WHAT IS LEFT OF CH AFTER YOU ADD COHEN REALS?

I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

Abstract. The principle CH* concerning elementary submodels is formulated and is shown to be valid in any generic extension obtained by adding any number of Cohen reals to a ground model satisfying CH.

CH* has interesting topological consequences, e.g.:

(i) Every initially ω_{1} compact, countably tight T_{3} space is compact.

(ii) Let X be a countably tight compact T_{2} space; then

(a) if S is G_{δ}-dense in X then every point of X is the limit of a converging ω_{1}-sequence from S;

(b) if Y ⊂ X with s(Y) ≤ ω_{1} then h(Y) ≤ ω_{1}

(c) X contains no complete binary tree of closed sets of height ω_{2}. (iii) If X is a compact T_{2} space with small diagonal then X is metrizable.

1. The Principle CH*

Our aim here is to formulate a principle that may be considered as a weakening of CH because it will be shown to hold in any generic extension of a ground model satisfying CH by adding any number of Cohen reals. This principle, that will be called CH*, will concern elementary submodels but will include no mention of forcing.

Let us start with the following trivial observation: If CH holds then for every uncountable regular cardinal λ there are ”many” countably closed elementary submodels of size ω_{1} of H(λ). More precisely, they form a stationary (hence cofinal) subset of [H(λ)]^{ω_{1}}. In fact, the existence of such countably closed elementary submodels is equivalent to CH.

What happens to these elementary submodels if Cohen reals are added? We are going to denote by C_{κ} the natural notion of forcing that adds κ many Cohen reals, i.e. C_{κ} = Fn(κ, 2). The following is well known (cf. e.g. [2] or [7]).

Fact 1.1. Assume that our ground model V satisfies CH, κ ∈ M ≺ H(λ), [M]^{ω} ⊂ M and G is C_{κ}-generic over V. Then, in V [G],

N = M[G] ≺ H(λ)^{V [G]},

Research supported by OTKA grant no. 16391.

where

M[G] = {val_{G}(τ): τ ∈ M}.

Of course, if κ > ω_{1} then |M| = ω_{1} implies |N| = |M| < κ hence N can no longer be countably closed. However, it will turn out that N has some properties that are weaker than countable closedness but still strong enough to yield interesting consequences. In order to formulate them we first need a definition.

Definition 1.2. Let N be any set and A ⊂ N. We say that B is an N-cover of A if the following three conditions are satisfied.

(a) A ⊂ B ⊂ N;

(b) N ∩ [B]^{ω} is cofinal in [B]^{ω};

(c) for every S ∈ N ∩ [B]^{ω} we have |S ∩ A| = ω.

Clearly, if [N]^{ω} ⊂ N then every set A ⊂ N is its own N-cover. Hence it is clear that both (i) and (ii) in the following theorem are weakenings of the property of countable completeness.

Theorem 1.3. If N = M[G] is as in 1.1 then

(i) N ∩ [N]^{ω} is cofinal in [N]^{ω}; (ii) every set X ∈ [N ∩ On]^{ω_{1}} has an uncountable subset A ∈ [X]^{ω_{1}}

that has an N-cover.

Proof. (i) is well-known and follows easily from the fact that in a CCC extension every countable set of ordinals is contained in a countable ground model set, moreover from the countable closedness of M.

To show (ii) it will suffice to prove that if p ⊧ Ẋ ∈ [M ∩ On]^{ω_{1}} then some extension q of p forces that some uncountable subset of Ẋ has an N-cover. To this end we first define distinct ordinals α_{ξ} ∈ M ∩ On and conditions p_{ξ} ∈ C_{κ} for ξ ∈ ω_{1} such that p_{ξ} ≤ p and p_{ξ} ⊧ α_{ξ} ∈ Ẋ. We may assume that {p_{ξ}: ξ ∈ ω_{1}} is a Δ-system with root q; of course then q ≤ p.

