Chain conditions and weak topologies

Abstract

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E → c 0 (ω 1 ) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C (K ), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space C p (K ) may be pcc even when C (K ) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example: -There exists a non-metrizable scattered compact Hausdorff space K with C (K ) weakly pcc. Lemma. Let Γ be a set, and let S be a subset of c 0 (Γ ) such that S is pcc in the relative weak topology. Then S is separable.Proof. It suffices to show that the set Γ = {γ ∈ Γ : y γ = 0 for some y ∈ S} is countable. Assume that Γ is uncountable.Then there exists r > 0 such that the set Γ = {γ ∈ Γ : |y γ | > r for some y ∈ S} is uncountable. For every γ ∈ Γ , let G γ = {y ∈ S: |y γ | > r}, and note that G γ is a non-empty relatively weakly open subset of S. For every y ∈ S, the set {γ ∈ Γ : y ∈ G γ } is finite. It follows that {G γ : γ ∈ Γ } is an uncountable point-finite family of relatively weakly open subsets of S, and this is a contradiction. 2 Proposition. The following are equivalent for a Banach space E:A. E is weakly pcc.B. For any Γ , every weak-to-weak continuous mapping E → c 0 (Γ ) has separable range. C. Every weak-to-weak continuous mapping E → c 0 (ω 1 ) has separable range.Proof. The implication A ⇒ B follows by Lemmas 1.2(a) and 1.4. C ⇒ A: Assume that E is not weakly pcc. Then there exists a point-finite family {U α : α ∈ ω 1 } consisting of distinct weakly open subsets of E. For every α ∈ ω 1 , there exists a non-constant, weakly continuous function f α : E → I such that

Copies of $c_{0}(Γ)$ in $C(K, X)$ spaces

“…Let K 2 be the compact Hausdorff spaces constructed recently by Dow, Junnila and Pelant [5,Example 2.16]. Since the set of isolated points of K 2 is uncountable, it follows that K 2 does not satisfy the ℵ 1 -chain condition.…”

“…Otherwise, according to [21, Lemma 1] every bounded linear operator T : X → c 0 (ℵ 1 ) has separable range. Thus a theorem of Holický, Smídek and Zajíček [9] (see also [5,Corollary 1.15]) would imply that dim(X) = ℵ 0 , a contradiction. Since X has JN 1 , we may apply Corollary 4.6 to finish the proof.…”

On the problem of compact totally disconnected reflection of nonmetrizability

“…It can be sometimes used to replace an application of ♣ by a ZFC argument. Its more complicated versions were used by A. Dow and coauthors in Examples 2.15 and 2.16 of [12] others were used in Section 5 of [2], however, we feel that despite their simplicity these principles are not widely known.…”

On complemented copies of $c_0(\omega _1)$ in $C(K^n)$ spaces

“…Their proof relies on Todorcevic's analysis of nonseparable Banach spaces ( [20,Corollary 6]) under the assumption of an additional set-theoretic axiom known as Martin's Maximum ( [6]). However A. Dow, H. Junnila and J. Pelant constructed in ZFC a Banach space (of the form C(K)) of density 2 ω which contains a copy of c 0 (ω 1 ) but such that its injective tensor square has no complemented copy c 0 (ω 1 ) (Example 2.16 of [5]). For more on complemented copies of c 0 (ω 1 ) see [1] and [10].…”

“…This notion plays a crucial role in the result of Galego and Hagler and of Todorcevic mentioned above ( [8,20]). Theorem 1.1 ( [5,8]). Let X be a Banach space.…”