H.H. Corson in [C] introduced the following version of Lindelöf property for convex closed subsets of Banach spaces:



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> 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.
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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]):



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> 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 $f\in\overline{\text{conv}}^{w^\*}A'$.
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The property (C) passes to closed subspaces and quotients. Thus, a natural question concerning products arises:



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> Assume $X$ has property (C). Does $X\times X$ have it as well?
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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:



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> Assume $C(K)$ has property (C). Do $C(K)\times C(K)$ and $C(K\times K)$ have also property (C)?
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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.



Productivity of Corson's property (C)
**Yes**, if $X$ and $Y$ have property (C), then so does $X\oplus Y$ because property (C) is a three-space property; this is a result of Pol (Proposition 1 in the paper you are referring to).


However, I am not sure if it is known whether property (C) passes from $C(K)$ to $C(K\times K)$. (If this is true, it should follow rather easily from the identification $C(K\times K) = C(K)\otimes\_{\varepsilon}C(K)$ and the structure of convex sets in the injective tensor products---otherwise perhaps there is a counter-example.)


