On the property (C) of Corson and other sequential properties of Banach Spaces

Abstract

A well-known result of R. Pol states that a Banach space $X$ has property ($\mathcal{C}$) of Corson if and only if every point in the weak*-closure of any convex set $C\subseteq B_{X^{\ast }}$ is actually in the weak*-closure of a countable subset of $C$. Nevertheless, it is an open problem whether this is in turn equivalent to the countable tightness of $B_{X^{\ast }}$ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under $\mathrm{MA}+\neg \mathrm{CH}$ for the class of $\mathcal{C}(K)$-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom ($\mathrm{PFA}$) the following conditions are equivalent for an arbitrary Banach space $X$: 1) $X$ has property ${\mathcal{E}}^{\prime }$; 2) $X$ has weak*-sequential dual ball; 3) $X$ has property ($\mathcal{C}$) of Corson; 4) $(B_{X^{\ast }},w^{\ast })$ has countable tightness. This provides a partial extension of a former result of Arhangel'skii. In addition, we show that every Banach space with property ${\mathcal{E}}^{\prime }$ has weak*-convex block compact dual ball.