PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 11, Pages 3379–3385 S 0002-9939(04)07475-1 Article electronically published on June 2, 2004

UNCOUNTABLE INTERSECTIONS OF OPEN SETS UNDER CPA_{prism}

KRZYSZTOF CIESIELSKI AND JANUSZ PAWLIKOWSKI

(Communicated by Alan Dow)

Abstract. We prove that the Covering Property Axiom CPA_{prism}, which holds in the iterated perfect set model, implies the following facts.

• If G is an intersection of ω_{1}-many open sets of a Polish space and G has cardinality continuum, then G contains a perfect set.

• There exists a subset G of the Cantor set which is an intersection of

ω_{1}-many open sets but is not a union of ω_{1}-many closed sets. The example from the second fact refutes a conjecture of Brendle, Larson, and Todorcevic.

1. Preliminaries and axiom CPA_{prism}

Our set-theoretic terminology is standard and follows that of [3]. In particular, |X| stands for the cardinality of a set X and 𝔠 = |ℝ|. The Cantor set 2^{ω} will be denoted by a symbol ℭ. We use term Polish space for a complete separable metric space without isolated points. For a Polish space X, the symbol Perf(X) will stand for a collection of all subsets of X homeomorphic to the Cantor set ℭ. For a fixed 0 < α < ω_{1} and 0 < β ≤ α a symbol π_{β} will stand for the projection from ℭ^{α} onto ℭ^{β}.

Axiom CPA_{prism} was introduced by the authors in [5], where it is shown that it holds in the iterated perfect set model. Also, CPA_{prism} is a simpler version of the axiom CPA, which is described in our monograph [6]. (See also [4].) For the reader’s convenience, we will restate the axiom in the next few paragraphs.

The main notions needed for the axiom are that of prism and prism-density. Let 0 < α < ω_{1} and let Φ_{prism}(α) be the family of all continuous injections f: ℭ^{α} → ℭ^{α} with the property that

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

or, equivalently, such that for every β < α

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

is a one-to-one function from ℭ^{β} into ℭ^{β}. Functions f from Φ_{prism}(α) were first introduced, in a more general setting, in [8], where they are called projection-keeping

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Received by the editors March 3, 2003 and, in revised form, July 27, 2003.

2000 Mathematics Subject Classification. Primary 03E35; Secondary 03E17.

Key words and phrases. Uncountable intersections of open sets.

The work of the first author was partially supported by NATO Grant PST.CLG.977652 and by a 2002/03 West Virginia University Senate Research Grant.

homeomorphisms. Note that Φ_{prism}(α) is closed under compositions and that, for every 0 < β < α, if f ∈ Φ_{prism}(α), then f ↾↾ β ∈ Φ_{prism}(β). Let

ℙ_{α} = {range(f): f ∈ Φ_{prism}(α)},

and note that if f ∈ Φ_{prism}(α) and E ∈ ℙ_{α}, then f[E] ∈ ℙ_{α}. In [11] the elements of ℙ_{α} are called I-perfect, where I is the ideal of countable sets.

The simplest elements of ℙ_{α} are perfect cubes, that is, the sets of the form C = ∏_{β<α} C_{β}, where C_{β} ∈ Perf(ℭ) for each β < α. (This is justified by a function f = ⟨f_{β}⟩_{β<α} ∈ Φ_{prism}(α), where each f_{β} is a homeomorphism from ℭ onto C_{β}.)

To state CPA_{prism} we need a few more definitions. For a fixed Polish space X we say that P ∈ Perf(X) is a prism if we consider it with an (implicitly given) continuous injection f from ℭ^{α}, 0 < α < ω_{1}, onto P. We say that Q is a subprism of a prism P ∈ Perf(X) provided Q = f[C], where f is as above and C ∈ ℙ_{α}. A family E ⊂ Perf(X) is prism-dense in X provided every prism in X contains a subprism Q ∈ E. Now we are ready to state the axiom.

