MORE ON SEQUENTIALLY COMPACT

IMPLYING PSEUDORADIAL

by A. Dow, I. Juhász, L. Soukup and Z. Szentmiklóssy

Abstract. We improve some results of [JSz] and answer the main problem raised there. Thus we show that if a compact T_{2} space X is not pseudoradial then there is a Y ⊂ X with |Y | < 𝔰 so that \overline{Y} is not pseudoradial. Also, we show that if any number of Cohen reals are added to a model of CH then CSC spaces are pseudoradial in the extension. As a byproduct, we also show that any ω_{1}-compact and sequentially compact separable T_{3} space is compact in such an extension.

The aim of this note is to answer the main open problem of [JSz]: Does 2^{ω} = 𝔠 > ω_{2} imply the existence of a compact T_{2} and sequentially compact (in short: CSC) space that is not pseudoradial? As we shall show, the answer to this is negative: No such spaces exist in generic extensions obtained by adding Cohen reals to a ground model satisfying CH.

We follow the notation of [JSz]. A set S in a space X is said to be relatively sequentially compact (or RSC) if every ω-sequence of points of S has a subsequence that converges in X (of course a space X is sequentially compact if every subset is RSC). For any cardinal number κ we denote by ℓ_{κ}(S) the set of limit points of all the converging κsequences consisting of points of S. A subset S of a space X is radially closed in X if ℓ_{κ}(S) ⊂ S for all cardinals κ and a space X is said to be pseudoradial if every radially closed subset is actually closed. Note that, for example, βω is not pseudoradial since the subset ω is radially closed and certainly not closed.

One of the main results of [JSz] states that if a compact T_{2} space X is not pseudoradial then there is a set S ∈ [X]^{<𝔠} such that \overline{S} is not pseudoradial either. Here we start with a theorem that easily yields that

Typeset by AMS-TEX

1980 Mathematics Subject Classification (1985 Revision). Primary 54A25, 54A35, 54D30; secondary 54A20.

Key words and phrases. Sequentially compact, pseudoradial, Cohen forcing.

NSERC of Canada and OTKA grant number 1805 .

the cardinal 𝔠 in this result can actually be replaced by the cardinal 𝔰. It is pointed out in [Sh] (P. 583 Theorem 3.11) that this result was actually first noted by Šapirovskiǐ. Let us recall that a set A in a space X is said to be λ-closed if for every B ∈ [A]^{λ} we have \overline{B} ⊂ A, moreoever A is < κ-closed if it is λ-closed for every λ < κ.

Theorem 1. Let σ be an uncountable cardinal and X be a compact T_{2} space that cannot be mapped (continuously) onto the Tychonov cube I^{σ}. Then every subset A ⊂ X which is both < σ-closed and radially closed is actually closed.

Proof. Assume, indirectly, that A ≠ \overline{A}. We may assume also that X = \overline{A}. Let K ⊂ \overline{A}\A be a non-empty compact set of minimum possible character, i.e. χ(K, X) = λ(A, X) using the notation and terminology of [JSz]. Let κ = cf(χ(K, X)), then clearly there is a κ-sequence of points of A that converges to K, hence as A is < σ-closed we must have σ ≤ κ.

Applying Šapirovskiǐ’s celebrated result (see [J1]) there is a point p ∈ K with πχ(p, K) = ρ < σ ≤ κ. So let {K^{ν} : ν ∈ ρ} be a family of closed G_{δ} sets in K that forms a local π-network at the point p in K. Then, by our choice of K, we have χ(K^{ν}, X) = χ(K, X) for each ν ∈ ρ, hence every such K^{ν} can be written as

K^{ν} = ∩{K_{α}^{ν} : α ∈ κ}

where K_{α}^{ν} is closed, χ(K_{α}^{ν}, X) = |α| + ω and so K_{α}^{ν} ∩ A ≠ ∅ for all α ∈ σ, hence lemma 3.7 of [J2] can be applied to get a κ-sequence of elements of A that converges to p. However this contradicts that A is radially closed. ⊣

Since no CSC space can be mapped onto I^{𝔰}, moreover a compact space that is not SC has a countable subset which is radially closed but not closed, we immediately get the above promised strengthening of Theorem 6 from [JSz]:

Corollary 2. If a compact T_{2} space is not pseudoradial then there is a set Y ⊂ X with |Y | < 𝔰 such that \overline{Y} is not pseudoradial. ⊣

The following observation is trivial but, as we are going to use it repeatedly, we state it separately.

Proposition 3. If the countable set S is RSC in X and F is a filter base of cardinality ℵ_{1} on S that converges to a point p ∈ X then p ∈ ℓ_{ω_{1}}(ℓ_{ω}(S)).