Let us set for each ξ ∈ ω_{1}

r_{ξ} = p_{ξ} ∩ M,

and define the C_{κ}-names Ȧ and Ḃ as follows:

Ȧ = {⟨p_{ξ}, α_{ξ}⟩: ξ ∈ ω_{1}}, Ḃ = {⟨r_{ξ}, α_{ξ}⟩: ξ ∈ ω_{1}}.

Clearly, q^{∥}—Ȧ ∈ [Ẋ]^{ω_{1}} & Ȧ ⊂ Ḃ ⊂ N, hence in view of (i) it will suffice to show that q forces part c) of 1.2. This, on the other hand is an easy consequence of the fact that every extension of q can only be incompatible with only finitely many of the conditions p_{ξ}.

Let us now introduce the following piece of notation.

Definition 1.4. For any regular cardinal λ we denote by N_{λ} the collection of those elementary submodels N ≺ H(λ) that satisfy |N| = ω_{1} ⊂ N as well as (i) and (ii) of 1.3.

Finally, we are ready to formulate our principle CH*.

Definition 1.5. CH* is the statement that for every large enough regular cardinal λ the collection N_{λ} is cofinal in [H(λ)]^{ω_{1}}.

Clearly, CH* is a consequence of CH, moreover from 1.1 and 1.3 we immediately get the following result.

Theorem 1.6. If V ⊧ CH then for any cardinal κ we have V^{C_{κ}} ⊧ CH*.

In fact, what can be shown is that for any regular λ > κ, in V^{C_{κ}}, the collection N_{λ} is even stationary in [H(λ)]^{ω_{1}}.

2. The Main Lemma

It will turn out that all the results that we deduce from CH* make use of a single lemma concerning initially ω_{1}-compact and countably tight T_{3} spaces belonging to an elementary submodel N ∈ N_{λ}. Therefore we call this our Main Lemma. Before formulating and proving it, however, we have to give several auxiliary results.

Lemma 2.1. Let N be an elementary submodel of some H(λ) with ω_{1} ⊂ N and X be a topological space such that X ∈ N. Assume that A, B are subsets of X with \overline{A} ∩ \overline{B} = ∅ and Ã, B̃ are N-covers of A and B, respectively. Then |Ã ∩ B̃ ∩ X| ≥ ω implies that there is an ω_{2}-dyadic system of closed sets in X.

Proof. By our assumption there is a set T ⊂ Ã ∩ B̃ ∩ X with |T | = ω. In view of 1.2 b) there are sets S_{A} ∈ N ∩ [Ã]^{ω} and S_{B} ∈ N ∩ [B̃]^{ω} with T ⊂ S_{A} ∩ S_{B}. Then, clearly, S = S_{A} ∩ S_{B} ∩ X is also in N and |S| = ω as T ⊂ S.

The ω_{2}-dyadic system will be of the form \{\overline{S_{α}^{i}}: ⟨α, i⟩ ∈ ω_{2} × 2\} with S_{α}^{i} ⊂ S for all ⟨α, i⟩. It can be defined by a straightforward transfinite recursion using the following claim.

Claim . If T ⊂ [S]^{ω} with |T | ≤ ω_{1} then there are sets S_{0}, S_{1} ⊂ S with \overline{S_{0}} ∩ \overline{S_{1}} = ∅ such that |S_{0} ∩ T | = |S_{1} ∩ T | = ω for each T ∈ T .

Assume, indirectly, that there is a collection T contradicting the claim. Since both X and S belong to N, by elementarity, we may assume that T ∈ N as well. But then ω_{1} ⊂ N and |T | ≤ ω_{1} imply that T ⊂ N. So for each T ∈ T we have T ∈ N ∩ [Ã]^{ω} and T ∈ N ∩ [B̃]^{ω}, hence part c) of 1.2 implies |T ∩A| = |T ∩B| = ω. But then S_{0} = S ∩A and S_{1} = S ∩ B clearly satisfy \overline{S_{0}} ∩ \overline{S_{1}} = ∅ and |S_{0} ∩ T | = |S_{1} ∩ T | = ω for each T ∈ T , contradicting the choice of T . This, in turn, completes the proof of the claim and thus of lemma 2.1.

