CARDINAL RESTRICTIONS ON

SOME HOMOGENEOUS COMPACTA

István Juhász*, Peter Nyikos**, and Zoltán Szentmiklóssy*

Abstract. We give restrictions on the cardinality of compact Hausdorff homogeneous spaces that do not use other cardinal invariants, but rather covering and separation properties. In particular, we show that it is consistent that every hereditarily normal homogeneous compactum is of cardinality 𝔠. We introduce property wD(κ), intermediate between the properties of being weakly κ-collectionwise Hausdorff and strongly κ-collectionwise Hausdorff, and show that if X is a compact Hausdorff homogeneous space in which every subspace has property wD(ℵ_{1}), then X is countably tight and hence of cardinality ≤ 2^{𝔠}. As a corollary, it is consistent that such a space X is first countable and hence of cardinality 𝔠. A number of related results are shown and open problems presented.

In this paper, “space” means “Hausdorff space” and “compactum” stands for “infinite compact (Hausdorff) space.” It is well known that every compactum without isolated points is of cardinality ≥ 𝔠(= 2^{ℵ_{0}}).

At the end of [vM], Jan van Mill posed the following problem:

1.1. Problem. Is every T_{5} [that is, hereditarily normal] homogeneous compactum of cardinality 𝔠?

In a July 2003 seminar at the Alfred Rényi Institute in Budapest, he conjectured that the answer to his question was positive. In this paper, we show that it is consistent that the answer to van Mill’s question is positive [Theorem 2.8]. We also show it consistent [Theorem 3.2] that every homogeneous compactum that is hereditarily wD(ℵ_{1}) is first countable and hence, by a famous theorem of Arhangel’skiĭ,

1991 Mathematics Subject Classification. (Updated to 2000.) Primary: 03E35, 54A25, 54D15, 54D30, 54F99; Secondary: 03E50, 54D45.

Key words and phrases. compactum, homogeneous, hereditarily, [weakly] [strongly] κ-cwH, property wD(κ), T_{5}, tightness, π-character, π-base, first countable, G_{δ}.

*Research of the first and third authors partially supported by OTKA gra nt no. 37758. **Research of the second author partially supported by a grant from the Erdős Center of the

János Bolyai Mathematical Society.

Typeset by AMS-TEX

of cardinality 𝔠. This theorem says that |X| ≤ 2^{χ(X)} for all compact Hausdorff X, where χ(X), as usual, stands for the character of X. When combined with the Čech-Pospišil theorem, this implies that every homogeneous compactum X is exactly of cardinality 2^{χ(X)}.

1.2. Definition. Let κ be a cardinal number. A space X is weakly κ-collectionwise Hausdorff [resp. satisfies property wD(κ)] if every closed discrete subspace D of cardinality κ has a subset D_{0} of cardinality κ which can be expanded to a disjoint [resp. discrete] collection of open sets U_{d} such that U_{d} ∩ D_{0} = {d} for all d ∈ D_{0}.

We use “κ-cwH” as an abbreviation for “κ-collectionwise Hausdorff”: without the modifier “weakly” this means that D_{0} can be taken to be D in the above definition:

1.3. Definition A space X is [strongly] κ-cwH if every closed discrete subspace D of cardinality ≤ κ can be expanded to a disjoint [resp. discrete] collection of open sets as in Definition 1.

In Cohen’s original model for the negation of the Continuum Hypothesis (CH), and indeed in any model obtained by adding ℵ_{2} Cohen reals to a model of ZFC, every locally compact normal space is ℵ_{1}-cwH. This well-known fact can be shown by applying the technique of proof of [W, Theorem 3] to [DTW, Theorem 1.4].

It is also well known and easy to prove that every normal κ-cwH space is strongly κ-cwH; similarly, every normal, weakly κ-cwH space has property wD(κ). Therefore, every T_{5}, locally compact space is hereditary strongly ω_{1}-cwH and hence hereditarily wD(ℵ_{1}) in any generic extension obtained by adding ℵ_{2} Cohen reals to an arbitrary ground model. This fact figures prominently in the proof of Theorem 2.8 that yields a consistent Yes answer to Problem 1.1.