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

If in the definition above we restrict our attention only to prisms which are perfect cubes in ℭ^{ω}, we get a notion of cube-density which is stronger than that of prismdensity. This naturally leads to a weaker version of CPA_{prism}, known as CPA_{cube}, which is obtained from CPA_{prism} by replacing the word “prism” with “cube.” Thus, any consequence of axiom CPA_{cube}, which has been studied in [7, 4, 10, 6], follows also from CPA_{prism}.

2. Intersections of ω_{1}-many open sets

For a Polish space X let G_{ω_{1}} be the collection of the intersections of ω_{1}-many open subsets of X. We are going to prove the following theorem.

Theorem 2.1. CPA_{prism} implies that the following property holds for every Polish space X.

(∗) If G is a G_{ω_{1}} subset of X and |G| = 𝔠, then G contains a perfect set.

Theorem 2.1 provides an affirmative answer to a question of Jörg Brendle, who asked us in [1] whether (∗) can be deduced from our axiom CPA. The fact that (∗) holds in the iterated perfect set model is proved in [2]. The argument presented below is considerably simpler.

Before we prove Theorem 2.1 we would like to notice, in Corollary 2.4, that in property (∗) we can replace the class of open sets by the considerably larger class Π_{2}^{1}. Here Σ_{1}^{1} stands for the class of analytic sets, that is, continuous images of Borel sets; Π_{1}^{1} for the class of co-analytic sets, the complements of analytic sets; Σ_{2}^{1} for continuous images of co-analytic sets, and Π_{2}^{1} for the class of all complements of Σ_{2}^{1} sets. For the argument that follows we need also to recall a theorem of Sierpiński that every Σ_{2}^{1} set is the union of ω_{1} Borel sets. (See, e.g., [9, p. 324].)

To argue for Corollary 2.4 we need the following two results.

Claim 2.2. If G ⊂ ℭ^{ω} is comeager in ℭ^{ω}, then it contains a perfect cube ∏_{i<ω} P_{i}.

An argument for the claim can be found in [7], [4], or [6].

Fact 2.3. Under CPA_{cube} (so, also under CPA_{prism}) the following holds: For every Σ_{2}^{1} subset B of a Polish space X there exists a family P of ω_{1}-many compact sets such that B = ⋃ P.

Proof. Since every Σ_{2}^{1} set is a union of ω_{1} Borel sets, we can assume that B is Borel. Let E be the family of all P ∈ Perf(X) such that either P ⊂ B or P ∩B = ∅. We claim that E is F_{cube}-dense. Indeed, if f: ℭ^{ω} → X is a continuous injection, then f^{−1}(B) is Borel in ℭ^{ω}. Thus, there exists a basic open set U in ℭ^{ω}, which is homeomorphic to ℭ^{ω}, such that either U ∩ f^{−1}(B) or U \ f^{−1}(B) is comeager in U. Apply Claim 2.2 to this comeager set to find a perfect cube P contained in it. Then f[P] ∈ E is a subcube of range(f). So, E is F_{cube}-dense.

By CPA_{cube} there exists an E_{0} ⊂ E such that |E_{0}| ≤ ω_{1} and |X \ ⋃ E_{0}| ≤ ω_{1}. Let P_{0} = {P ∈ E_{0} : P ⊂ B} and P = P_{0} ∪ {{x}: x ∈ B \ ⋃ E_{0}}. Then P is as desired. □

Corollary 2.4. Assume that CPA_{prism} holds and X is a Polish space.

• If G is an intersection of ω_{1}-many Π_{2}^{1} sets from X and |G| = 𝔠, then G contains a perfect set.

Proof. Let G = ⋂_{ξ<ω_{1}} T_{ξ}, where each T_{ξ} ⊂ X is a Π_{2}^{1} set. Then,

X \ G = ⋃_{ξ<ω_{1}}(X \ T_{ξ})

and each set X \ T_{ξ} is in the class Σ_{2}^{1}, so, by Fact 2.3, it is a union of ω_{1}-many compact sets. Thus, each T_{ξ} is an intersection of ω_{1} open sets. □

Theorem 2.1 follows easily from the following combinatorial fact concerning iterated perfect sets.

