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 f° is not a-fragmented.
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