With one exception that we know of, the theorems in this paper represent the first restrictions on the cardinality of homogeneous compacta that do not assume restrictions on other well-known cardinal functions (character, cellularity, tightness, etc.). The exception is that every monotonically normal homogeneous compactum is first countable (and therefore of cardinality 𝔠). This has been part of the folklore for some time, but we give a quick proof below (Theorem 3.6). Monotone normality is a very strong property, and Problem 1.1 asks whether at least the weaker part of this result can be extended to the much wider class of T_{5} homogeneous compacta. Perhaps (see Problems 3.4 and 3.5 below) the stronger part can be extended also.

A dichotomy and a consistency result for Problem 1.1

We begin with a lemma which we hope to be of independent interest. Recall that a (local) π-base for a point x of a space X is a family U of non-empty open sets such that every neighborhood of x contains a member of U. The least cardinality of a π-base for x is called the π-character of x in X and is denoted by πχ(x, X).

2.1. Lemma. Let Y be a locally compact space. The set of points y which fail to satisfy at least one of the following two properties is dense in Y.

(a) πχ(y, Y) = ω.

(b) Every G_{δ} containing y has nonempty interior.

In particular, if Y has both points satisfying (a) and points satisfying (b), then Y cannot be homogeneous.

Proof. If Y has a dense set of isolated points, then (a) fails for these and we are done. [Some papers use the convention that πχ(y, Y) = ω even if y is isolated, but not this paper.] Otherwise, let A be the closure of the set of isolated points. It is enough to show that the (locally compact) subspace Y \ A has a dense set of points where at least one of (a) or (b) fails. In other words, it only remains to prove our Lemma for the case where Y has no isolated points, and we assume this from now on.

Let U_{0} be any nonempty open subset of Y with compact closure. We define a descending well-ordered family of such sets by induction. If α is an ordinal and U_{α} has been defined, let U_{α+1} be any nonempty open set whose closure is a proper subset of U_{α}.

If α is a limit ordinal and U_{ξ} has been defined for all ξ < α, then C = ⋂{U_{ξ} : ξ < α} is nonempty because it is the intersection of the chain of compact sets \{\overline{U_{ξ}} : ξ < α\}. If α is of countable cofinality, and y is a point of C satisfying (b), then C, being a G_{δ}, has nonempty interior and we may let U_{α} be the interior of C. If α is of uncountable cofinality and y is a point of C satisfying (a), then let {V_{n} : n ∈ ω} be a countable π-base for y. Since α is of uncountable cofinality, there must exist n such that V_{n} ⊂ U_{ξ} for cofinally many ξ ∈ α, but then V_{n} ⊂ U_{ξ} for all ξ ∈ α because the U_{ξ}’s are nested, hence V_{n} ⊂ C. So now we can let U_{α} = V_{n} and continue the induction.

Eventually the chain stops at some limit ordinal α, i. e. the intersection C has empty interior. But then all the points of C fail to satisfy (b) if α is of countable cofinality and to satisfy (a) if α is of uncountable cofinality. Since U_{0} was arbitrary, this completes the proof. □

2.2. Corollary. If Y is a locally compact space in which every point is non-isolated and of countable π-character, then the union of the closed nowhere dense G_{δ}-subsets of X is dense in X. □

By a theorem of Shapirovskiĭ, every compact T_{5} space has a point of countable π-character, so that (a) in Lemma 2.1 is automatically satisfied by some points of any T_{5} compactum. This theorem of Shapirovskiĭ is a corollary of his remarkable theorem that a compactum cannot be mapped continuously onto [0, 1]^{ℵ_{1}} iff every subcompactum has a point of countable relative π-character. This gives us many classes of compacta with points of countable π-character, including the class of countably tight compacta. Two other classes are given by the following two theorems, which seem to be new:

2.3. Theorem. If X is a hereditarily weakly ω_{1}-cwH compact space, then X cannot be mapped continuously onto [0, 1]^{ℵ_{1}}.

