ARE INITIALLY ω_{1}-COMPACT SEPARABLE REGULAR SPACES COMPACT?

ALAN DOW AND ISTVAN JUHÁSZ

Abstract. In this paper we investigate the question of the title. While it is immediate that CH yields a positive answer we discover that the situation under the negation of CH holds some surprises.

1. Introduction

A topological space is said to be initially ω_{1}-compact if every open cover of cardinality at most ω_{1} has a finite subcover. This is well-known to be equivalent (in the context of Hausdorff spaces) to the property that the space is countably compact and each subset A of cardinality ω_{1} has a complete accumulation point, that is, a point whose every neighbourhood contains a set of points of A with cardinality |A|. Analogous to converging ω-sequences, we say that, for a cardinal κ, a sequence ⟨x_{α} : α < κ⟩ converges to a point x, if for each neighbourhood U of x, there is a β < κ such that {x_{α} : β < α < κ} ⊂ U. We say that a point x is a κ limit if there is some κ sequence of points distinct from x which converges to x.

Since a regular separable space has a base for the topology of cardinality at most 𝔠, it follows immediately from CH that a separable regular space is compact if it is initially ω_{1}-compact. If the space is in addition first countable (or even has countable tightness) then this implication has been shown to hold in several models of ¬CH and only recently been shown to consistently fail. However there is a very well known class of examples, discussed below, which in many models of ¬CH are themselves non-compact initially ω_{1}-compact separable regular spaces. Hence it seemed reasonable to suppose that the failure of CH would imply the existence of such a space. In fact if 𝔠 has cofinality greater than ω_{1} then this is indeed the case and the question is not decided by the assertion that 𝔠 has cofinality ω_{1}.

For the remainder of the paper we let (I)_{C} denote the assertion “every separable initially ω_{1}-compact regular space in the class C is compact”. If we omit the subscript C then we interpret C as the entire class of regular spaces. We will use LC to denote the class of locally compact spaces, SC to denote the class of sequentially compact spaces, and CT will be the class of spaces with countable tightness. A space is sequentially compact if every infinite set contains a converging sequence. A space is said to have countable tightness if the closure of every set is equal to the union of the closures of its countable subsets.

2. On (I)_{C} for special C

As mentioned above the following result is immediate.

Research supported by the Hungarian National Foundation for Scientific Research grant no.

16391.

Proposition 2.1 (CH). (I), i.e. every initially ω_{1}-compact separable regular space is compact.

Part of the interest in this topic stems from the recent investigations into spaces of countable tightness which we now summarize.

Proposition 2.2. (I)CT follows from PFA ([2]) and also holds in any model obtained by adding Cohen reals to a model of CH ([4]). However (I)CT can fail ([10]) and there can even be a first countable example ([8]).

A well-known class of spaces are the so-called Franklin-Rajagopalan spaces [11], see also [13, §7] or [14, 2.11].

These are constructed as follows.

Example 1. Fix a maximal strictly increasing mod finite well-ordered chain of coinfinite subsets of ω, say {a_{α} : α < κ}. We assume that κ is a regular cardinal. Let B be the Boolean subalgebra of P(ω) which is generated by this chain together with the finite subsets of ω and set K to be the Stone space of ultrafilters on B. The space K can be regarded as ω ∪{x_{α} : α ≤ κ} in which ω is dense, open and discrete, while {x_{α} : α ≤ κ} is homeomorphic to the ordinal space κ + 1. By the maximality of the chain, it follows that every infinite subset of ω has a subsequence converging to some x_{α} with α < κ. It follows that both K and K \ {x_{κ}} are sequentially compact.

Proposition 2.3. If there is a maximal chain mod finite of co-infinite subsets of ω with cofinality greater than ω_{1} then there is a non-compact initially ω_{1}-compact separable regular space which is also locally compact and sequentially compact, i.e. (I)LC and (I)SC both fail.

Proof. If there is such a chain of length κ > ω_{1} then let K be the space constructed in Example 1 and let x_{κ} denote the same point as above. Simply set X equal to the space K with the point x_{κ} removed. To see that X is initially ω_{1}-compact we just note that no subset of X of cardinality at most ω_{1} converges, in K, to x_{κ}.

Clearly then 𝔱 > ω_{1}, hence MA(ω_{1}), implies the failure of (I)SC and (I)LC. It is well known that if Cohen reals are added to a model of CH then there are no such well-ordered mod finite chains. In addition, (I)SC holds in such models because of the following result which is taken from Theorem 6 of [5].