From 2.1 we easily deduce the following lemma.

Lemma 2.2. Assume N ∈ N_{λ} for some regular λ, moreover X is a regular space with X ∈ N and not containing an ω_{2}-dyadic system of closed sets. We also assume that the underlying set of X is an ordinal. Then, if A ∈ [N ∩X]^{ω_{1}} and Ã ⊂ X is an N-cover of A, every complete accumulation point of Ã in X belongs to \overline{A}, the closure of A.

Proof. Let x be a complete accumulation point of Ã and assume that x ∉ \overline{A}. Then, by the regularity of X, we have an open set U with x ∈ U and \overline{U} ∩ \overline{A} = ∅.

We also have |U ∩ Ã| = ω_{1}, hence applying (ii) of 1.3 there is a set B ∈ [U ∩Ã]^{ω_{1}} that has an N-cover B̃. But then we also have \overline{A}∩\overline{B} = ∅, moreover B ⊂ Ã∩B̃∩X, hence |Ã∩B̃∩X| = ω_{1}. Consequently Lemma 2.1 applies and yields an ω_{2}-dyadic system of closed sets in X, which is impossible.

Remark . It can be shown that in 2.2 every complete accumulation point of Ã is actually a complete accumulation point of A, however we will not use this stronger result.

The following purely topological lemma is folklore, but we supply its proof because perhaps it has not been published in this generality.

Lemma 2.3. Assume that X is a topological space with t(X) ≤ F (X) = ω and F is an ω_{1}-directed family of closed sets in X. Then for any A ⊂ X with \overline{A} ∈ F^{+} (i.e. \overline{A}∩F ≠ ∅ for all F ∈ F) there is a countable set S ⊂ A with \overline{S} ∈ F^{+}.

Proof. Assume, indirectly, that \overline{A} ∈ F^{+} but for every countable S ⊂ A we have \overline{S} ∉ F^{+}. Then by transfinite recursion on α ∈ ω_{1} we define points p_{α} ∈ \overline{A}, sets S_{α} ∈ [A]^{≤ω} and closed sets F_{α} ∈ F as follows.

If α ∈ ω_{1} and p_{β}, S_{β}, F_{β} have been defined for all β ∈ α then \overline{∪\{S_{β}: β ∈ α\}} ∉ F^{+}, hence we can pick F_{α} ∈ F^{+} such that F_{α} ∩ \overline{∪\{S_{β}: β ∈ α\}} = ∅ and F_{α} ⊂ ∩\{F_{β}: β ∈ α\} by the ω_{1}-directedness of F.

Let p_{α} ∈ F_{α} ∩ \overline{A} and, using t(X) ≤ ω, choose S_{α} ∈ [A]^{≤ω} such that p_{α} ∈ \overline{S_{α}}. This completes the recursion.

Now, it is obvious that {p_{α}: α ∈ ω_{1}} is a free sequence in X, contradicting F (X) = ω.

We are now ready to prove our main lemma.

Lemma 2.4 (Main Lemma). Let X be an initially ω_{1}-compact T_{3} space with t(X) ≤ ω and F be an ω_{1}-directed family of closed sets in X. We also assume that the underlying set of X is an ordinal. Assume that N ∈ N_{λ} and X, F ∈ N, moreover set

F_{N} = \{\overline{F ∩ N}: F ∈ F ∩ N\}.

Then

∩F_{N} ⊂ ∩F.

Proof. It is well-known that the assumptions on X imply F (X) = ω. Also, there is no ω_{1}-dyadic system of closed sets in X. Indeed, otherwise we could find a continuous map f of some closed subspace Y of X onto 2^{ω_{1}}, as e.g. in [3], p. 70. Since f preserves initial ω_{1}-compactness and w(2^{ω_{1}}) = ω_{1} it would follow that f is also a closed map. However, as is well-known, closed maps preserve countable tightness, hence we are led to the absurd conclusion that t(2^{ω_{1}}) = ω!