Proof. Suppose f : X → [0, 1]^{ℵ_{1}} is a continuous onto function. Let F be a closed subset of X such that the restriction of f to F is irreducible; that is, no proper closed subset of F maps onto [0, 1]^{ℵ_{1}}. Let Q be a countable dense subset of [0, 1]^{ℵ_{1}}. By irreducibility, a subset of F which meets each point-inverse f^{−1}(q) (q ∈ Q) is dense in F; hence F is separable. By the hereditarily weakly ω_{1}-cwH property, F cannot have an uncountable discrete subspace; but [0, 1]^{ℵ_{1}} does have one, and this contradicts surjectivity of f. □

An alternative proof of Theorem 2.3 uses the fact that [0, 1]^{ℵ_{1}} satisfies the countable chain condition, and that a closed irreducible map has the property that a pair of disjoint nonempty open sets in the domain have images with nonempty interiors, and these interiors are disjoint.

2.4. Theorem. If X is a compact space satisfying wD(ℵ_{0}) hereditarily, then X cannot be mapped continuously onto [0, 1]^{ℵ_{1}}.

Proof. Let Y be the Tychonoff plank, (ω_{1} × ω + 1) ∪ (ω_{1} + 1 × ω), let S be a copy of [0, 1]^{ℵ_{1}} that contains Y, and let f : X → S be a continuous onto function. Let Z = f^{−1}Y and for each point ⟨α, n⟩ in ω_{1} × ω, let x(α, n) ∈ f^{−1}{⟨α, n⟩}. Since the preimage of (ω_{1} + 1 × {n}) is compact, there is a condensation point z_{n} of {x(α, n) : α ∈ ω_{1})}, and z_{n} ∈ f^{−1}{⟨ω_{1}, n⟩}. Clearly {z_{n} : n ∈ ω} is closed discrete in Z.

Suppose A is an infinite subset of ω and {U_{n} : n ∈ A} is a family of open sets satisfying z_{n} ∈ U_{n} for all n ∈ A. For every n ∈ A, the closure of U_{n} contains the preimage of C_{n} × {n} for some club subset C_{n} of ω_{1}. Fix α ∈ ⋂_{n∈A} C_{n} and let x_{n} ∈ f^{−1}\{⟨α, n⟩\} ∩ \overline{U_{n}} for all n ∈ A. Since the preimage of \{α\} × ω + 1 is a compact subset of Z, there is an accumulation point of {x_{n} : n ∈ A}, which shows that {U_{n} : n ∈ A} is not a discrete collection. □

2.5. Corollary. If X is a compact space that is either hereditarily weakly ω_{1}-cwH or hereditarily wD(ℵ_{0}), then X has a dense set of points of countable π-character. □

The condition on discrete subsets that appears in our next lemma is clearly satisfied by any Lindelöf space. Further information on it will be given in and following Theorem 4.1.

2.6. Lemma. Let X be a hereditarily wD(ℵ_{1}) space, such that every discrete subset of X of cardinality ℵ_{1} has a complete accumulation point. Let D be a discrete subset of cardinality ℵ_{1} and let H be a closed G_{δ}-subset of X containing all the complete accumulation points of D. Then H has nonempty interior.

Proof. Obviously, D \H is countable. Let Y = X \(D ∩H)′, where (D ∩H)′ stands for the derived set of all limit points of D ∩H. Since D is discrete, D ∩H is a closed discrete subset of Y of cardinality ℵ_{1}. Let U be a discrete (in Y) family of ℵ_{1} open subsets of Y expanding ℵ_{1} points of D ∩H. Since H is closed, (D ∩H)′ ⊂ H and so Y \ H = X \ H is a countable union of closed subsets F_{n} (n ∈ ω) of X. So all but countably many members of U are subsets of H: otherwise, we could pick points of uncountably many of them in some fixed set F_{n} and thus have an uncountable closed discrete subset of F_{n} and hence of X. □

2.7. Theorem. If X is a homogeneous compact space satisfying wD(ℵ_{1}) hereditarily, then X is countably tight and hence of cardinality ≤ 2^{𝔠}.

