing that the existence of 2^{ω_{1}} different selective ultrafilters can be also deduced from a slightly stronger version of CPA_{prism}^{game} also in this case. This can be found in [8].

Recall that the number 𝔲 is defined as the smallest cardinality of the base for a non-principal ultrafilter on ω. Thus Theorem 4.8 and Corollaries 2.7 and 2.8 imply that

Corollary 4.11. CPA_{prism}^{game} implies that 𝔲 = 𝔯_{σ} = ω_{1}.

4.4. Non-selective P -points and number 𝔦.

Recall that an ultrafilter F on ω is a P -point provided for every partition P of ω either P ∩ F ≠ ∅ or there is an F ∈ F such that |F ∩ P | < ω for all P ∈ P. Clearly every selective ultrafilter is a P -point. Thus, CPA_{prism}^{game} implies the existence of a P -point. On the other hand, Shelah proved that there are models with no P -points. (See e.g. [1, thm. 4.4.7].) Hart in [16] proved that in a model obtained by adding many side-by-side Sacks reals there is a P -point which is not selective. Next, we will prove that this follows also from CPA_{prism}^{game}. The main idea of the proof is the same as that used in [16].

For m < ω let P_{m} = {n < ω: 2^{m} − 1 ≤ n < 2^{m+1} − 1} and define a partition P of ω by P = {P_{m}: m < ω}. Consider the following ideal Ī on ω

Ī = {A ⊂ ω: lim sup_{m→∞} |A ∩ P_{m}| < ω}  (4.14)

and notice the following simple fact.

Fact 4.12. If A ∈ [Ī^{+}]^{≤ω} is linearly ordered by ⊂^{∗} then there is a C(A) ∈ Ī^{+} such that C(A) ⊂^{∗} A for all A ∈ A.

Proof. Let {A_{n}: n < ω} ⊂ A be a ⊂^{∗}-decreasing sequence coinitial with A. For every i < ω choose m_{i} < ω and C_{i} ∈ [P_{m_{i}}]^{i} such that C_{i} ⊂ ⋂_{j≤i} A_{j}. Then C(A) = ⋃_{i<ω} C_{i} is as desired.

To construct a nonselective P -point we are going to prove the following theorem.

Theorem 4.13. If CPA_{prism}^{game} holds then there exists a ⊂^{∗}-decreasing sequence B = {B_{ξ} ∈ Ī^{+} : ξ < ω_{1}} such that the filter F generated by B is an ultrafilter on ω.

Notice that from this we will immediately deduce the required result.

Corollary 4.14. If CPA_{prism}^{game} holds then there exists a nonselective P -point.

Proof. Let F be as in Theorem 4.13. Clearly F is nonselective, since P is disjoint with Ī^{+} ⊃ F and every selector of P is in Ī ⊂ P(ω) \ F. The fact that F is a P -point follows from the fact that F has a base linearly ordered by ⊂^{∗}. Indeed, if {S_{n}: n < ω} ⊂ P(ω) \ F is a partition of ω then for every m < ω there is ξ_{m} < ω_{1} such that B_{ξ_{m}} ⊂^{∗} ω \ ⋃_{n≤m} S_{n}. Let β < ω_{1} be such that B_{β} ⊂^{∗} B_{ξ_{m}} for all m < ω. Then F = B_{β} ∈ F is such that |F ∩ S_{n}| < ω for all n < ω.

The proof of Theorem 4.13 will be based on the following lemma, which is analogous to Lemma 4.9. Note, that although the statement of this lemma is identical to that of Lemma 4.9(b), we cannot apply this lemma here, since the ideal Ī is not weakly selective.

Lemma 4.15. Let Ī be as in (4.14). Then for every A ∈ Ī^{+} and a prism P in 2^{ω} there exist a B ∈ Ī^{+}, B ⊂ A, a subprism Q of P, and a j < 2 such that (∘) g ↾ B is constant equal to j for every g ∈ Q.

Proof. Fix an A ∈ Ī^{+} and an f ∈ F_{prism}(2^{ω}) from ℭ^{α} onto P. Since A∈Ī^{+}, for every k < ω we can find an m_{k} < ω such that |A∩P_{m_{k}}| ≥ k 2^{2^{k}}. First we will construct a subprism Q_{0} of P and a sequence ⟨A_{k} ∈ [A ∩ P_{m_{k}}]^{k}: k < ω⟩ such that for every k < ω

g ↾ A_{k} is constant for every g ∈ Q_{0}.  (4.15)

This will be done using Lemmas 4.4 and 4.3. So, for each k < ω let D_{k} be the collection of all pairwise disjoint families E ∈ [ℙ_{α}]^{<ω} such that for every E ∈ E

f(h) ↾ P_{m_{k}} = f(h′) ↾ P_{m_{k}} for all h, h′ ∈ E.  (4.16)

Clearly each D_{k} satisfies condition (†) from Lemma 4.4, so by an easy induction we can find a sequence ⟨E_{k} ∈ D_{k}: k < ω⟩ satisfying the assumptions of Lemma 4.3. Let E_{0} = ⋂_{k<ω} ⋃ E_{k} ∈ ℙ_{α}. We will show that Q_{0} = f[E_{0}] satisfies (4.15).

