Lindelof type of generalization of separability in Banach spaces

Abstract

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: For each subset F of X, which separates X, there exists a countable separating subset F0 of F. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Markucevic-bases, Corson property and related geometric issues are discussed.