Proof. If X is of uncountable tightness, then by the main theorem of [JSz], X has a free ω_{1}-sequence converging to some point y. A free ω_{1}-sequence in a space X is a family {x_{α} : α < ω_{1}} of points such that, for each α < ω_{1}, the closure of {x_{ξ} : ξ < α} does not meet the closure of {x_{η} : η ≥ α}. It is easily seen that every free sequence is a discrete subspace. To say that it converges to y is to say that every neighborhood of y contains all but countably many of its points. Equivalently, y is the only complete accumulation point of {x_{α} : α < ω_{1}}. Now, every G_{δ} containing y shrinks to a closed G_{δ} that also contains y by regularity of X, and so, by Lemma 2, has nonempty interior. Consequently, y satisfies condition (b) in Lemma 2.6. However, this contradicts the homogeneity of X because of Corollary 2.5 and Lemma 2.1.

For the second conclusion, we use the theorem in [J2] that every compact space of countable tightness has a point of character ≤ 𝔠. By homogeneity, this implies every point is of character ≤ 𝔠, and so |X| ≤ 2^{𝔠} by the Arhangel’skiĭ Theorem. □

We are now ready to give the consistent affirmative answer to Problem 1.1 promised in the introduction.

2.6. Theorem. If (at least) ℵ_{2} Cohen reals are added to any model V of ZFC, the resulting extension W has the property that every homogeneous T_{5} compactum is countably tight and of character ≤ ℵ_{1}, hence of cardinality ≤ 2^{ℵ_{1}}. In particular, if (2^{ℵ_{1}})^{V} Cohen reals are added, then in the resulting extension every homogeneous, countably tight compactum, hence every homogeneous T_{5} compactum is of cardinality 𝔠.

Proof. By the remarks following Definition 1.3, every homogeneous compact T_{5} space in W satisfies property wD(ℵ_{1}) hereditarily, and hence is countably tight by Theorem 2.7. Moreover, in [J2] it is shown that every compact space of countable tightness has a point of character ≤ ℵ_{1} in an extension obtained by adding just ℵ_{1} Cohen reals to an arbitrary model of ZFC. [It is also conjectured there that this may actually be true in ZFC.] But first adding ℵ_{2} Cohen reals and then adding ℵ_{1} is equivalent to adding ℵ_{2} Cohen reals all at once. Thus if one adds (at least) ℵ_{2} Cohen reals then in the resulting extension every homogeneous compact space of countable tightness is of character ≤ ℵ_{1}, hence of cardinality ≤ 2^{ℵ_{1}} by the Arhangel’skiĭ Theorem. Finally, if (2^{ℵ_{1}})^{V} Cohen reals are added, then the resulting model of ZFC also satisfies 𝔠 = 2^{ℵ_{1}}. □

3. Obtaining first countability in some homogeneous compacta Arhangel’skiĭ has conjectured [A] that evey homogeneous, countably tight com-

pactum is of cardinality 𝔠. Theorem 2.8 proves the consistency of this, but the conjecture is also a consequence of the Proper Forcing Axiom (PFA), which implies that every countably tight compactum has a point of first countability [D1], [J3, Theorem 3.3].

Our next theorem uses a different, much weaker axiom to get first countability of homogeneous, hereditarily wD(ℵ_{1}) compacta. For convenience, we adopt the following terminology.

Axiom I: Every locally compact space of countable spread has a point of first countability.

Axiom L: Every locally compact space of countable spread is hereditarily Lindelöf.

