On some generalizations of compactness in spaces C-p(X, 2) and Cp(X, Z)

Abstract

We discuss topological properties of a space X which imply that the spaces C-p(X, 2) and C-p(X, Z) have properties similar to compactness, such as sigma-compactness and sigma-countable compactness. In particular, for a zero-dimensional space X, we prove: (1) X is normal and Cp(X, 2) is a-compact iff X is an Eberlein-Grothendieck space and the set of non-isolated points in X is Eberlein compact, and (2) Cp(X, Z) is sigma-compact iff X is an Eberlein compact space.