Proposition 2.5. Let 0 < α < ω_{1} < 𝔠 and H = ℭ^{α} \ ⋃_{ξ<ω_{1}} F_{ξ}, where each set F_{ξ} is compact. Then either H contains a perfect set P, or else there exists an E ∈ ℙ_{α} disjoint with H.

Proof of Theorem 2.1. Let G = X \ ⋃_{ξ<ω_{1}} T_{ξ}, where each set T_{ξ} is closed in X, and assume that G does not contain a perfect set.

Let E = {P ∈ Perf(X): P ∩ G = ∅}. We will show that E is prism-dense. This will finish the proof since then, by CPA_{prism}, X \⋃ E ⊃ G has cardinality ≤ ω_{1} < 𝔠.

So, let f: ℭ^{α} → X be a continuous injection. We need to find an E ∈ ℙ_{α} for which f[E] ∈ E, that is, f[E] ∩ G = ∅. Let F_{ξ} = f^{−1}(T_{ξ}) for ξ < ω_{1}. Then H = ℭ^{α} \ ⋃_{ξ<ω_{1}} F_{ξ} is equal to f^{−1}(G). If H contains a perfect set P, then so does G ⊃ f[P], contradicting our assumption. Thus, by Proposition 2.5, there exists an E ∈ ℙ_{α} disjoint with H = f^{−1}(G). So, f[E] is disjoint with G. □

To prove Proposition 2.5 we need some auxiliary terminology. For a proper ideal I of subsets of ℭ and an ordinal 0 < α < ω_{1} we say that a tree T ⊂ ℭ^{≤α} (ordered by inclusion) is a co-I tree, T ∈ T_{I}^{α}, provided

• for every β < α and t ∈ T ∩ ℭ^{β} we have ℭ \ succ_{T} (t) ∈ I, where succ_{T} (t) = {s(β): t ⊂ s ∈ T ∩ ℭ^{β+1}} is the set of all immediate successors of t in T, and

• for every limit ordinal λ ≤ α the λ-th level T ∩ ℭ^{λ} of T consists all the branches of T ∩ ℭ^{<λ}, that is, t ∈ T ∩ ℭ^{λ} if and only if t ↾ γ ∈ T for every γ < λ.

Also let K_{I}^{α} = {T ∩ ℭ^{α}: T ∈ T_{I}^{α}} and let I^{α} = ⋃_{K∈K_{I}^{α}} P(ℭ^{α} \ K). It is easy to see that I^{α} is an ideal on ℭ^{α}. We will call I^{α} the α-th Fubini power of the ideal I. This notion, with slightly different emphasis, appears also in Zapletal [11].

We will be interested in these notions only for I of the form I_{κ} = [ℭ]^{≤κ}, where κ < 𝔠 is equal to either ω or ω_{1}. It is easy to see that the families T_{I_{κ}}^{α} and K_{I_{κ}}^{α} are closed under the intersection of κ-many of their elements. So (I_{κ})^{α} is a κ^{+}-additive ideal on ℭ^{α}.

A big part of the difficulty behind our proof of Theorem 2.1 lies in the following lemma generalizing the Cantor-Bendixon theorem that an uncountable closed set contains a perfect set. In fact, this result is true not only for closed sets, but also for analytic sets. This version of the lemma, which can be viewed as a generalization of the Suslin theorem that an uncountable analytic set contains a perfect set, can be found in [11, Lemma 4.1.29(2)]. (See also [11, Lemma 3.3.1].)

Lemma 2.6. Let 0 < α < ω_{1}. If F is a closed subset of ℭ^{α} and F ∉ (I_{ω})^{α}, then F contains an iterated perfect set, that is, there exists an E ∈ ℙ_{α} such that E ⊂ F.

