Topological Properties of Banach Spaces

Weakly compact sets in Banach spaces are fragmentable. We introduce the weaker notion of the a-fragmentability of a subset of a Banach space, a notion that has a far wider range of applicability. We prove that all weakly Cech-analytic subsets of a Banach space are a-fragmented. This implies that all Banach spaces having Kadec norms are a-fragmented. However l∞ is not σ-fragmented