It makes no difference whether we leave out “locally” in either axiom, since countable spread (viz., the property that every discrete subspace is countable), the hereditary Lindelöf property, and the existence of a point of first countability are obviously preserved on taking the one-point compactification. Going in the opposite direction, one observes that in a locally compact space, a point has a countable local base iff it is a G_{δ}, and every point of a locally compact space is contained in a compact G_{δ}. Also, since every point is a G_{δ} in a regular hereditarily Lindelöf space, Axiom L implies Axiom I.

A well-known old result of the third author is that Axiom L is true under MA(ω_{1}) [Sz], [R, 6.4.]. It is also true in various models obtained from a countable support iteration of forcing with all ccc [or all proper] posets that do not destroy a certain coherent Souslin tree and then forcing with the tree. However, we do not know if Axiom I is strictly weaker than Axiom L.

Before giving the theorem using Axiom I, however, we need to prove the following ZFC result.

3.1. Lemma. In a locally compact, hereditarily wD(ℵ_{1}) space, the boundary of any open Lindelöf subset has countable spread.

Proof. Let H be an open Lindelöf subset of a space X as described. Let D be a discrete subspace of the boundary of H; since H is open, its boundary is \overline{H} \ H. Let D′ be the derived set of D; since D is discrete, D′ = \overline{D} \ D. Also, D′ is disjoint from H because \overline{H} \ H is closed. Hence H is a subset of W = \overline{H} \ D′, and D is closed in the relative topology of W.

If D were uncountable, there would be an uncountable subset D_{0} of D with a discrete-in-W open expansion {U_{d} : d ∈ D_{0}}. But then {U_{d} ∩ H : d ∈ D_{0}} would be an uncountable discrete-in-H collection of subsets of H, and this is a contradiction since H is Lindelöf. Hence D is countable. □

3.2. Theorem. Assume Axiom I. Then every homogeneous, hereditarily wD(ℵ_{1}) compactum is first countable and hence of cardinality 𝔠.

Proof. Let X be a compactum as above. By homogeneity and Corollaries 2.2 and 2.5, every point in X is contained in a nowhere dense closed G_{δ} set B. The complement of B is an open Lindelöf subspace whose boundary is B, and so B is of countable spread by Lemma 2.6. Now we use Axiom I to conclude that B has a point x which is a relative G_{δ}-subset of B. But since B is itself a G_{δ}-subset of X, it follows that x is a G_{δ}-point of X and hence a point of first countability. Another application of homogeneity gives us first countability of X. □

Although Axiom I does not hold in the models obtained by adding Cohen reals, the following conjecture seems reasonable in the light of Theorems 2.8 and 3.2.

3.3. Conjecture. It is consistent that every homogeneous T_{5} compactum is first countable.

Jan van Mill has observed that in any model of 2^{ℵ_{0}} < 2^{ℵ_{1}}, a Yes answer to Problem 1.1 would imply that every T_{5} homogeneous compactum is first countable, since every compact space of cardinality < 2^{ℵ_{1}} has a point of first countability by the Čech-Pospišil Theorem. Of course, the models of Theorem 2.8 that do give a Yes answer to Problem 1.1 satisfy 2^{ℵ_{0}} = 2^{ℵ_{1}}. At the moment, the best candidate for confirmation of our conjecture is one of the models whose construction involves a coherent Souslin tree. Positive answers to both of the following questions in the same model of set theory would also confirm the conjecture:

3.4. Problem. Is every homogeneous T_{5} compactum countably tight?

3.5. Problem. Is every homogeneous countably tight compactum first countable?

As we have seen, the PFA implies a positive answer to Problem 3.5, and the answer to Problem 2 is positive in models from Theorem 2.8, but we would also like to have it the other way around!

As indicated in the introduction, Conjecture 3.3 is provable in ZFC for monotonically normal compacta.

Theorem 6. Every monotonically normal homogeneous compactum is first countable and hence of cardinality 𝔠. Moreover, it is hereditarily paracompact.

Proof. This is a quick corollary of the following theorem of Williams and Zhou [WZ], [J3, 3.12]:

Let P (p, X) denote min{κ : p is not a P_{κ^{+}}-point of X} (i.e., P (p, X) is the smallest size of a family of neighbourhoods of p in X whose intersection is not a neighbourhood of p). Then in a monotonically normal compact space X, the set of all points p such that χ(p, X) = P (p, X) is dense in X.