Proposition 2.4. If Cohen reals are added to a model of CH then every ω_{1}- compact, sequentially compact, separable regular space is compact. That is, (I)SC holds in all such models.

It is considerably easier to get (I)LC to fail. If p is any point of βω \ ω then the space βω \ {p} is an example of a locally compact initially ω_{1}-compact space which is not compact if and only if p is not an ω_{1}-limit in βω. More generally, if K is a separable compact space in which there is a non-isolated point, say x, which is not an ω-limit nor an ω_{1}-limit in K then X = K \ {x} is also such an example. However we have the following two results.

Theorem 2.5. If uncountably many Cohen reals are added (to any model of ZFC) then every point p of βω \ ω is an ω_{1}-limit in βω.

Theorem 2.6. If any number of Cohen reals are added to a model of CH then (I)LC holds. In fact, for every compact separable space K every point p of K is an ω_{1}-limit in K if and only if p does not have a countable local base.

For a non-empty set S, we let Fn(S, 2) denote the usual Cohen poset consisting of finite partial functions from S into 2. For a Fn(S, 2)-name, τ, of a subset of ω, let the support of τ be the smallest (countable) subset R of S such that for each n ∈ ω and p ∈ Fn(S, 2),

p ⊩ n ∈ τ iff p ↾ R ⊩ n ∈ τ .

Proof of Theorem 2.5. For this result we may assume that we are forcing with the poset Fn(ω_{1}, 2). In the extension, let p be an ultrafilter on ω and for each α < ω_{1}, let F_{α} denote the filter base consisting of all those members F of p for which there is a Fn(ω_{1}, 2)-name in the ground model with support contained in α (such that F is equal the interpretation of the name by the generic filter). It is easily seen that we can, for each α < ω_{1}, choose an ultrafilter q_{α} ∈ βω \{p} such that F_{α} ⊂ q_{α}. The sequence ⟨q_{α} : α < ω_{1}⟩ converges to p since for each A ∈ p, there is an γ (containing the support of a name for A) such that A ∈ q_{α} for all α ≥ γ.

Proof of Theorem 2.6. We first establish that it suffices to show the following Lemma.

Lemma 2.7. Let V be a model of CH and G be a Fn(κ, 2)-generic filter. If, in V [G], F is a filter on ω which is not countably generated then there are filters F_{α} (α < ω_{1}) such that every member of F is a member of all but countably many of the F_{α} and there is some a ∈ F_{α} such that ω − a ∈ F.

Indeed, suppose that X is a compact space and that ω is a dense subset of X. Also suppose that x in X does not have a countable local base. Set F to be the filter of all subsets F of ω such that x is not in the closure of ω\F, i.e. F is the trace of the neighbourhood filter of x on ω. It should be clear that F is not countably generated. Let {F_{α} : α ∈ ω_{1}} be as above. For each α, choose a point x_{α} in X which is in the intersection ⋂{cl_{X}(a) : a ∈ F_{α}}. We now verify that the sequence ⟨x_{α} : α ∈ ω_{1}⟩ converges to x. We prove this by showing that x_{α} ∈ U for all but countably many α (it should be clear that x_{α} ≠ x for each α). Of course if W ⊂ U is a closed neighbourhood of x, then a = W ∩ ω is a member of F. In addition, for each α such that a ∈ F_{α}, we have that x_{α} ∈ cl_{X}(a) ⊂ W, hence x_{α} ∈ U.

Proof of Lemma 2.7. Let F be a Fn(κ, 2)-name of a filter on ω and, with no loss of generality, assume that 1 forces that F has no base of cardinality less or equal ω_{1}. Recursively choose ω_{2} names for members of F as follows. For each α, let 1 force that Ḟ_{α} ∈ F and Ḟ_{α} does not contain any member of F which has a Fn(I_{α}, 2)- name, where I_{α} is a set of size at most ℵ_{1} which contains the support of each Ḟ_{β} for β < α.

Since V ⊧ CH, we can assume that the above sequence has been thinned down so that the supports of the Ḟ_{α} form a Δ-system with root R and that all the Ḟ_{α} are identical wrt R. Let J_{α} denote the countable set which is equal to the support of Ḟ_{α} with R removed. Hence the family {J_{α} : α < ω_{1}} are pairwise disjoint. Note that our assumption on the enumeration ensures that Ḟ_{α} does not contain any member of F whose support is contained in R. By passing to the extension obtained by forcing with Fn(R, 2) we can assume that the support of Ḟ_{α} is equal to J_{α} (hence the supports are pairwise disjoint ).

We are ready to define F_{ξ} for ξ < ω_{1}.