Now let x ∈ ∩F_{N}; to show that x ∈ ∩F by the regularity of X it clearly suffices to show that for any open neighbourhood U of x we have \overline{U} ∈ F^{+}. To this end let us fix a smaller neighbourhood V of x with \overline{V} ⊂ U.

Using 1.3 (i) and the ω_{1}-directedness of F it is easy to see that F∩N is also ω_{1}-directed. Consequently one can easily define a sequence {F_{α}: α ∈ ω_{1}} ⊂ F ∩ N such that α < β implies F_{α} ⊃ F_{β} and for every F ∈ F ∩N there is α ∈ ω_{1} with F_{α} ⊂ F (note that |F ∩N| ≤ |N| = ω_{1}).

Now let us pick for each α ∈ ω_{1} a point x_{α} ∈ V ∩ F_{α} ∩ N, this is possible because x ∈ \overline{F_{α} ∩ N} and x ∈ V. Note that then for every set a ∈ [ω_{1}]^{ω_{1}} we have {x_{α} : α ∈ a} ∈ F_{N}^{+}. Thus if there is a point y with |{α ∈ ω_{1} : x_{α} = y}| = ω_{1}, then {y} ∈ F_{N}^{+}, hence using y ∈ N also \{y\} ∈ F^{+}. But then y ∈ V ⊂ U implies \overline{U} ∈ F^{+} as well.

If, on the other hand, for all y we have |{α ∈ ω_{1} : x_{α} = y}| ≤ ω then Y = {x_{α}: α ∈ ω_{1}} ∈ [N ∩ X]^{ω_{1}} hence using (ii) of 1.3 there is a set A ∈ [Y]^{ω_{1}} that has an N-cover Ã. Since X ∈ N we can clearly assume that Ã ⊂ X, because Ã ∩ X is also an N-cover of A.

Now all the assumptions of Lemma 2.2 are satisfied, hence every complete accumulation point of Ã is contained in \overline{A} ⊂ \overline{V} ⊂ U. Consequently, as X is initially ω_{1}-compact, we must have |Ã \ U| ≤ ω. Let us fix a set T ∈ N ∩ [Ã]^{ω} such that Ã \ U ⊂ T.

Then Ã\ T ∈ F_{N}^{+}, hence applying Lemma 2.3 and property b) of the N-covers we can find a set S ∈ N ∩ [Ã \ T]^{ω} such that \overline{S} ∈ F_{N}^{+} as well. But then S ∈ N, hence \overline{S} ∈ N imply that \overline{S} ∈ F^{+}, consequently from S ⊂ U we also get \overline{U} ∈ F^{+}, and thus the proof is completed.

3. Applications of CH*

In this section we are going to present several interesting consequences of CH*. Since CH* implies CH, it is natural that most of these consequences, though not all, have been known to follow from CH. Many of them were also proven to hold in generic extensions obtained by adding ω_{2} Cohen reals to a ground model satisfying CH, especially in [1] that gave the main impetus to our present work.

The first result of this section was proved under CH, independently, by van Dowen and Dow, see e.g. [1].

Theorem 3.1. (CH*) Every initially ω_{1}-compact T_{3} space X of countable tightness is compact.

Proof. We may of course assume that the underlying set of X is an ordinal. Now let F be any maximal filter of closed sets in X, we shall show that ∩F ≠ ∅.

