Metrizable Bounded Sets in C(X) Spaces and Distinguished C-p(X) Spaces

摘要

Quite recently W. Ruess [17] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's l(1)-dichotomy. Being motivated by this fact we show that for a Tychonoff space X the bounded sets of C-p(X) are metrizable (respectively, the bounded sets of C-k(X) are weakly metrizable) if and only if X is countable. If X is a P-space we show that every bounded set in C-p(X) is metrizable if and only if X is countable and discrete. The second part of the paper deals with distinguished C-p(X) spaces. Among other things we show that C-p(X) is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that C-p(X) is always distinguished whenever X is countable.