Since a compact P-space is finite, P (p, X) is countable for all points of a compact homogeneous monotonically normal space, and so the space is first countable.

For the hereditary paracompactness, just use the Balogh-Rudin theorem that a monotonically normal space is hereditarily paracompact iff it contains no homeomorphic copy of a stationary subset of ω_{1} [BR], and the elementary fact that no first countable compact space can contain such a copy. □

If we assume MA(ω_{1}) + Axiom R, we can relax “monotonically normal” in Theorem 3.6 to “hereditarily strongly ω_{1}-cwH.” [Recall that every monotonically normal space is hereditarily collectionwise normal.] Theorem 3.2 gives us first countability since MA(ω_{1}) implies Axiom I, and hereditary paracompactness follows from Balogh’s theorem [B] that MA(ω_{1}) + Axiom R implies every hereditarily strongly ω_{1}-cwH space is either hereditarily paracompact or contains a perfect preimage of ω_{1}.

The following question lies at the opposite extreme of generality. It is completely open—we have no consistency results in either direction.

3.7. Problem. Is every homogeneous compactum either first countable or continuously mappable onto [0, 1]^{ℵ_{1}}?

For homogeneous ccc (that is, countable cellularity) compacta the answer is Yes if one assumes 2^{ℵ_{0}} < 2^{ℵ_{1}}. This is because |X| ≤ 2^{c(X)πχ(X)} for homogeneous spaces [A], and so any homogeneous ccc compactum of cardinality > 𝔠 is also of uncountable π-character and hence continuously mappable onto [0, 1]^{ℵ_{1}} by Shapirovskiĭ’s theorem. On the other hand, since |X| = 2^{χ(X)} for every homogeneous compactum, 2^{ℵ_{0}} < 2^{ℵ_{1}} implies that every homogeneous compactum of cardinality 𝔠 is first countable. This reasoning extends to higher cardinals, and results in such considerations as the following: if, as some suspect, every homogeneous compactum is of cellularity ≤ 𝔠, then 2^{𝔠} < 2^{𝔠^{+}} implies that every homogeneous compactum is either of character ≤ 𝔠 or admits a continous map onto [0, 1]^{𝔠^{+}}.

Problem 3.7 is closely related to the following general problem about compacta:

3.7′ Problem. Is it consistent that every compactum either has a point of first countability or admits a continuous map onto [0, 1]^{ℵ_{1}}?

There are compacta satisfying neither condition in various models of set theory, including those satisfying CH [F] and those satisfying ♣ or its weakening (t), which holds whenever a single Cohen real is added [J1].

Our next theorem features a use of the property that every open Lindelöf subspace has hereditarily Lindelöf boundary. This property is satisfied by every locally compact space satisfying wD(ℵ_{1}) in any model of ZFC where Axiom L holds, by Lemma 3.1. It is already strong enough to imply first countability in any homogeneous compactum [Theorem 3.11]. We will make use of two lemmas, the first of which is very easy, cf. [Ny] or [B].

3.8. Lemma. In a locally compact space, every point has an open Lindelöf (equivalently: σ-compact) neighborhood. □

3.9. Lemma. Let X be a locally compact space such that every Lindelöf subset has Lindelöf closure. Then any two disjoint closed subsets of X, one of which is Lindelöf, can be put into disjoint open sets. Hence X has Property wD(ℵ_{0}).

Proof. Let A and B be disjoint closed subsets of X. If A is Lindelöf, then Lemma 3.8 gives a countable cover U of A by open Lindelöf sets, none of which meets B. Then U = ⋃ U is Lindelöf, and so A and B ∩ \overline{U} are disjoint closed Lindelöf subsets of X and hence can be put into disjoint open subsets V and W of X. Then V and W ∪ (X \ \overline{U}) are disjoint open subsets of X containing A and B, respectively.