Indeed, fix a k < ω and notice that E_{k} = {E_{i}: i < 2^{k}}. For each i < 2^{k} choose an h_{i} ∈ E_{i} and define φ: A∩P_{m_{k}} → 2^{2^{k}} by φ(p)(i) = f(h_{i})(p). Since |A ∩ P_{m_{k}}| ≥ k 2^{2^{k}}, by the pigeon hole principle we can find an s ∈ 2^{2^{k}} such that |φ^{−1}(s)| ≥ k. Choose an A_{k} ∈ [φ^{−1}(s)]^{k}. Then for every i < 2^{k} and p ∈ A_{k} we have f(h_{i})(p) = φ(p)(i) = s(i). So, f(h_{i}) ↾ A_{k} is constant equal to s(i). Combining this with the inclusion A_{k} ⊂ P_{m_{k}} and the condition (4.16) we obtain (4.15).

To finish the proof fix a selector Ā from the family {A_{k}: k < ω}. Then Ā ∈ I^{+}, where I is the ideal of finite subsets of ω. Applying Lemma 4.9(a) to Ā and Q_{0} we can find a j < 2, an S ∈ [Ā]^{ω}, and a subprism Q of Q_{0} such that g ↾ S is constant equal to j for every g ∈ Q. Put B = ⋃_{k∈S} A_{k}. Then, by (4.15), g ↾ B is constant equal to j for every g ∈ Q.

It is clear that B ∈ Ī^{+} ∩ P(A), since it is a union of infinitely many sets A_{k} ∈ [A ∩ P_{m_{k}}]^{k}.

Note also that, similarly as for Remark 4.10, the conclusion of the following fact holds in a model obtained by adding many side-by-side Sacks reals. This was first noticed by Hart in [16].

Remark 4.16. CPA_{prism}^{game} implies that there are ω_{2} different nonselective P -points.

The existence of 2^{ω_{1}} different such ultrafilters follows also from a slightly stronger version of CPA_{prism}^{game}. This can be found in [8].

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 convenient to express this definition in a slightly different notation. Thus, for W ⊂ ω let W^{0} = W and W^{1} = ω \ W. A family J ⊂ [ω]^{ω} is independent provided the set

⋂_{W ∈J_{0}} W^{τ(W)}

is infinite for every finite subset J_{0} of J and τ: J_{0} → {0, 1}.

The independence cardinal 𝔦 is defined as follows:

𝔦 = min{|J |: J is infinite maximal independent family}.

According to Andreas Blass [3, Section 11.5] the fact that the equation 𝔦 = ω_{1} holds in the iterated perfect set model was first proved by Eisworth and Shelah (unpublished).

Theorem 4.17. CPA_{prism}^{game} implies that 𝔦 = ω_{1}.

The proof of the theorem 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 4.18. For every countable independent family W ⊂ [ω]^{ω} separating points and a prism P in [ω]^{ω} there exist W ∈ [ω]^{ω} and a subprism Q of P such that W ∪ {W } is independent but W ∪ {W, x} is not independent for every x ∈ Q.

Proof. Let W = {W_{i}: i < ω} and let φ: ω → 2^{ω} be a Marczewski function for W, that is, for i, k < ω

φ(k)(i) = \{ \begin{matrix}1 & for k ∈ W_{i} \\ 0 & for k ∉ W_{i}. \end{matrix}

Note that φ is one-to-one, since W separates points. Notice also that for every k, n < ω and τ ∈ 2^{n}

k ∈ ⋂_{i<n} W_{i}^{τ(i)} ⇔ (∀i < n) k ∈ W_{i}^{τ(i)} ⇔ (∀i < n) φ(k)(i) = τ(i) ⇔ τ ⊂ φ(k).

Now, if [τ] = {t ∈ 2^{ω}: τ ⊂ t} then sets {[τ]: τ ∈ 2^{<ω}} form a base for 2^{ω} and

k ∈ ⋂_{i<n} W_{i}^{τ(i)} ⇔ φ(k) ∈ [τ].  (4.17)

Thus, independence of W implies that φ[ω] is dense in 2^{ω}, so it is homeomorphic to the set ℚ of rational numbers. Note also that from (4.17) it follows immediately that

(a) if W ⊂ ω is such that φ[W] and φ[ω] \ φ[W] are dense then W ∪ {W } is independent;

(b) if W, x ⊂ ω are such that for some τ ∈ 2^{<ω} either φ[x ∩ W] ∩ [τ] = ∅ or φ[W] ∩ [τ] ⊂ φ[x] then W ∪ {W, x} is not independent.

Let I be the ideal of nowhere dense subsets of φ[ω]. Then, by Fact 4.6, I is weakly selective, since φ[ω] is homeomorphic to ℚ. So, identifying φ[ω] with ω and applying Lemma 4.9(b), we can find a subprism Q 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]. Trimming V, if necessary, we can assume that φ[V] ⊂ [τ] and that φ[ω \ V] is also dense in [τ]. Now let W ⊃ V be such that φ[W] ∩ [τ] = φ[V] and both φ[W] and φ[ω \ W] are dense in φ[ω]. Then, by (a) and (b), W ∪ {W } is independent while W ∪ {W, x} is not independent for every x ∈ Q.