To this end we apply CH* and choose a large enough regular λ and an elementary submodel N ∈ N_{λ} with both X, F ∈ N. Now, since X is initially ω_{1}-compact and F is maximal, F is clearly even ω_{2}-directed. Also, if F is a closed set with F ∈ F^{+} then actually F ∈ F. So using Lemma 2.3 for F we get that for every F F there is an S ∈ [F]^{≤ω} such that \overline{S} ∈ F as well. Reflecting this statement down to N we may now conclude that for every F ∈ F ∩ N there is a set S ∈ N ∩ [F]^{≤ω} such that \overline{S} ∈ F ∩ N as well. But from S ∈ N we have S ⊂ N as well, so \overline{S} = \overline{\overline{S} ∩ N} ∈ F_{N}. Consequently we clearly have ∩(F ∩ N) = ∩F_{N}, and here the left hand side is non-empty because |F ∩ N| ≤ ω_{1} and X is initially ω_{1}-compact. Thus we have by our Main Lemma 2.4 that ∩F ⊃ ∩F_{n} ≠ ∅ and this completes the proof that X is compact.

Our next result seems to be new even under CH. It concerns compact T_{2} spaces of countable tightness (in s spaces), but of course, in view of 3.1, it could be formulated for initially ω_{1}-compact T_{3} spaces instead of compact T_{2} ones.

Theorem 3.2. (CH*) Let X be a CCT space and S ⊂ X be G_{δ}- dense in X, i.e. S ∩ H ≠ ∅ for every non-empty G_{δ} set H ⊂ X. Then X = ℓ_{ω_{1}}(S), i.e. every point x of X is the limit of a converging ω_{1}-sequence of points of S.

Proof. If x ∈ S this is trivial, so assume x ∈ X \S. (Of course, we may assume again that X ⊂ On.) Let F be the family of all those closed G_{δ} sets in X that contain x.

Using CH* choose a regular λ and an elementary submodel N ∈ N_{λ} such that X, F, x, S ∈ N. Since F is clearly ω_{1}-directed, the Main Lemma 2.4 can be applied, hence we have

∅ ≠ ∩F_{N} ⊂ ∩F = {x}.

Consequently, for each F ∈ F ∩N we have x ∈ \overline{F ∩ N}. In fact, we claim that x ∈ \overline{F ∩ N ∩ S} is also true. Indeed, first of all we have x ∈ \overline{F ∩ S} since otherwise we had an open set G with x ∈ G and G ∩ F ∩ S = ∅ which is impossible as G ∩ F is a non-empty G_{δ} set. So by countable tightness of X there is a countable set T ⊂ F ∩ S with x ∈ \overline{T} and by elementarity and x, S, F ∈ N we may assume that T ∈ N as well. But then T ⊂ N is also true, hence indeed

x ∈ \overline{T} ⊂ \overline{F ∩ N ∩ S}.

Now, similarly as in the proof of 2.4 we can define a decreasing ω_{1}- sequence {F_{α}: α ∈ ω_{1}} ⊂ F ∩ N such that for every F ∈ F ∩ N we have F_{α} ⊂ F for some α ∈ ω_{1}. But then we clearly have

x ∈ ∩\{\overline{F_{α} ∩ N ∩ S}: α ∈ ω_{1}\} ⊂ ∩F_{N} = \{x\},

hence if x_{α} ∈ F_{α}∩N ∩S for each α ∈ ω_{1} then the ω_{1}-sequence {x_{α}: α ∈ ω_{1}} ⊂ S must converge to x, and the proof is completed.

The following two results were established in [1] in V^{C_{ω_{2}}} for a ground model V satisfying CH.

Corollary 3.3. (CH*) If X is CCT and p ∈ X with χ(p, X) > ω then p ∈ ℓ_{ω_{1}}(X \ {p}), i.e. p is a nontrivial ω_{1}-limit in X.

Proof. This is immediate from 3.2 because χ(p, X) > ω is clearly equivalent to X \ {p} being G_{δ}-dense in X.

Corollary 3.4. (CH*) If X is CCT and the subspace Y ⊂ X is pseudocompact, then \overline{Y} = ℓ_{ω_{1}}(Y).

Proof. Indeed, this follows from 3.2 because if Y is pseudocompact then it is G_{δ}-dense in \overline{Y}.

The following result was know problem under CH and was proven, under CH, in [6]. As it turns out almost the same proof now yields the same conclusion from CH*.

