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 (C) of Corson if and only if every point in the weak*-closure of any convex set C⊆BX⁎ 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 BX⁎ with respect to the weak*-topology. Frankiewicz, Plebanek and Ryll-Nardzewski provided an affirmative answer under MA+¬CH for the class of C(K)-spaces. In this article we provide a partial extension of this latter result by showing that under the Proper Forcing Axiom (PFA) the following conditions are equivalent for an arbitrary Banach space X:(1)X has property E′;(2)X has weak*-sequential dual ball;(3)X has property (C) of Corson;(4)(BX⁎,w⁎) 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 E′ has weak*-convex block compact dual ball.