Proof of Theorem 4.17. For a countable independent family W ⊂ [ω]^{ω} separating points and an f ∈ F_{prism}([ω]^{ω}) define P = range(f) and let W (W, P) ∈ [ω]^{ω} and a subprism Q(W, P) of P be as in Lemma 4.18. If f ∈ C_{prism}([ω]^{ω}) and P = range(f) = {x} then we put Q(W, P) = P and define W (W, P) as an arbitrary W such that W ∪{W } is independent while W ∪ {W, x} is not.

Let A_{0} ⊂ [ω]^{ω} be an arbitrary countable independent family separating points and consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q(A_{0} ∪ {W_{η}: η < ξ}, P_{ξ}),

where sets W_{η} are defined inductively by W_{η} = W (A_{0} ∪ {W_{ζ}: ζ < η}, P_{η}).

By CPA_{cube}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which and Player II loses, that is, [ω]^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}.

Now, notice that the family J = A_{0} ∪ {W_{ξ}: ξ < ω_{1}} is a maximal independent family. It is clear that J is independent, since every set W_{ξ} was chosen so that A_{0} ∪ {W_{ζ}: ζ ≤ ξ} is independent. To see that J is maximal it is enough to note that every x ∈ [ω]^{ω} belongs to a Q_{ξ} for some ξ < ω_{1}, and so A_{0} ∪ {W_{ζ}: ζ ≤ ξ} ∪ {x} is not independent.

By Theorem 4.17 we see that CPA_{prism}^{game} implies the existence of an independent family of size ω_{1}. Next, answering a question of Michael Hrušák [17] we show that such a family can be simultaneously a splitting family. This is similar in flavor to Theorem 3.4. In the proof we will use the following lemma.

Lemma 4.19. For every countable family V ⊂ [ω]^{ω} and a perfect set P in [ω]^{ω} there exists a W_{1} ∈ [ω]^{ω} such that V ∪ {W_{1}} is independent and W_{1} splits every A ∈ P.

Proof. We follow the argument from [11, p. 121] that 𝔰 ≤ 𝔡.

For every A ∈ [ω]^{ω} let b_{A} be a strictly increasing bijection from ω onto A. Then b: [ω]^{ω} → ω^{ω} defined by b(A) = b_{A} is continuous. In particular b[P] = {b_{A}: A ∈ P } is compact, so there exists a strictly increasing f ∈ ω^{ω} such that b_{A}(n) < f(n) for every A ∈ P and n < ω. For n < ω let f^{n} denote the n-fold composition of f and let S_{n} = {m < ω: f^{n}(0) ≤ m < f^{n+1}(0)}. Then f^{n}(0) ≤ b_{A}(f^{n}(0)) < f(f^{n}(0)) = f^{n+1}(0) for every A ∈ P and n < ω. In particular, for every A ∈ P

S_{n} ∩ A ≠ ∅.

So, if T ⊂ ω be infinite and co-infinite and W_{1} = ⋃_{n∈T} S_{n} then W_{1} splits every A ∈ P. Thus, it is enough to take infinite and co-infinite T ⊂ ω such that V ∪ {W_{1}} is independent.

Theorem 4.20. CPA_{prism}^{game} implies that there exists a family F ⊂ [ω]^{ω} of cardinality ω_{1} which is simultaneously independent and splitting.

Proof. The proof is just a slight modification of that for Theorem 4.17. (Compare also Theorem 3.4.)

For a countable independent family W ⊂ [ω]^{ω} separating points and an f ∈ F_{prism}([ω]^{ω}) put P = range(f) and let W_{0} ∈ [ω]^{ω} and a subprism Q of P be as in Lemma 4.18. Let W_{1} be as in Lemma 4.19 used with P = Q and V = W ∪ {W_{0}}. We put Q(W, P) = Q_{1} and W(W, P) = {W_{0}, W_{1}}.

If f ∈ C_{cube}([ω]^{ω}) and P = range(f) = {x} then we put Q(W, P) = P and W(W, P) = {W_{0}, W_{1}}, where W_{0} and W_{1} are such that W ∪ {W_{0}, W_{1}} is independent and W_{1} splits P = {x}.

Let A_{0} ⊂ [ω]^{ω} be an arbitrary countable independent family separating points and consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q(A_{0} ∪ ⋃{W_{η}: η < ξ}, P_{ξ}),

where W_{η}’s are defined inductively by W_{η} = W(A_{0} ∪ ⋃{W_{η}: η < ξ}, P_{η}).

By CPA_{prism}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which and Player II loses, that is, [ω]^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}. Then the family F = A_{0} ∪ ⋃{W_{ξ}: ξ < ω_{1}} is independent and splitting.

4.5. Crowded ultrafilters on ℚ.

Let Perf(ℚ) stand for the family of all closed subsets A of ℚ without isolated points, that is, such that their closures cl_{ℝ}(A) in ℝ are perfect sets. Recall that 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. [12, 10]) in connection with the reminder β ℚ \ ℚ of the Čech-Stone compactification β ℚ of ℚ.

In what follows we will also use the following simple fact, in which a non-scattered subset of ℚ is understood as a set containing a subset dense in itself.