Theorem 3.5. (CH*) If a compact T_{2} space X has small diagonal then X is metrizable.

Proof. If t(X) > ω held then, by the main result of [6], X would contain a nontrivial convergent ω_{1}-sequence which contradicts having small diagonal.

So X is CCT and consequently so is X_{Δ} obtained by collapsing the diagonal Δ to a single point in X^{2}. Indeed, this is so because the

8 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

CCT property is preserved under products and continuous images. But then we must have χ(Δ, X_{Δ}) = ω, since otherwise, by 3.3, X_{Δ} \ {Δ} would contain an ω_{1}-sequence converging to Δ, hence, equivalently, X^{2} \ Δ would contain such a sequence converging to the diagonal Δ, contradicting that X has small diagonal. However, we clearly have χ(Δ, X^{2}) = χ(Δ, X_{Δ}), hence X has a G_{δ} diagonal and so is metrizable.

The following consequence of CH* is again easily deduced from our main lemma. It will, however, play an essential role below.

Theorem 3.6. (CH*) Assume that X is CCT and F is an ω_{1}-directed family of closed sets in X such that d(F) ≤ ω_{1} for each F ∈ F. Then there is a subfamily F′ ∈ [F]^{≤ω_{1}} such that ∩F′ = ∩F.

Proof. We may assume again that the underlying set of X is an ordinal and from CH* we can find some regular λ and N ∈ N_{λ} with both X, F ∈ N.

Now, for each F ∈ F ∩ N by elementarity there is a dense subset S of F with |S| ≤ ω_{1} and S ∈ N. From ω_{1} ⊂ N then we also have S ⊂ N, hence \overline{F ∩ N} = \overline{S} = F. In other words, we have F ∩ N = F_{N}, so by the main lemma F′ = F ∩ N is as required.

An immediate consequence of 3.6 is that in a CCT space X there is no strictly decreasing ω_{2}-sequence of closed sets of density ≤ ω_{1}. This fact is used in the next result.

Theorem 3.7. (CH*) If X is CCT and Y ⊂ X satisfies s(Y) ≤ ω_{1} then h(Y) ≤ ω_{1} as well.

Proof. Applying [3, 2.26] just as in the proof of [3, 3.13], we have z(Y) = hd(Y) ≤ s(Y) · t(X)^{+} ≤ ω_{1} by our assumptions. Assume, indirectly, that h(Y) > ω_{1}, then there is a strictly decreasing ω_{2}- sequence {H_{α}: α ∈ ω_{2}} of closed subsets of Y. But then the sequence \{\overline{H_{α}}: α ∈ ω_{2}\} is also strictly decreasing, moreover we have d(\overline{H_{α}}) ≤ d(H_{α}) ≤ ω_{1}, and this is impossible by our above remark.

Our last consequence of CH* concerns complete binary trees of closed sets in a CCT space X. The proof will use 3.6 and some ZFC results that seem to be new and interesting in themselves.

A complete binary tree of closed sets of height κ in a space X (or simply a Cantor κ-tree in X) is a collection {F_{t}: t ∈ 2^{<κ}} where each F_{t} is a nonempty closed set in X such that t ⊃ s implies F_{t} ⊂ F_{s} and

Ft0^{⌢} ∩ Ft1^{⌢} = ∅ for each t ∈ 2^{<κ}.

We denote by T r_{κ}(X) the family of all Cantor κ-trees in X. Our aim now is to show that if κ is regular and X is a CCT space in which there is a Cantor κ-tree, then there is also one whose members are all separable. First, however, we prove the following lemma.

Lemma 3.8. Assume cf(κ) > ω, X is a CCT space and T = {F_{t}: t ∈ 2^{<κ}} ∈ T r_{κ}(X). Then there is a countable set S ⊂ X and an index t ∈ 2^{<κ} such that for each s ∈ 2^{<κ} with s ⊃ t we have \overline{S} ∩ F_{s} ≠ ∅.

