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

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We consider covering properties of weak topologies of Banach spaces, especially of weak or point‐wise topologies of function spaces C(K), for compact spaces K. We answer questions posed by A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Our main results are the following. A Banach space of density at most ω1 is hereditarily metaLindel of in its weak topology. If the weight of a compact spaceK is at most ω1, then the spaces Cw(K) and Cp(K) are hereditarily metaLindel of. Let Tfalse¯ be the one‐point compactification of a treeT. Then the space Cpfalse(Tfalse¯false) is hereditarily σ‐metacompact. If T is an infinitely branching full tree of uncountable height and of cardinality bigger than c, then the weak topology of the unit sphere of Cfalse(Tfalse¯false) is not σ‐fragmented by any metric. The space Cp(rβω1) is neither metaLindel of nor σ‐relatively metacompact. The space Cp(rβω2) is not σ‐relatively metaLindel of. Under the set‐theoretic axiom ♦, there exists a scattered compact space K1 such that the space Cp(K1) is not σ‐relatively metacompact, and under a related axiom ◊, there exists a scattere compact space K2 such that the space Cp(K2) is not σ‐relatively metaLindel of. 1991 Mathematics Subject Classification: 54C35, 46B20, 54E20, 54D30.

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