The Lindelöf property and fragmentability

Abstract

Let K K be a compact Hausdorff space and C ( K ) C(K) the space of continuous real functions on K K . In this paper we prove that any t p ( K ) t_{p}(K) -Lindelöf subset of C ( K ) C(K) which is compact for the topology t p ( D ) t_{p}(D) of pointwise convergence on a dense subset D ⊂ K D\subset K is norm fragmented; i.e., each non-empty subset of it contains a non-empty t p ( D ) t_{p}(D) -relatively open subset of small supremum norm diameter. Several applications are given.