Proof. Assume, indirectly, that there are no such S and t. We may then define by transfinite recursion on α ∈ ω_{1} points p_{α} ∈ X and indices t_{α} ∈ 2^{<κ} with p_{α} ∈ F_{t_{α}} and t_{α_{1}} ⊂ t_{α_{2}} for α_{1} < α_{2} as follows.

Suppose that α ∈ ω_{1} and {p_{β}, t_{β}: β ∈ α} have already been chosen with these properties. From cf(κ) > ω it follows that t = ∪{t_{β} : β ∈ α} ∈ 2^{<κ}, hence by the indirect assumption there is an extension t_{α} ⊃ t such that

\overline{\{p_{β}: β ∈ α\}} ∩ F_{t_{α}} = ∅.

We then pick p_{α} ∈ F_{t_{α}} arbitrarily.

Clearly, this construction goes through for all α ∈ ω_{1}, and it is also obvious that the sequence {p_{α}: α ∈ ω_{1}} obtained in this way is free. But this contradicts that X is CCT.

Note that this proof only used F (X) = ω. In the following result, however, the compactness of X is used essentially.

Theorem 3.9. If κ is an uncountable regular cardinal and X is a CCT space with T r_{κ}(X) ≠ ∅ then there is a Cantor κ-tree T̃ ∈ T r_{κ}(X) such that every element of T̃ is separable.

Proof. Let T = {F_{t}: t ∈ 2^{<κ}} be a Cantor κ-tree in X. We denote by H the collection of all pairs ⟨H, t⟩ where H is a closed set in X, t ∈ 2^{<κ} moreover for any s ∈ 2^{<κ} with s ⊃ t we have F_{s} ∩ H ≠ ∅. We also consider the following partial order ≺ on H:

⟨H, t⟩ ≺ ⟨H′, t′⟩ if H ⊂ H′ and t ⊃ t′.

Note that for any t ∈ 2^{<κ} we clearly have ⟨F_{t}, t⟩ ∈ H.

Let us now list several properties of this partial ordering ⟨H, ≺⟩.

1) For every ⟨H, t⟩ ∈ H there are ⟨H_{0}, t_{0}⟩, ⟨H_{1}, t_{1}⟩ ∈ H such that H_{0} ∪ H_{1} ⊂ H, H_{0} ∩ H_{1} = ∅ and t ⊂ t_{0}, t ⊂ t_{1}.

Indeed, we may simply put t_{i} = ti^{⌢} and H_{i} = H ∩ F_{t_{i}} for i ∈ 2.

2) If ν ∈ κ and {⟨H_{α}, t_{α}⟩: α ∈ ν} is a decreasing sequence in H (i.e. α ∈ β ∈ ν imply ⟨H_{β}, t_{β}⟩ ≺ ⟨H_{α}, t_{α}⟩) then there is some ⟨H, t⟩ ∈ H with ⟨H, t⟩ ≺ ⟨H_{α}, t_{α}⟩ for all α ∈ ν. In other words, the poset ⟨H, ≺⟩ is κ-closed.

Indeed, this is immediate from the regularity of κ and the compactness of X; we may just set H = ∩{H_{α}: α ∈ ν} and t = ∪{t_{α}: α ∈ ν}.

10 I. JUHÁSZ, L. SOUKUP, AND Z. SZENTMIKLÓSSY

3) Finally, for any ⟨H, t⟩ ∈ H there is a ⟨K, s⟩ ∈ H with ⟨K, s⟩ ≺ ⟨H, t⟩ such that K is separable.

Indeed, by our definition, {F_{r} ∩ H: r ⊃ t} is clearly a Cantor κ-tree in H. But then we may apply Lemma 3.8 which yields a countable set S ⊂ H and a sequence s ⊃ t such that F_{r} ∩\overline{S} ≠ ∅ for each r ⊃ s, hence ⟨\overline{S}, s⟩ is as required.

Of course, 3) just says that K = {⟨K, s⟩ ∈ H: d(K) = ω} is dense in the poset H.

