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 → 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:– exists a non-metrizable scattered compact Hausdorff space K with C ( K ) weakly pcc.