Fact 4.21. Every non-scattered set B ⊂ ℚ contains a subset from Perf(ℚ).

Proof. Since B is non-scattered, decreasing it, if necessary, we can assume that B is dense in itself. Let {k_{n} ≤ n: n < ω} be an enumeration of ω with infinitely many repetitions and let ℚ \ B = {a_{n}: n < ω}. By induction construct a sequence ⟨⟨p_{n}, U_{n}⟩ ∈ B ∩ P(ℚ): n < ω⟩ such that p_{n} ∈ B \ ⋃_{i<n} U_{i} and |p_{n} − p_{k_{n}}| < 2^{−n} while U_{n} ∋ a_{n} is a clopen subset of ℚ \ {p_{i}: i ≤ n}. Then ℚ \ ⋃_{n<ω} U_{n} ⊂ B is as desired.

The following theorem answers in positive a question of M. Hrušák [17] on whether there exists a crowded ultrafilter in the iterated perfect set model.

Theorem 4.22. CPA_{prism}^{game} 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 ∪ {I_{n} ∩ ℚ: n < ω}, where I_{n} = [p − 2^{−n}, p + 2^{−n}]. The proof of the theorem is based on the following lemma, in which [ℚ]^{ω} is considered with the same topology as [ω]^{ω} upon natural identification.

Lemma 4.23. Let D ⊂ Perf(ℚ) be a countable family such that F (D) is non-trivial. Then for every prism P in [ℚ]^{ω} there exist a subprism Q of P and a Z ∈ Perf(ℚ) such that F (D ∪ {Z}) is non-trivial 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 2^{ℚ}, the identification mapping given by the characteristic function. Thus, we will consider P as a prism in 2^{ℚ}. Fix an f ∈ F_{prism}(2^{ℚ}) from ℭ^{α} onto P.

Let {D_{n} ∈ Perf(ℚ): n < ω} be a cofinal sequence in F (D) with a property that D_{n+1} ⊂ D_{n} ⊂ I_{n} for every n < ω. Choosing a subsequence, if necessary, we can find disjoint intervals J_{n} such that K_{n} = D_{n} ∩ J_{n} ∈ Perf(ℚ).

First we will show that there exist a sequence ⟨B_{n} ⊂ K_{n}: n < ω⟩ of non-scattered sets and a subprism P_{0} of P such that

g ↾ B_{n} is constant for every g ∈ P_{0} and n < ω.  (4.18)

For each n < ω let D_{n} be the collection of all pairwise disjoint families E ∈ [ℙ_{α}]^{<ω} for which there exists a non-scattered set B_{n} ⊂ K_{n} with the property that

g ↾ B_{n} is constant for every g ∈ f [⋃ E].

To see that the families D_{n} satisfy condition (†) from Lemma 4.4 it is enough to notice that for every non-scattered set B ⊂ ℚ and every prism P_{1} there is a subprism Q_{1} of P_{1} and a non-scattered subset B′ of B such that g ↾ B′ is constant for every g ∈ Q_{1}. But B contains a subset W homeomorphic to ℚ. So, by Fact 4.6, the ideal I of nowhere dense subsets of W is weakly selective. So, applying Lemma 4.9 to this ideal and the prism P_{1} we can find a B′ ∈ I^{+}, which clearly is not scattered, and a Q_{1} as desired.

Thus, using Lemma 4.4, we can find a sequence ⟨E_{n} ∈ D_{n}: n < ω⟩ satisfying the assumptions of Lemma 4.3. Let E_{0} = ⋂_{n<ω} ⋃ E_{n} ∈ ℙ_{α}. It is easy to see that the sets B_{n} witnessing E_{n} ∈ D_{n} and Q_{0} = f[E_{0}] satisfy (4.18). Notice also that by Fact 4.21 we can assume that B_{n} ∈ Perf(ℚ) for every n < ω.

Now let A be a selector from the family {B_{n}: n < ω}. Then A ∈ I^{+}, where I is the ideal of finite subsets of ℚ. Applying Lemma 4.9(a) to A and P_{0} we can find i < 2, S ∈ [A]^{ω}, and a subprism Q of P_{0} such that g ↾ S is constant equal to i for every g ∈ Q. Put Z = ⋃_{n∈S} B_{n}. Then, by (4.18), g ↾ Z is constant for every g ∈ Q. Finally, note that Z ∈ Perf(ℚ) since Z is closed, as B_{n} → p ∉ ℚ.

Proof of Theorem 4.22. For a prism P in [ℚ]^{ω} and a countable family D ⊂ Perf(ℚ) for which F (D) is non-trivial let Z(D, P) ∈ Perf(ℚ) and a subprism Q(D, P) of P be as in Lemma 4.23. Consider the following strategy S for Player II:

S(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P_{ξ}) = Q({Z_{η}: η < ξ}, P_{ξ}),

where sets Z_{η} are defined inductively by Z_{η} = Z({Z_{ζ}: ζ < η}, P_{η}).

By CPA_{prism}^{game} strategy S is not a winning strategy for Player II. So there exists a game ⟨⟨P_{ξ}, Q_{ξ}⟩: ξ < ω_{1}⟩ played according to S in which Player II loses, that is, [ℚ]^{ω} = ⋃_{ξ<ω_{1}} Q_{ξ}.