Now, it is standard to construct using properties 1)-3) a full binary subtree {⟨K_{t}, s_{t}⟩: t ∈ 2^{<κ}} of K such that for each t ∈ 2^{<κ} we have Kt0^{⌢} ∩ Kt1^{⌢} = ∅. Consequently, the first ”co-ordinates” {K_{t}: t ∈ 2^{<κ}} of this tree yield us a Cantor κ-tree in X consisting of separable sets, as was required.

Since the weight of a separable T_{3} space is at most c, it may not contain a well-ordered decreasing sequence of closed sets of length c^{+}. Consequently we immediately have the following ZFC result from 3.9.

Corollary 3.10. If X is a CCT space then T r_{c^{+}}(X) = ∅.

It should be noted that, as is shown e.g. by the Alexandrov one-point compactifications of large discrete spaces, there is no upper bound for the lengths of decreasing sequences of closed sets in CCT spaces.

Of course, under CH we have c^{+} = ω_{2} in 3.10. Combining 3.6 with 3.9 we can see that ω_{2} works as an upper bound under CH^{∗} as well.

Corollary 3.11. (CH*) For every CCT space X we have T r_{ω_{2}}(X) = ∅.

The standard proof of the Čech–Pospišil theorem (see e.g. [3, 3.16],) yields the following statement: If X is a compact T_{2} space such that χ(p, X) ≥ κ holds for each p ∈ X then T r_{κ}(X) ≠ ∅. Hence 3.11 implies that, under CH*, in every CCT space there is a point of character at most ω_{1}. It was conjectured in [4] that this consequence may be provable in ZFC. As a matter of fact, it may turn out that even 3.11 is provable in ZFC!

Also, the above conjecture was shown to hold in V^{C_{ω_{1}}} for any ground model V. We don’t know if the same can be proved about 3.11.

We conclude this paper with a result that says that under PFA even ω_{1} is a (strict) upper bound for the height of a Cantor tree in a CCT space. In view of our above remarks this constitutes a strengthening of A. Dow’s result saying that, under PFA, every CCT space has a point of countable character. The proof is quite similar to the proof of this result of Dow given in [5] for .

Theorem 3.12. (PFA) If X is a CCT space then T r_{ω_{1}}(X) = ∅.

Proof. Assume, indirectly, that T ∈ T r_{ω_{1}}(X). Then, if P is the standard countably closed notion of forcing that collapses |X| to ω_{1}, X is no longer compact in V^{P}. Indeed, there will be many branches of T with empty intersection. On the other hand, as was shown in the proof of 3.3 in [5], X remains both countably compact and countably tight in V^{P}. Hence, by Balogh’s lemma (see 3.1 in [5]), there is a proper poset Q that adds a free sequence of length ω_{1} to X. But then PFA applied to P ∗ Q implies that X contains such a sequence already in V, a contradiction.

References

[1] A. Dow, Compact spaces of countable tightness in the Cantor models, Set Theory and its Applications, (Toronto, 1987), Lectures Notes in Math., 1401 (1989), 55-67, Springer, Berlin–New York,1989.

[2] A. Dow, F. Tall, B. Weiss, New proofs of the consistency of the normal Moore space conjecture I., Top. Appl. 37 (1990), no 1, 33-51.

[3] I. Juhász, Cardinal functions – ten years later, Math. Centre Tract, 123, Amsterdam, 1980.

[4] I. Juhász On the minimum character of points in compact spaces,in: Proc. 1989 Top. Conf. Pécs, Coll. Math. Soc. J. Bolyai 55. 1993, 365-371.

[5] I. Juhász, Cardinal functions, In: Recent Progress in General Topology, Papers from Prague Toposym 1991, ed.: M. Husěk and J. van Mill, North Holl., Amsterdam, (1992) 419–441.

[6] I. Juhász, Z. Szentmiklóssy, Convergent free sequences in compact spaces, Proc. AMS. vol 116 (1992) 1153-1160.

[7] S. Shelah, Proper Forcing.