Combining Proposition 3 with Corollary 2 one immediately obtains a proof of Šapirovskiǐ’s result (see [Š]) that under CH every CSC space X is pseudoradial, in fact

(1) \{ \begin{matrix}for any S ∈ [X]^{ω} one has \\ \overline{S} = ℓ_{ω_{1}}(ℓ_{ω}(S)) = \overline{S}^{r} . \end{matrix}

The main result of [JSz] improves Šapirovskiǐ’s since it says that even under 2^{ω} = 𝔠 ≤ ω_{2} every CSC space is pseudoradial. It was also shown there that this result is sharp in the sense that 𝔠 = ω_{3} is consistent with the existence of a CSC space that is not pseudoradial. Our main result here implies that this is indeed the most one could hope for: If we add any number of Cohen reals to a ground model satisfying CH, then in the extension every CSC space X is pseudoradial. In fact we show that every S ∈ [X]^{ω} satisfies (1), hence as s = ω_{1} holds in such an extension (see [F]), Corollary 2 applies. What we will actually show is that the extension satisfies the following property (∗) which, in addition to (1), has other interesting consequences.

(∗) If X is an ω_{1}-compact space and S ∈ [X]^{ω} is RSC in X then for any ultrafilter u on S there is a point x ∈ X and a subfamily v ∈ [u]^{ω_{1}} such that v → x.

It is interesting to compare the property (∗) with Michael’s bisequential property (see [M] or 6.12 of [N]): a space X is bisequential if for each S ∈ [X]^{ω} and ultrafilter u on S converging to a point x ∈ X there is a \underline{countable} subfamily v ⊂ u such that v → x.

Our main theorem now reads as follows. (We use C_{κ} to denote the standard notion of forcing that adds κ Cohen reals.)

Theorem 4. If V ⊨ CH and κ is any cardinal then V^{C_{κ}} ⊨ (∗). In addition, in V^{C_{κ}}, every CSC space is pseudoradial.

Proof. Let G ⊂ C_{κ} be generic over V. For any regular cardinal λ > κ consider H_{λ} and let M ≺ H_{λ} be an elementary submodel that is closed under countable sequences (i.e. M^{ω} ⊂ M). It is well-known that then N = M[G] ≺ H_{λ}[G] = H_{λ}^{V [G]}, moreover |N| = |M| and N satisfies the following property (i):

(i) N ∩ [N]^{ω} is cofinal in ⟨[N]^{ω}, ⊂⟩.

A key point of our proof is the following, actually quite easy, folklore lemma that establishes another property of N (see Lemma 5.1 in [D]).

Lemma 5. If N is as above then it satisfies the following in V [G]:

(ii) for every set A ⊂ ω we have a countable subfamily of P(A) ∩ N that is cofinal in it w.r.t. inclusion, in short

cf(P(A) ∩ N, ⊂) ≤ ω .

Proof. We first establish that

P(A) ∩ N = P(A) ∩ M[G] = P(A) ∩ V [G ↾ (κ ∩ M)].

First of all, if B ∈ N ∩ P(A) = M[G] ∩ P(A), it follows from the definition of M[G] that B has a name, Ḃ, in M. We may assume that Ḃ is a nice name (see 5.11 of Kunen’s text [K]) and so Ḃ is actually a subset of M. Therefore B is actually a member of V [G ↾ (κ ∩ M)]. For the other inclusion, since M is countably closed, it follows that every member of P(ω) ∩ V [G ↾ (κ ∩ M)] has a name which is an element of M.

Since A ⊂ ω, it has a nice name Ȧ whose support is some I ∈ [κ]^{ω}. Now we work with W = V [G ↾ (κ ∩ M)] as our ground model using the standard forcing factoring results. Thus we see that a set B (in W) will be a member of P(A) if and only if there is a condition p ∈ C_{I} ∩ G such that p ⊩ B ⊂ Ȧ. But for such a p and ground model B, we clearly have B ⊂ A_{p} = {n ∈ ω : p ⊩ n ∈ Ȧ} ⊂ A. Consequently, {A_{p} : p ∈ C_{I} ∩ G} ⊂ P(A) ∩ W is the required countable cofinal family in P(A) ∩ N = P(A) ∩ W. ⊣

Let us now get back to the proof of Theorem 4. We first show that V^{C_{κ}} is a model of (∗). By the above, for each regular λ > κ the collection of those elementary submodels N of H_{λ} (in V^{C_{λ}}) that satisfy both (i) and (ii) are cofinal (in fact even stationary) in [H_{λ}]^{ω_{1}}. So given X, S and u as in (∗), first note that we may assume that S = ω and then choose λ such that X, u ∈ H_{λ} and an elementary submodel N ∈ [H_{λ}]^{ω_{1}} satisfying (i) and (ii) such that {X, u} ∪ ω_{1} ⊂ N.