Now, let F = F ({Z_{ξ}: ξ < ω_{1}}). Then clearly F is a crowded nonprincipal filter. To see that it is maximal, take an x ∈ [ℚ]^{ω}. Then there is a ξ < ω_{1} such that x ∈ Q_{ξ}. Then either Z_{ξ} ∩ x = ∅ or Z_{ξ} ⊂ x. Thus, either x or its complement belong to F.

Note also that similarly as for Remarks 4.10 and 4.16 we can argue that there are many non-principal crowded ultrafilters.

Remark 4.24. CPA_{prism}^{game} implies that there are ω_{2}-many different crowded non-principal ultrafilters.

The existence of 2^{ω_{1}} different such ultrafilters follows also from a slightly stronger version of CPA_{prism}^{game}. This can be found in [8].

The construction of crowded ultrafilters is quite similar to that of selective ultrafilters and of nonselective P -points. This similarity suggests that it may be possible to construct a crowded ultrafilter which is also selective. This, however, cannot be done:

Proposition 4.25. There is no non-principal crowded ultrafilter on ℚ which is also a P -point.

Proof. Let F be a non-principal crowded ultrafilter on ℚ and let {x} = ⋂_{F ∈F} cl_{ℝ}(F) ∈ ℝ \ ℚ. Then I_{n} = (x − 2^{−n}, x + 2^{−n}) ∩ ℚ belongs to F for every n < ω. Let P = {I_{n} \ I_{n+1} : n < ω} ∪ {ℚ \ I_{0}}. Then P is a partition of ω disjoint with F. It is also easy to see that if F ⊂ ω is such that |F ∩ P | < ω for every P ∈ P then F ∉ F.

It is also not difficult to show that no non-principal crowded ultrafilter on ℚ can be a Q-point.

It is worth mentioning that CPA_{prism}^{game} implies also the existence of many other kinds of ultrafilters, like those constructed in [16]. In fact, many constructions that are done under CH can be carried out also under CPA_{prism}^{game}. However, this always needs some combinatorial lemma, such as Lemma 4.9, which allows replacing points with prisms.

5. Proof of fusion Lemma 4.4 and of consistency of CPA_{prism}^{game}

Proof of Lemma 4.4. For s ∈ 2^{A_{k}} and j < 2 let sˆ j stand for the function s ∪ {⟨⟨β_{k}, n_{k}⟩, j⟩} ∈ 2^{A_{k+1}}.

Let {s_{i}: i < 2^{k+1}} be an enumeration of 2^{A_{k+1}}. By induction on i < 2^{k+1} we will construct a sequence ⟨x_{s_{i}} ∈ ℭ^{α}: i < 2^{k+1}⟩ such that for every i < 2^{k+1}

(a) if s_{i} = sˆ j, where s ∈ 2^{A_{k}} and j < 2, then x_{s_{i}} ∈ E_{s},

(b) if m < i and β = max{β̄: s_{i} ↾ (β̄ × ω) = s_{m} ↾ (β̄ × ω)} then x_{s_{i}} ↾ β = x_{s_{m}} ↾ β and x_{s_{i}}(β) ≠ x_{s_{m}}(β).

The point x_{s_{0}} is chosen arbitrarily from E_{s_{0}↾A_{k}}. To make an inductive step, if for some 0 < i ≤ 2^{k+1} points {x_{s_{m}}: m < i} are already constructed choose an m̄ < i for which β as in (b) is maximal. Notice that by the inductive assumption and the condition (ag) we have x_{sm̄} ↾ β ∈ π_{β}[E_{sm̄↾A_{k}}] = π_{β}[E_{s_{i}↾A_{k}}]. So we can choose an x_{s_{i}} ∈ E_{s_{i}↾A_{k}} extending x_{sm̄} ↾ β and such that x_{s_{i}}(β) ≠ x_{s_{m}}(β) for all m < i. It is easy to see that such x_{s_{i}} satisfies (a) and the condition (b) for m = m̄. For other m < i condition (b) follows from the maximality of β and the assumption that E_{k} satisfies (ag) and (sp).

Conditions (a) and (b) imply that E′_{k+1} = {{x_{s}}: s ∈ 2^{A_{k+1}}} satisfy (A) except for being a subset of ℙ_{α}. Let ε ∈ (0, 2^{−(k+1)}] be small enough that for every m < i < 2^{k+1} and β as in (b) we have π_{β+1}[B_{α}(x_{s_{i}}, ε)] ∩ π_{β+1}[B_{α}(x_{s_{m}}, ε)] = ∅. For s ∈ 2^{A_{k}} and j < 2 define

E_{sˆ j} = E_{s} ∩ B_{α}(x_{sˆ j}, ε).

Then E_{k+1} = {E_{s}: s ∈ 2^{A_{k+1}}} is a subset of ℙ_{α} by (4.9). Conditions (i) and (ii) are clear from the construction, while (ag) for E_{k+1} follows from (b) and (4.7). Property (sp) holds by (b) and the choice of ε, since (sp) was true for E′_{k+1}. We have completed the proof of (A).

