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.