Let us now put v = u ∩ N. By (i) we may pick for each α ∈ ω_{1} a family v_{α} ∈ [v]^{ω} ∩ N such that α < β < ω_{1} imply v_{α} ⊂ v_{β}, moreoever v = ∪{v_{α} : α ∈ ω_{1}}. By elementarity and v_{α} ∈ N, for each α ∈ ω_{1} there is a point x_{α} ∈ X ∩N and a sequence S_{α} ∈ [S]^{ω} ∩N such that S_{α} → x_{α}, moreover |S_{α}\V | < ω for every V ∈ v_{α}.

Now, applying that X is ω_{1}-compact, we can choose a point x ∈ X that is a complete accumulation point of {x_{α} : α ∈ ω_{1}}. We do not claim that x ∈ N, but we will show that v → x.

Indeed, let U be an arbitrary open set containing x and set A = S ∩ U = ω ∩ U. Since |{α : x_{α} ∈ U}| = ω_{1} there is a fixed n ∈ ω such that |{α : S_{α}\n ⊂ A}| = ω_{1} as well. But for each such α we also have S_{α}\n ∈ N, hence by (ii) applied to A there is some set B ∈ P(A) ∩ N such that |{α : S_{α}\n ⊂ B ⊂ A}| = ω_{1}. But then we clearly have B ∩ V ≠ ∅ for each V ∈ v = u ∩ N, consequently we must have B ∈ u, hence B ∈ v, because B, u ∈ N ≺ H_{λ}. However, this is exactly what we needed to show that v → x. Noticing that |v| = ω_{1} we have completed the proof of (∗). It is useful to note that by applying Proposition 3 we may also conclude that x ∈ ℓ_{ω_{1}}(ℓ_{ω}(S)), hence any X as above will also satisfy (1).

Now let X be a CSC space. Note that any countable S ⊂ X is certainly RSC in X and X is ω_{1}-compact, hence X satisfies (1). This means, of course, that every separable subspace of X is pseudoradial. Finally we recall that V^{C_{κ}} ⊧ 𝔰 = ω_{1} and conclude, by Corollary 2, that X is pseudoradial. ⊣

In our next result we point out another interesting consequence of (∗).

Theorem 6. If (∗) holds (hence in V^{C_{κ}} if V ⊧ CH), the closure of a countable RSC set S in an ω_{1}-compact T_{3} space is compact. In particular, every ω_{1}-compact, SC, separable T_{3} space is compact.

Proof. Since X is T_{3}, it is clearly sufficient to show that every ultrafilter u on S converges in X. This, however, is immediate from (∗). ⊣

Another result from [JSz] states that under MA(σ-centered) (i.e. 𝔭 = 𝔠) every countably tight CSC space is pseudoradial. As one more application of Theorem 1, we present here the following strengthening of this result. Note that in this case 𝔭 = 𝔰 = 𝔠.

Theorem 7. If 𝔭 = 𝔠 then for every countable set S in a CSC space X we have

(1′)  \overline{S} = ℓ_{c}(ℓ_{ω}(S)) = \overline{S}^{r} .

In particular, if X does not map onto I^{ω_{1}} (e.g. t(X) = ω or X is T_{5}) then X is pseudoradial.

Proof. We may assume that X = \overline{S}. It is well known that if for some compact set K ⊂ X\S we have χ(K, X) < 𝔠 then there is an ω-sequence of points of S that converges to K. In particular, for every p ∈ X with χ(p, X) < 𝔠 we have p ∈ ℓ_{ω}(S).

So now assume that p ∈ X\S with χ(p, X) = 𝔠. Then there is a decreasing sequence {K_{α} : α ∈ c} of compact sets such that {p} = ∩{K_{α} : α ∈ 𝔠} and χ(K_{α}, X) = |α|+ω for every α ∈ 𝔠. According to our above remark, and using that X is SC, for every α ∈ 𝔠 we have a point x_{α} ∈ K_{α} ∩ ℓ_{ω}(S). Since x_{α} → p this clearly implies that p ∈ ℓ_{𝔠}(ℓ_{ω}(S)), and the proof of (1′) is completed.

The second part of the theorem now follows immediately from Theorem 1. ⊣

References

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Department of Mathematics and Statistics, York University, North York, Ontario, Canada, M3J 1P3

Math. Inst. Hung. Acad. Sci., P.O. Box 127, 1364 Budapest, Hungary