Chain conditions and weak topologies

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→c₀(ω₁) 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 Cp(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.