For the final conclusion, note that X is pseudonormal (that is, every pair of disjoint closed subsets, one of which is countable, can be put into disjoint open sets) and use the easy fact [vD, 12.1] that every pseudonormal space satisfies wD(ℵ_{0}). □

Before going on to Theorem 3.11, we note an interesting “stepping-down” consequence of Lemmas 3.1 and 3.9 in the presence of Axiom L.

3.10. Corollary. [Axiom L] Let X be a locally compact space. If X is hereditarily wD(ℵ_{1}), then X is hereditarily wD(ℵ_{0}).

Proof. First, note that the property of being wD(κ) is inherited by all subspaces if it is inherited by all open subspaces. This is because, if Y is a subspace of X and D is a closed discrete subspace of Y, then D is also closed discrete in the (open, in X) subspace X \ D′, where D′ again stands for the derived set of D in X. So if we define D_{0} and the sets U_{d} as in Definition 1, with X \ D′ playing the role of X, then the traces U_{d} ∩ Y will be a discrete collection of open subsets of Y expanding D_{0} in Y, as desired.

Next, let Y be an open subspace of X. Since Y is locally compact, it follows from Lemma 3.1 and Axiom L that every open subspace of Y has hereditarily Lindelöf boundary, and hence that every Lindelöf subspace of Y has Lindelöf closure. Now use Lemma 3.9 to conclude that Y has Property wD(ℵ_{0}). □

Corollary 3.10 requires more than just ZFC: it is easy to modify Ostaszewski’s space using ♢ to get a hereditarily separable locally compact locally countable space which is not wD(ℵ_{0}); yet it is vacuously (hereditarily) wD(ℵ_{1}). It is also not possible to drop local compactness from Corollary 3.10. Let X be the space with underlying set ω_{1} + 1 and in which all points except ω_{1} are isolated, while a set containing ω_{1} is open iff its complement is nonstationary. Let Y be the subspace of X × ω + 1 obtained by removing the corner point ⟨ω_{1}, ω⟩. In Y, no infinite subset of the countable closed discrete subspace E = {ω_{1}} × ω can be expanded to a discrete collection of open sets. Hence Y is not wD(ℵ_{0}). On the other hand, if D is a closed discrete subset of Y of cardinality ℵ_{1}, the non-isolated points of D \E are contained in ω_{1} × {ω}, and if there are uncountably many of them, there is a nonstationary subset N of ω_{1} such that (N × {ω}) ∩ D is uncountable. Then (N × {ω}) ∩ D can easily be expanded to a discrete collection of open sets. Since the non-isolated points of Y form a closed discrete subspace, the wD(ℵ_{1}) property is clearly inherited by all subspaces of Y.

3.11. Theorem. Let K be a homogeneous compactum in which every open Lindelöf subspace has hereditarily Lindelöf boundary. Then K is first countable.

Proof. Let p ∈ K and let X = K \ {p}. If X is Lindelöf then p is a G_{δ}-point in K, hence a first countability point, and we are done. So we may assume that X is not Lindelöf.

Next we show K satisfies wD(ℵ_{0}) hereditarily. Indeed, the property that every open Lindelöf subspace of K has hereditarily Lindelöf boundary is inherited by open subspaces of K. It follows that if Y is an open subspace of K, then every Lindelöf subspace of Y has Lindelöf closure; hence Y satisfies wD(ℵ_{0}), and so every subspace of K satisfies it (see the first paragraph in the proof of Corollary 3.10).

From Corollary 2.5 we get that K has a dense set of points of countable πcharacter. Since p is non-isolated, X is noncompact and so contains a noncompact, Lindelöf open subset X_{0}, as can be easily shown by repeated applications of Lemma 3.8. Continue inductively through the countable ordinals, defining a strictly increasing chain of Lindelöf open subsets of X as follows. We get X_{ξ+1} once X_{ξ} is defined, by covering the boundary of X_{ξ} with countably many open Lindelöf subspaces of X. Note that X_{ξ+1} can be chosen to be strictly larger than cℓ_{X}(X_{ξ}); this is where we use the assumption that X is not Lindelöf. At limit ordinals α let X_{α} = ⋃{X_{ξ} : ξ < α}. By induction, X_{ξ} is Lindelöf and has hereditarily Lindelöf boundary for all countable ξ.

