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

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 C-p (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 Go-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. (C) 2017 Elsevier B.V. All rights reserved.