F_{ξ} is generated by those ȧ such that either ȧ = ω − Ḟ_{ξ} or some condition forces that ȧ ∈ F and the support of ȧ is disjoint from J_{ξ}.

Since the generators of the F_{ξ} of the latter kind are members of F, they are closed under finite intersections (the support of an intersection is contained in the support of the union). Therefore to show that F_{ξ} has the finite intersection property it suffices to show that ȧ − Ḟ_{ξ} is not empty for each ȧ ∈ Ḟ such that the support of ȧ is disjoint from J_{ξ}. Otherwise there is a condition p which forces that ȧ is contained in F_{ξ}. Since these sets have disjoint supports, it follows that p forces that F_{ξ} contains the set {n : (∃q < p) q ⊩ n ∈ ȧ}. However this latter set is in the ground model V [G ∩ Fn(R, 2)] and is a member of F. By our assumption, 1 ⊩ F_{ξ} does not contain any member of F which is a member of V [G ∩ Fn(R, 2)].

3. More on the Failure of (I)

This section culminates with the result that (I) fails whenever the cofinality of 𝔠 is greater than ω_{1}. However we first study some other conditions which imply that (I) fails and which seem to be of independent interest. When we use the symbol 2^{κ} we will mean the usual product topological space. In this context we will use (2^{κ})_{0} to denote all those members of the product which are eventually equal to 0 and the obvious analogue, (2^{κ})_{f}, for arbitrary f ∈ 2^{κ}.

The following lemma is our main tool for constructing examples to witness the failure of (I).

Lemma 3.1. Suppose that cf(κ) > ω_{1} and that S ∈ [2^{κ}]^{ω} is such that 1. for all infinite T ⊂ S, cl(T) ∩ (2^{κ})_{0} is not empty, and

2. S has a limit point not in (2^{κ})_{0}. Then, for some g ∈ 2^{κ}, the subspace X = (S \ {g}) ∪ (cl(S) ∩ (2^{κ})_{0}) is a witness to the failure of (I).

Proof. We choose g ∈ 2^{κ} \ (2^{κ})_{0} to be any limit of S (which exists by 2). It follows that X is not compact since it is not closed in 2^{κ}. Obviously X is separable since S \ {g} is dense in X. By the hypothesis 1, every infinite subset of S has a limit point in X ∩ (2^{κ})_{0}. Therefore, to show that X is initially ω_{1} compact, it suffices to show that (2^{κ})_{0} is initially ω_{1} compact. This follows immediately from the fact that if A ⊂ (2^{κ})_{0} has cardinality at most ω_{1} then \overline{A} ⊂ (2^{κ})_{0}. Indeed, since cf(κ) > ω_{1}, there is a λ < κ such that every member of A is constantly 0 on [λ, κ). Therefore A is contained in the closed set 2^{λ} × ⟨0, 0, 0, … ⟩.

It is a natural strengthening of the hypotheses of the previous Lemma to ask that cl(T) ∩ (2^{κ})_{f} ≠ ∅ for all infinite T ⊂ S, for all f ∈ 2^{κ}. This is very closely related to that of an independent splitting family on ω (see [12]). As we see by the following definition we also have a very close parallel to the notion of a κ-HFD (see [7, 6.1]).

Definition 3.2. An infinite subset S of 2^{κ} is a (κ, λ)-HFD if for every infinite T ⊂ S, there is set J ⊂ κ such that |κ \ J| < λ and {t ↾ J : t ∈ T } is dense in 2^{J}.

We will say that S is a (κ, →)-HFD if for every infinite T ⊂ S, there is an α < κ such that {t ↾ (κ \ α) : t ∈ T } is dense in 2^{(κ\α)}. Of course the notion of a (κ, κ)-HFD coincides with that of a (κ, →)-HFD in the case that κ is regular.

The hypotheses of Lemma 3.1 are easily seen to be fulfilled if S is a (κ, →)-HFD, hence we have the following result.

Corollary 3.3. The failure of (I) follows from the existence of a (κ, →)-HFD for any cardinal κ with cofinality greater than ω_{1}.

The existence of (κ, ω_{1})-HFD’s has been established in models obtained by Cohen forcing (see [6]).

Proposition 3.4. Forcing with Fn(κ, 2) yields a model in which there is a (κ, ω_{1})- HFD.

By combining the previous two results we deduce that (I) fails in any model in which at least ω_{2} Cohen reals have been added.

Corollary 3.5. Forcing with Fn(ω_{2}, 2) yields a model in which (I) fails. Therefore the failure of (I) is consistent with 𝔠 > ω_{1} combined with any other cardinal arithmetic.

