Open discrete shrinkings in function spaces

Abstract

Given a sequence U={Un:n∈ω} of non-empty open subsets of a space X, a family {Vn:n∈ω} is a shrinking of U if Vn is a non-empty open set and Vn⊂Un for every n∈ω. A space X has discrete shrinking property if every sequence of non-empty open subsets of X has a discrete shrinking. We show that if L is a non-metrizable locally convex topological vector space and Lw is the set L with the weak topology of the space L, then Lw has the discrete shrinking property. In particular, a space X is uncountable if and only if Cp(X) has the discrete shrinking property. The same statement is not true for Cp(X,[0,1]) so we study when the discrete shrinking property holds in Cp(X,[0,1]). We prove, among other things, that Cp(X,[0,1]) features discrete shrinking property if X is an essentially uncountable space.