To prove condition (B), fix an enumeration {s_{i}: i < 2^{k}} of 2^{A_{k}} and define γ = max{β_{0}, …, β_{k}} < α. Also for i, m < 2^{k} put E_{s_{i}}^{−1} = E_{s_{i}} and

β_{i}^{m} = max{β ≤ γ: s_{i} ↾ (β × ω) = s_{m} ↾ (β × ω)}.

By induction we will construct the sequences ⟨{E_{s_{i}}^{m} ∈ ℙ_{α}: i < 2^{k}}: m < 2^{k}⟩ and ⟨P_{m} ∈ ℙ_{α}: m < 2^{k}⟩ such that for every j, m < 2^{k}

(a) E^{m} = {E_{s_{i}}^{m}: i < 2^{k}} satisfies (ag),

(b) E_{s_{j}}^{m} ⊂ E_{s_{j}}^{m−1} and if x ∈ E_{s_{j}}^{m−1} and π_{γ}(x) ∈ π_{γ}[E_{s_{j}}^{m}] then x ∈ E_{s_{j}}^{m},

(c) π_{γ}[P_{m}] = π_{γ}[E_{s_{m}}^{m}],

(d) P_{m} ⊂ E_{s_{m}}^{m−1} and {P_{i}: i ≤ m} ∈ D. So, assume that for some m < 2^{k} the sequence ⟨P_{i}: i < m⟩ and the family E^{m−1} satisfying (ag) are already constructed. Notice that, by (b), sets in E^{m−1} are pairwise disjoint, since this was the case for E^{−1} = E_{k}. So, by condition (†) applied to E = {P_{i}: i < m}, we can choose P_{m} ∈ ℙ_{α}∩P(E_{s_{m}}^{m−1}) such that {P_{m}} ∪ {P_{i}: i < m} ∈ D. This guarantees (d).

Next, for i < 2^{k} define

E_{s_{i}}^{m} = E_{s_{i}}^{m−1} ∩ π_{β_{i}^{m}}^{−1}(π_{β_{i}^{m}}[P_{m}]) = {x ∈ E_{s_{i}}^{m−1} : x ↾ β_{i}^{m} ∈ π_{β_{i}^{m}}[P_{m}]}

and notice that π_{β_{i}^{m}}[P_{m}] ⊂ π_{β_{i}^{m}}[E_{s_{m}}^{m−1}] = π_{β_{i}^{m}}[E_{s_{i}}^{m−1}]. So, by (4.11), E_{s_{i}}^{m} ∈ ℙ_{α}. Also, the definition ensures (b) since β_{i}^{m} ≤ γ.

Note that, by the inductive assumption (a), for all i < 2^{k} we have

π_{β_{i}^{m}}[E_{s_{i}}^{m}] = π_{β_{i}^{m}}[E_{s_{i}}^{m−1}] ∩ π_{β_{i}^{m}}[P_{m}] = π_{β_{i}^{m}}[E_{s_{m}}^{m−1}] ∩ π_{β_{i}^{m}}[P_{m}] = π_{β_{i}^{m}}[P_{m}].

Since β_{m}^{m} = γ, this implies (c). To prove (a) pick β < α and different i, j < 2^{k} such that s_{i} ↾ (β × ω) = s_{j} ↾ (β × ω). If β ≤ β_{i}^{m} then also β ≤ β_{j}^{m} and π_{β}[E_{s_{i}}^{m}] = π_{β}[P_{m}] = π_{β}[E_{s_{j}}^{m}]. So, assume that β > β_{i}^{m} and β > β_{j}^{m}. Then β_{i}^{m} = β_{j}^{m} and

π_{β}[E_{s_{i}}^{m}] = {π_{β}(x): x ∈ E_{s_{i}}^{m−1} & x ↾ β_{i}^{m} ∈ π_{β_{i}^{m}}[P_{m}]}
= {π_{β}(x): x ∈ E_{s_{j}}^{m−1} & x ↾ β_{j}^{m} ∈ π_{β_{j}^{m}}[P_{m}]}
= π_{β}[E_{s_{j}}^{m}].

So E^{m} satisfies (a). This finishes the construction.

Notice that by the maximality of γ and the properties (a) and (c) the family E′_{k} = {P_{m}: m < 2^{k}} satisfies (ag). Since it is a refinement of E_{k} it also satisfies (sp). So (B) is proved.

To find E_{k+1} as in (C) first take an E′_{k+1} satisfying (A) and then use (B) to find its refinement E′_{k+1} ∈ D satisfying (ag) and (sp).

In what follows we will show that CPA_{prism}^{game} holds in the generic extension V [G] of a model V of ZFC+CH, where G is a V -generic filter over 𝕊_{ω_{2}}, the ω_{2} countable support iteration of Sacks forcing. We will use here terminology from [1].

Let ℙ = ⟨Perf(ℭ), ⊂⟩. Recall that perfect set (Sacks) forcing 𝕊 is usually defined as the set of all trees T (P) = {x ↾ n ∈ 2^{<ω}: x ∈ P & n < ω}, where P ∈ ℙ, and is ordered by 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: ℙ → 𝕊 establishes isomorphism between forcings ℙ and 𝕊. Also if for s ∈ 𝕊 we define lim(s) = {x ∈ 2^{ω}: ∀n < ω (x ↾ n ∈ s)} then lim: 𝕊 → ℙ is the inverse of T.