It is also standard to show that Martin’s Axiom for countable posets implies that there is a (𝔠, →)-HFD. This will follow from the next result. We would like to relate the existence of a (𝔠, →)-HFD to another less familiar cardinal function on ω. A π-base for an ultrafilter on ω is a family of infinite subsets of ω with the property that every member of the ultrafilter contains one (mod finite). The π-character of an ultrafilter is the least cardinal of a π-base. It was proven by Balcar and Simon [1] that the minimum possible π-character of a free ultrafilter on ω is equal to the reaping number 𝔯. A family A ⊂ [ω]^{ω} is reaped by b ∈ [ω]^{ω} if a \ b and a ∩ b are infinite for all a ∈ A. Then 𝔯 is the least cardinal of a family that cannot be reaped. It is routine to check that Martin’s Axiom for countable posets implies that 𝔯 is equal to 𝔠.

Proposition 3.6. The existence of a (𝔠, →)-HFD follows from 𝔯 = 𝔠.

Proof. Let {T_{α} : α ∈ 𝔠} enumerate the infinite subsets of ω. We inductively choose a family {A_{α} : α ∈ 𝔠} of subsets of ω. We will set S = {s_{n} : n ∈ ω} ⊂ 2^{𝔠} where s_{n}(α) = 1 if and only if n ∈ A_{α}. In order to ensure that S is a (𝔠, →)-HFD we will ensure that, for each α < 𝔠, the family {A_{β} ∩ T_{α} : α < β < 𝔠} is an independent family in P(T_{α}). This is easily done. Let B_{α} be the Boolean subalgebra of P(ω) which is generated by {T_{γ} : γ < α} together with {A_{γ} : γ < α}. Since |B_{α}| < 𝔯, there is a set A_{α} such that b ∩ A_{α} and b \ A_{α} are infinite for each infinite b ∈ B_{α}. It follows easily by induction that S is a (𝔠, →)-HFD.

Question 1. Is there a (𝔠, →)-HFD? Is there a (κ, →)-HFD for some κ?

Of course the existence of a (κ, →)-HFD with cf(κ) > ω_{1} is much stronger than the hypothesis of Lemma 3.1 and therefore of the failure of (I). We established that the failure of (I) followed from the existence of an ultrafilter with π-character equal to 𝔠 together with the assumption that cf(𝔠) > ω_{1} (recall that 𝔯 = 𝔠, the hypothesis of Lemma 3.6, is the assertion that all free ultrafilters on ω have π-character equal to 𝔠). Let us note also that such an ultrafilter was shown to exist if 𝔠 > ω_{1} is regular by Bell and Kunen [3]. (In fact, they showed that there is an ultrafilter of π-character ≥ cf𝔠.) Our result however follows from the following stronger one supplied by S. Shelah. The result can equally well be stated in terms of infinite subsets of ω or in the terms of Lemma 3.1.

Theorem 3.7 (S. Shelah). There is set S ∈ [2^{𝔠}]^{ω} such that 1. for all infinite T ⊂ S, cl(T) ∩ (2^{𝔠})_{0} is not empty, and

2. the constant function 1 is a limit point of S.

Proof. As in Lemma 3.6, it is equivalent to find a special family of subsets of ω.

Indeed, we construct a centered family {A_{α} : α ∈ 𝔠} such that for each infinite T ⊂ ω, there is a γ ∈ 𝔠 such that {T ∩ (ω \ A_{α}) : α ∈ 𝔠 \ γ} is also centered.

Having done so, we set S = {s_{n} : n ∈ ω} with s_{n}(α) = 1 if and only if n ∈ A_{α}. We check that properties 1 and 2 of Lemma 3.1 hold. Since the family of A_{α}’s is centered, it follows that the constant sequence 1 ∈ 2^{𝔠} is a limit point of S. Similarly, for each infinite T̃ ⊂ S, set T = {n : s_{n} ∈ T̃}. Let γ be such that {T \A_{α} : α ∈ 𝔠\γ} is centered. It follows then that ⋂ {cl ({s_{n} : n ∈ T \ A_{α}}) : γ < α < 𝔠} is not empty, so fix any g ∈ 2^{𝔠} in this intersection. For each α ≥ γ, g is a limit of {s_{n} : n ∉ A_{α}}, hence g(α) = 0. Therefore g ∈ (2^{𝔠})_{0}.

