Completeness and compactness properties in metric spaces, topological groups and function spaces

Abstract

We prove that many completeness properties coincide in metric spaces, precompact groups and dense subgroups of products of separable metric groups. We apply these results to function spaces Cp(X,G) of G-valued continuous functions on a space X with the topology of pointwise convergence, for a separable metric group G.A space X is weakly pseudocompact if it is Gδ-dense in at least one of its compactifications. A topological group G is precompact if it is topologically isomorphic to a subgroup of a compact group. We prove that every weakly pseudocompact precompact topological group is pseudocompact, thereby answering positively a question of M. Tkachenko.