Perfect set forcing is usually represented as 𝕊 rather that in its more natural form ℙ since the conditions in 𝕊 are absolute, unlike those in ℙ. However, in light of our axiom, it is important to think of this forcing in terms of ℙ.

Recall also that for a countable A ⊂ ω_{2} we defined Φ_{prism}(A) as the family of all projection-keeping homeomorphisms f: ℭ^{A} → ℭ^{A}. For A ⊂ α ≤ ω_{2} define

ℙ_{A}^{α} = {[range(f)]_{α}: f ∈ Φ_{prism}(A)},

where [E]_{α} = {g ∈ ℭ^{α}: g ↾ A ∈ E} for every E ⊂ ℭ^{A}. Also, we put

ℙ_{α} = ⋃ {ℙ_{A}^{α}: A ∈ [α]^{≤ω}}

and order it by inclusion. (Thus, for a countable α we have two different definitions of ℙ_{α}. However, it is not difficult to see that they describe the same family.)

It is known that forcing ℙ_{α} is equivalent to 𝕊_{α}, a countable support iteration of 𝕊 of length α. This fact is stated explicitly by Kanovei in [19], though it was also used, in less explicit form, in earlier papers of Miller [23] and Steprāns [25]. More precisely, in [23] and [25] the authors consider the family 𝕊_{α}^{D} ⊂ ℙ_{α} of determined conditions in ℙ_{α}, which form a dense subset of ℙ_{α}, and notice that 𝕊_{α}^{D} is equivalent to 𝕊_{α}. This fact is most precisely described by the following fact, whose explicit proof can be found in [8].

Fact 5.1. 𝕊_{α} is order isomorphic to 𝕊_{α}^{D} for every 0 < α ≤ ω_{2}. In particular, forcings 𝕊_{α} and ℙ_{α} are equivalent.

In the proof of the consistency of CPA_{prism}^{game} we will use the following proposition, which is of interest by its own. In its statement the symbol CPA_{prism}^{game}[X] stands for CPA_{prism}^{game} for a fixed Polish space X.

Proposition 5.2. For any Polish space X axiom CPA_{prism}^{game}[X] implies the full axiom CPA_{prism}^{game}.

Proof. Let X be a Polish space. First notice the following two facts. (F1) If Y is a Polish subspace of X then CPA_{prism}^{game}[X] implies CPA_{prism}^{game}[Y].

To see it, by way of contradiction assume that Player II has a winning strategy S in GAME_{prism}(Y). For each prism P in X let Q_{P} be its subprism such that either Q_{P} ∩ Y = ∅ or Q_{P} ⊂ Y. Such a subprism can be found by Claim 2.3 since Y is a G_{δ} subset of X. Define a strategy S̄ for Player II in the game GAME_{prism}(X) by putting

S̄(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P) = S(⟨⟨P_{η}, Q_{η}⟩: η < ξ & Q_{P_{η}} ⊂ Y ⟩, Q_{P})

provided Q_{P} ⊂ Y, and S̄(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P) = Q_{P} otherwise. It is easy to see that S̄ is a winning strategy for Player II in GAME_{prism}(X), contradicting CPA_{prism}^{game}[X]. So (F1) is proved.

(F2) If a Polish space Y is a 1 − 1 continuous image of X then CPA_{prism}^{game}[X]

implies CPA_{prism}^{game}[Y]. Indeed, let f be a continuous bijection from X onto Y and by way of contradiction assume that Player II has a winning strategy S in GAME_{prism}(Y). Define a strategy S̄ for Player II in GAME_{prism}(X) by putting

S̄(⟨⟨P_{η}, Q_{η}⟩: η < ξ⟩, P) = S̄(⟨⟨f[P_{η}], f[Q_{η}]⟩: η < ξ⟩, f[P]).

(Here if h is a coordinate function for a prism P then prism f[P] is considered with a coordinate system h ∘ f.) It is easy to see that S̄ is a winning strategy for Player II in GAME_{prism}(X), contradicting CPA_{prism}^{game}[X]. So (F2) is proved.

To finish the proof take a Polish space X for which CPA_{prism}^{game}[X] holds and recall that the Baire space ω^{ω} is homeomorphic to a subspace of X (since X contains a copy of ℭ and ℭ contains a copy of ω^{ω}). Thus, by (F1), CPA_{prism}^{game}[Z] holds for an arbitrary Polish subspace Z of ω^{ω}. Now, if Y is an arbitrary Polish space then there exists a closed subset F of ω^{ω} such that Y is a one-to-one continuous image of F. (See e.g. [20, Theorem 7.9].) So, by (F2), CPA_{prism}^{game}[Y] holds as well.

Theorem 5.3. CPA_{prism}^{game} holds in the iterated perfect set model. In particular, it is consistent with ZFC set theory.

Proof. Start with a model V of ZFC+CH and let V [G] be a generic extension of V with respect to forcing ℙ_{ω_{2}}. By Fact 5.1 forcing ℙ_{ω_{2}} is equivalent to 𝕊_{ω_{2}}, so it preserves cardinals and 𝔠 = ω_{2} in V [G]. For α ≤ ω_{2} let G_{α} = G ↾ α. Then G_{α} is V -generic over ℙ_{α}. By Proposition 5.2 it is enough to prove only CPA_{prism}^{game}[X] for X = ℭ.