We construct the family {A_{α} : α ∈ 𝔠} by induction on α with the aid of Kunen’s 𝔠 × 𝔠-independent matrix ([9]) (similar to, but easier than, their construction of an ultrafilter of π-character ≥ cf𝔠) . A family {a(α, γ) : α ∈ R, γ ∈ 𝔠} is said to be an R × 𝔠-independent matrix if for each α ∈ R and β, γ ∈ 𝔠, a(α, γ) ∩ a(α, β) is finite while for each finite H ⊂ R and each function φ ∈ 𝔠^{H}, ⋂{a(α, φ(α)) : α ∈ H} is infinite. Kunen proved in [9], that there is a 𝔠×𝔠-independent matrix, let {a(α, β) : α, β ∈ 𝔠} be such a matrix.

Fix an enumeration, {T_{β} : β < 𝔠}, of [ω]^{ω}. Suppose that α < 𝔠 and we have chosen ρ_{β} ∈ 𝔠 for each β < α and set A_{β} = a(β, ρ_{β}) so that {T_{ξ} ∩(ω\A_{β}) : β ∈ α\ξ} is centered for each ξ < α. Note that {A_{β} : β < α} is centered because the matrix is independent.

For each ξ ≤ α and for each H ∈ [α\ξ]^{<ω}, choose g(ξ, H) ∈ 𝔠 so that, if possible,

T_{ξ} ∩ ⋂_{ζ∈H}(ω \ A_{ζ}) ⊂ a(α, g(ξ, H)) .

Let ρ_{α} be any member of 𝔠 which is not equal to g(ξ, H) for any ξ ≤ α and H ∈ [α\ξ]^{<ω}. We finish by verifying that, for each ξ ≤ α, {T_{ξ}}∪{ω\A_{β} : ξ ≤ β ≤ α} is centered mod finite. If it were not, then there would be some finite H ⊂ (α \ ξ) such that T_{ξ}∩⋂_{ζ∈H}(ω\A_{ζ}) is disjoint (mod finite) from ω\A_{α}. Of course this means that a(α, ρ_{α}) contains mod finite this intersection. However this contradicts that a(α, g(ξ, H)) is almost disjoint from a(α, ρ_{α}) since, by induction, T_{ξ} ∩⋂_{ζ∈H}(ω \A_{ζ}) is infinite.

As indicated, this section culminates with the following quite satisfactory result.

Corollary 3.8. If 𝔠 has cofinality greater than ω_{1}, then (I) fails.

4. (I) does not imply CH

In the previous section we established that (I) implies that 𝔠 has cofinality ω_{1}, and, by Corollary 3.5, (I) can fail even if cf(𝔠) is ω_{1}. We now establish that there is a model of the failure of CH in which (I) holds.

Theorem 4.1. It is consistent with the negation of CH that every separable initially ω_{1}-compact regular space is compact.

Proof. Consider the Bell-Kunen model constructed in [3]. In this model of the failure of CH, the following property holds (which implies that every ultrafilter on ω has π-character equal to ω_{1}):

For each filter F on ω, there is a family {a_{α} : α < ω_{1}} of infinite subsets of ω such that for each A ∈ F, there is a β < ω_{1} such that a_{α} − A is finite for each α > β.

The model is constructed by iteratively constructing models M_{α} (α < ω_{1}) of Martin’s Axiom in which M_{α} ⊧ 𝔠 = ω_{α+1}. The final model M is the union of these inner models. If F is a filter in the final model, then F ∩ M_{α} will be “diagonalized” by some a_{α} in the model M_{α+1}, i.e. a_{α} \ A is finite for each A ∈ F ∩ M_{α}. If A is any member of F, then A will appear in some model M_{β} and a_{α} will be almost contained in A for all α > β.

Now suppose that X is an initially ω_{1}-compact regular space which has ω as a dense subset. If X is not compact, then let W be any open cover which has no finite subcover. Since X is regular we may assume that no finite subcollection of W has dense union. Let F be the following filter base on ω

{ω \ ⋃ W′ : W′ ∈ [W]^{<ω}} .

Let {a_{α} : α < ω_{1}} be the above mentioned converging π-base family for F. For each α < ω_{1}, there is an x_{α} ∈ X such that x_{α} is a limit point of a_{α} since X is countably compact. But now, fix x ∈ X and we show that x is not a complete accumulation point of {x_{α} : α < ω_{1}}. Let W ∈ W be a neighbourhood of x and set A = ω \ W. By the choice of the a_{α}’s, there is a β < ω_{1} such that x_{α} ∈ \overline{A} for all α ≥ β. Therefore W ∩ {x_{γ} : γ ∈ ω_{1}} ⊂ {x_{γ} : γ < β}.

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