Proof of Proposition 2.5. If there is a ξ < ω_{1} such that F_{ξ} ∉ (I_{ω})^{α}, then, by Lemma 2.6, there is an E ∈ ℙ_{α} such that E ⊂ F_{ξ}. Clearly this E is disjoint with H = ℭ^{α} \ ⋃_{ξ<ω_{1}} F_{ξ}. So, assume that F_{ξ} ∈ (I_{ω})^{α} for every ξ < ω. We will show that H contains a perfect set.

For every ξ < ω_{1}, since F_{ξ} ∈ (I_{ω})^{α}, there is a T_{ξ} ∈ T_{I_{ω}}^{α} disjoint with F_{ξ}. But then T = ⋂_{ξ<ω_{1}} T_{ξ} ∈ T_{I_{ω_{1}}}^{α} is disjoint with ⋃_{ξ<ω_{1}} F_{ξ}, so T is a subset of H. Thus, it is enough to show that T contains a perfect set. But this follows immediately from the assumption that ω_{1} < 𝔠.

If α is a successor ordinal, say α = β + 1, then take t ∈ π_{β}[T] and a perfect set C ⊂ succ_{T} (t). Then P = {t} × C is a perfect subset of T.

If α is a limit ordinal, take an increasing sequence ⟨α_{n}: n < ω⟩ cofinal with α. By induction on n < ω choose a sequence ⟨t_{σ} ∈ π_{α_{n}}[T]: n < ω & σ ∈ 2^{n}⟩ such that t_{σˆ 0} and t_{σˆ 1} are distinct extensions of t_{σ}. Then

P = {t ∈ ℭ^{α}: t ↾ α_{n} ∈ {t_{σ}: σ ∈ 2^{n}} for every n < ω}

is a perfect subset of T. □

Note that Lemma 2.6 (see [11, Lemma 3.3.1]) and appropriate CPA versions hold also for many other “nicely definable” forcings. (See [11, chapter 5].) The form of these CPA’s remains the most similar to our axiom for the forcings for which Lemma 3.4 remains valid. This, for example, includes Miller forcing. For these forcings the argument remains intact, so appropriate CPA’s also imply (∗).

3. The complements of G_{ω_{1}} sets

Brendle, Larson, and Todorcevic conjectured in [2, conj. 5.11] that, in the iterated perfect set model, the complement of a G_{ω_{1}} set is also G_{ω_{1}}, that is, that any G_{ω_{1}} set is a union of ω_{1}-many closed sets. In this section we will prove that the conjecture is false, by showing that the existence of a counterexample follows from CPA_{prism}. More precisely, this follows from the following theorem, which is interesting in its own right.

Let X be the family of all continuous functions f from an uncountable G_{δ} subset of ℭ into ℭ.

Theorem 3.1. CPA_{prism} implies that there exists a sequence ⟨g_{α} ∈ X : α < ω_{1}⟩ such that for every g ∈ X there is an α < ω_{1} such that |g ∩ g_{α}| = 𝔠.

Before we prove the theorem we notice that it easily implies the following corollary, which refutes [2, conj. 5.11]. It also gives a negative answer to [2, Question 5.13]. Note that in the proof of the corollary we use from Theorem 3.1 only the fact that g ∩ g_{α} ≠ ∅.

Corollary 3.2. If ω_{1} < 𝔠 and G = ℭ^{2} \ ⋃_{α<ω_{1}} g_{α}, where the g_{α}’s are from Theorem 3.1, then G is not a union of ω_{1}-many closed sets.

In particular, CPA_{prism} implies that G is G_{ω_{1}} but ℭ^{2} \ G is not.

Proof. To see that G is not a union of ω_{1}-many closed sets, notice that

(1)  if P ⊂ G is compact then π[P] is countable.