Let {c_{α}: α < ω_{2}} be an enumeration, in V [G], of ℭ such that for every α-th element ω_{1} α of Γ, α > 0, we have:

• {c_{ξ}: ξ < ω_{1} α} = ℭ ∩ V [G_{α}];

• c_{ω_{1} α} is the Sacks generic real in V [G_{α+1}] over V [G_{α}].

We will show that this sequence satisfies CPA_{prism}^{game} in V [G].

So let S be a strategy for Player II. Thus, S is a function from a sub-

set of D = ⋃_{ξ<ω_{1}} (F_{prism}^{∗}(X) × F_{prism}^{∗}(X))^{ξ} × F_{prism}^{∗}(X) into F_{prism}^{∗}(X). (Here each prism is considered with its explicit coordinate function from F_{prism}^{∗}(X).) Since ℙ_{ω_{2}} is ω_{2}-cc and satisfies axiom A, there is an α ∈ Γ such that ω_{1} α = α and^{4}

S ∩ V [G_{α}] = S ∩ [(D × F_{prism}^{∗}(X)) ∩ V [G_{α}]] ∈ V [G_{α}].  (5.1)

Since the quotient forcing ℙ_{ω_{2}}/ℙ_{α} is equivalent to ℙ_{ω_{2}} we can assume that α = 0, that is, that V [G_{α}] is our ground model V.

Let ⟨⟨f_{ξ}, g_{ξ}⟩: ξ < ω_{1}⟩ be a game played according to the strategy S in which Player I plays in such a way that {f_{ξ}: ξ < ω_{1}} = F_{prism}^{∗}(X) ∩ V. Then G = {g_{ξ}: ξ < ω_{1}} ∈ V is F_{prism}^{∗}-dense. It is enough to show that

X \ V ⊂ ⋃_{g_{ξ}∈F_{prism}} range(g_{ξ}).

So, take an r ∈ X \ V. Then there exists a ℙ_{ω_{2}}-name τ for r such that

ℙ_{ω_{2}} ∥− τ ∈ X \ V.

We can also choose τ such that it is a ℙ(A)-name for some A ∈ [ω_{2}]^{≤ω} with 0 ∈ A. Then all the information on r is coded by G_{A} = G ↾ A. Therefore r ∈ V [{c_{ω_{1} ξ}: ξ ∈ A}]. Assume that A has an order type α. Clearly α < ω_{1} and ℙ(A) is isomorphic to ℙ_{α}. Applying this isomorphism we can assume that τ is a ℙ_{α}-name for r and ℙ_{α} ∥− τ ∈ X \ V. Picking the smallest α with this property, we can also assume that for every β < α we have

ℙ_{α} ∥− τ ∈ X \ V [G_{β}].

Now, for any such a name τ and any R ∈ ℙ_{α} there exist P ∈ ℙ_{α}, P ⊂ R, and a continuous injection function f: P → X (so f ∈ F_{prism}(X)∩V) which “reads τ continuously” in the sense that

Q ∥− τ ∈ f[Q]  (5.2)

for every Q ⊂ P, Q ∈ ℙ_{α}. (See [25, Lemma 3.1] or [23, Lemma 6, p. 580]. This also can be deduced from [8, Lemma 3.2.2].) So, the set

D = {Q ∈ ℙ_{α}: (∃ξ < ω_{1}) Q = dom(g_{ξ}) & Q ∥− τ ∈ g_{ξ}[Q]} ∈ V

is dense in ℙ_{α}. (For R ∈ ℙ_{α} take f as in (5.2), find ξ < ω_{1} with f = f_{ξ}, and notice that Q = dom(g_{ξ}) justifies the density of D.)

4Formally no f ∈ F_{prism}^{∗}(X) ∩ V [G_{ω_{2}}] belongs to V [G_{α}] with α < ω_{2}. However in this proof the expression “f ∈ F_{prism}^{∗}(X) ∩ V [G_{α}]” will be understood as “Code(f) belongs to V [G_{α}]”, where for dom(f) = P ⊂ ℭ^{α} with P = range(g), g ∈ Φ_{prism}(α), and D_{α} ∈ V being a fixed countable dense subset of ℭ^{α} we define Code(f) = f ↾ g[D_{α}].

Take Q ∈ D ∩ G_{ω_{1}} and ξ < ω_{1} such that Q = dom(g_{ξ}). Then there is a z ∈ Q such that g_{ξ}(z) = r. This finishes the proof.

Acknowledgment. The authors thank Dr. Elliott Pearl for proofreading a version of this paper.

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5Preprints marked by ^{⋆} are available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/˜ kcies/STA/STA.html [21] Kunen, K., Some points in βN, Math. Proc. Cambridge Philos. Soc. 80(3) (1976),

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Krzysztof Ciesielski Janusz Pawlikowski Department of Mathematics Department of Mathematics West Virginia University University of Wrocław Morgantown, WV 26506-6310 pl. Grunwaldzki 2/4 USA 50-384 Wrocław, Poland e-mail: K Cies@math.wvu.edu e-mail: pawlikow@math.uni.wroc.pl

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