FRAGMENTABILITY OF FUNCTION SPACES Cp(T) FOR PSEUDOCOMPACT SPACES T

Abstract

For a compact space \(T\) it is known that the space \(C_p(T)\) (of all continuous functions in \(T\), endowed with the pointwise convergence topology \(p\)) is fragmentable by a metric that majorizes \(p\) if and only if it is fragmentable by another metric which majorizes the sup-norm topology in \(C(T)\). We show that this fact remains valid for pseudocompact spaces \(T\). For pseudocompact and for strongly pseudocompact spaces \(T\) we give characterizations of fragmentability of \(C_p(T)\) by means of a topological game which is a modification of a game used earlier for characterization of fragmentability. The results are based on a recent generalization of the theorem of Eberlein.