To see this, take a compact subset P of ℭ^{2} for which π[P] is uncountable. We need to show that P ∩ ⋃_{α<ω_{1}} g_{α} ≠ ∅. So, take a Borel selection g: π[P] → ℭ for P. (For example, put g(x) = min{y ∈ ℭ: ⟨x, y⟩ ∈ P }.) Then, there exists a dense G_{δ} subset D of π[P] such that g ↾ D is continuous. Thus, g ↾ D ∈ X and (g ↾ D) ∩ g_{α} ≠ ∅ for some α < ω_{1}. In particular, P ∩ ⋃_{α<ω_{1}} g_{α} ⊃ (g ↾ D) ∩ g_{α} ≠ ∅.

Next, let P = {P_{ξ}: ξ < ω_{1}} be a family of compact subsets of G. We need to show that G ≠ ⋃ P. But, by (1), there exists an x ∈ ℭ \ ⋃_{ξ<ω_{1}} π[P_{ξ}]. Let p ∈ ({x} × ℭ) \ ⋃_{α<ω_{1}} g_{α}. Then p ∈ G \ ⋃ P.

To see the additional part, first note that ℭ^{2} \ G is not G_{ω_{1}}, by the above argument. To see that G is G_{ω_{1}}, it is enough to show that its complement ⋃_{α<ω_{1}} g_{α} is a union of ω_{1}-many compact sets. But every g_{α} is a G_{δ} subset of ℭ^{2}, so it is a Polish space. Thus g_{α} is a union of ω_{1} compact sets — this follows directly from the formulation of CPA_{prism} (as well as from CPA_{cube}). □

The proof^{1} of Theorem 3.1 is based on the following simple observation. Note also that if we are willing to prove the theorem only with the conclusion that g ∩ g_{α} ≠ ∅ (i.e., a version needed to prove Corollary 3.2), then in Lemma 3.3 one need only require that F ∩ f ≠ ∅ for every continuous f: ℭ → ℭ. The argument for this version is a bit simpler, and in this form Lemma 3.3 is more likely to be known.

Lemma 3.3. There exists a continuous function F from a G_{δ} subset T of ℭ into ℭ such that |F ∩ f| = 𝔠 for every continuous f: ℭ → ℭ.

Proof. Let C be the family of all continuous functions f: ℭ → ℭ, considered with the sup norm. Then C is homeomorphic to the Baire space ω^{ω}, and so is C × ℭ. (To see this use, for example, [9, thm. 7.7].) Let T be a G_{δ} subset of ℭ homeomorphic to ω^{ω} and let h = ⟨h_{1}, h_{2}⟩: T → C × ℭ be a homeomorphism. Define F: T → ℭ by

F (t) = [h_{1}(t)](t).

Clearly F is continuous. To see the other part, take an f ∈ C and notice that P = (h_{1})^{−1}(f) is uncountable. It is enough to show that F ↾ P = f ↾ P.

Indeed, if t ∈ P, then h_{1}(t) = f and F (t) = [h_{1}(t)](t) = f(t). □

1It was pointed to us by the referee that “Theorem 3.1 is of the syntactical form isolated in [11] for the consequences of CPA, while Corollary 3.2 is not of this form.” This certainly sheds more light on our work. However, in order to use the results from [11] to deduce Theorem 3.1, in addition to checking “the syntactical form,” one needs also to know how to force the conclusion of Theorem 3.1. So, one cannot use [11] without actually using forcing technique.

One of the most important properties of prisms, distinguishing them from cubes, is the following fact, which is a particular case of [8, thm. 20]. In its current form it has been used in [5]. Its proof can be also found in [6, Lemma 3.2.2].

Lemma 3.4. For every 0 < α < ω_{1}, a Polish space X, and a continuous function f: ℭ^{α} → X there exist 0 < β ≤ α and E ∈ ℙ_{α} such that f ∘ π_{β}^{−1} is a function on π_{β}[E] ∈ ℙ_{β} which is either one-to-one or constant.

We will also need the following result.