Having finished the induction, X_{ω_{1}} = ⋃{X_{ξ} : ξ < ω_{1}} is a non-Lindelöf open subset of X. If X_{ω_{1}} is clopen in X, then for every open neighbourhood U of p in K the compact set X_{ω_{1}} \ U is contained in X_{α} for some α < ω_{1}. Consequently, every G_{δ}-set containing p contains X_{ω_{1}} \ X_{α} for some α < ω_{1} and so has nonempty interior. This means that p can play the role of y in Lemma 2.1 (b). Otherwise, since X satisfies wD(ℵ_{0}) hereditarily, any point on the boundary of X_{ω_{1}} in X is isolated in Bd_{X}(X_{ω_{1}}) [Ny2, Lemma 1.6], and so can play the role of y in (b) similarly to p in the case where X_{ω_{1}} is clopen in X. This, by Lemma 2.1, contradicts the homogeneity of K and shows that X is indeed Lindelöf. □

Our next problem involves condition (b) in Lemma 2.1, which in a homogeneous space is equivalent to every nonempty G_{δ} having nonempty interior. Spaces with this latter property are often called pseudo-P spaces.

3.12. Problem. Is there a homogeneous pseudo-P compactum?

The Stone-Čech remainder of ω is a pseudo-P compactum, but is well known not to be homogeneous. Jan van Mill has observed that any pseudo-P compactum of weight ≤ ω_{1} has a dense set of P-points (just use W. Rudin’s argument that CH implies βω − ω has P-points) and so an example for Problem 3.12 would have to have weight > ω_{1}.

Concluding remarks

Finally, here is some information about the hypothesis used in Lemma 2.6. It shows that, at least consistently, the assumption that all discrete subspaces of size ℵ_{1} have condensation points is strictly weaker than the same for all subspaces of size ℵ_{1}.

4.1. Theorem. The following are equivalent.

(1) There is an S-space.

(2) There is a regular locally countable uncountable space of countable spread.

(3) There is a regular space X in which every discrete subspace of cardinality ℵ_{1} has a complete accumulation point, yet there is a subspace of cardinality ℵ_{1} without a complete accumulation point.

Proof. (1) ⟹ (2) is well-known [cf. the proof of Theorem 3.1 in [R]], and (2) ⟹ (3) is obvious. To see (3) ⟹ (1), take Y ⊂ X of size ℵ_{1} with no complete accumulation point. Then Y must have countable spread and is not Lindelöf, hence it contains an uncountable right separated subspace Z. But then Z is an S-space [R, proof of 3.3.]. □

The condition that every subset of cardinality ℵ_{1} has a complete accumulation point is known as C[ℵ_{1}, ℵ_{1}] and is equivalent to every open cover of cardinality ℵ_{1} having a countable subcover. More generally, Saks [S1] defined a space to be C[𝔫, 𝔪] if every subset A satisfying 𝔫 ≤ |A| ≤ 𝔪 has a complete accumulation point. Saks noted that the outline of [K, Exercise 5I] can easily be modified to show that C[𝔫, 𝔪] implies every open cover of cardinality ≤ 𝔪 has a subcover of cardinality < 𝔫. The converse is also true if 𝔪 < ℵ_{ω} as can be shown by a slight modification of the proof that a space is initially 𝔪-compact iff every infinite subset of cardinality ≤ 𝔪 has a complete accumulation point [S2, Theorem 2.2].

We close by recalling a quirk of terminology. The expression, “finally compact in the sense of complete accumulation points,” is usually taken to mean that every infinite subset of regular cardinality has a complete accumulation point. This is equivalent to the space being linearly Lindelöf [H].

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