H.H. Corson in [C] introduced the following version of Lindelöf property for convex closed subsets of Banach spaces:
A Banach space $X$ has property (C) if every family of convex closed subsets of $X$ with empty intersection contains a countable subfamily with empty intersection.
For instance, all weakly Lindelöf Banach spaces have property (C), but there is an example of a space with property (C) which is not weakly Lindelöf (see [C]).
There are several characterisations of spaces with property (C), eg. in terms of duals (cf. [P]):
A space $X$ has property (C) if and only if for every $A\subseteq B_{X^*}$ and $f\in\overline{A}^{w^*}$ there exists countable $A'\subseteq A$ such that $f\in\overline{\text{conv}}^{w^*}A'$.
The property (C) passes to closed subspaces and quotients. Thus, a natural question concerning products arises:
Assume $X$ has property (C). Does $X\times X$ have it as well?
I haven't been able to find any answers or hints concerning this question. However it seems to be basic... I am especially interested in the case of spaces $C(K)$ of continuous real-valued functions on a compact space $K$ with the supremum norm:
Assume $C(K)$ has property (C). Do $C(K)\times C(K)$ and $C(K\times K)$ have also property (C)?
Thank you very much for any answer or comment.
References
[C] H.H. Corson, The weak topology of a Banach space, Trans. Amer. Math. Soc. 101 (1961), 1–15.
[P] R. Pol, On a question of H.H. Corson and some related problems, Fund. Math. 109 (1980), 143–154.