Proposition 3.5. CPA_{prism} implies that there exists a family F = {f_{α}: α < ω_{1}} of continuous injections from ℭ^{2} into ℭ such that {f_{α}[{x} × ℭ]: x ∈ ℭ & α < ω_{1}} is dense in Perf(ℭ).

Proof. Let C be the family of all continuous functions from ℭ into ℭ endowed with the uniform convergence topology, and let X be a subspace of C consisting of all continuous injections from ℭ into ℭ. Note that X is a G_{δ} subset of X and that X is a Polish space. Thus, there exists a metric on X which makes it a Polish space.

For every prism P in X witnessed by a continuous injection f from ℭ^{α} onto P there exists a subprism P^{∗} defined as follows. Let f^{∗}: ℭ^{α+1} → ℭ be defined by f^{∗}(x) = f(x ↾ α)(x(α)). Clearly f^{∗} is continuous. Apply Lemma 3.4 to the function f^{∗} to find a β ≤ α + 1 and an E ∈ ℙ_{α+1} such that f^{∗} ∘ π_{β}^{−1} is a function on π_{β}[P] ∈ ℙ_{β} which is either one-to-one or constant. Note that

β = α + 1 and f^{∗} is one-to-one on E,

since there are distinct x, y ∈ E with x ↾ α = y ↾ α for which we have f^{∗}(x) ≠ f^{∗}(y). Let P^{∗} = f[π_{α}[E]] and f_{P^{∗}} = f^{∗} ↾ E.

Since E = {P^{∗}: P is a prism in X} is prism-dense, by CPA_{prism} we can find an E_{0} ∈ [E]^{≤ω_{1}} such that Y = X \ ⋃ E has cardinality at most ω_{1}.

For Q ∈ E_{0}, if f_{Q} is defined on E ∈ ℙ_{α+1}, let g ∈ Φ_{prism}(α + 1) be a projectionkeeping homeomorphism from ℭ^{α+1} into E. If in ℭ^{α+1} = ℭ^{α} × ℭ we identify ℭ^{α} with ℭ, then g_{Q} = f_{Q} ∘ g is an injection from ℭ^{2} into ℭ, and for every h ∈ Q there is an x ∈ ℭ such that g_{Q}[{x} × ℭ] ⊂ h[ℭ]. Now, if the domains of the functions from Y are identified with ℭ^{2}, then F = Y ∪ {g_{Q}: Q ∈ E_{0}} is as desired. □

Proof of Theorem 3.1. Let {f_{α}: α < ω_{1}} be as in Proposition 3.5 and let F: T → ℭ be as in Lemma 3.3. For α < ω_{1} let K_{α} = f_{α}[ℭ × T]. Define g_{α}: K_{α} → ℭ by

g_{α}(f_{α}(x, t)) = F (t) for every ⟨x, t⟩ ∈ ℭ × T .

Clearly the functions g_{α} are continuous and defined on G_{δ} sets.

Fix a g ∈ X. We need to find an α < ω_{1} such that |g ∩g_{α}| = 𝔠. Since the domain of g is uncountable, it contains a perfect set P. So, there are α < ω_{1} and x_{0} ∈ ℭ such that f_{α}[{x_{0}} × ℭ] ⊂ P. Let f ∈ C be defined by f(y) = g(f_{α}(x_{0}, y)). Then, by Lemma 3.3, there is a Q ∈ Perf(T) such that F ↾ Q = f ↾ Q. Thus, for every t ∈ Q ⊂ T we have

g_{α}(f_{α}(x_{0}, t)) = F (t) = f(t) = g(f_{α}(x_{0}, t)).

Thus, g_{α} ↾ f_{α}[{x_{0}} × Q] = g ↾ f_{α}[{x_{0}} × Q]. □

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Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310

E-mail address: K Cies@math.wvu.edu
URL: http://www.math.wvu.edu/~kcies

Department of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail address: pawlikow@math.uni.wroc.pl