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

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.