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

TL;DR

This paper explores the equivalence of various completeness properties across metric spaces, topological groups, and function spaces, providing new proofs and applying these results to G-valued function spaces and answering a longstanding question about precompact groups.

Contribution

It introduces novel proofs for the equivalence of completeness properties and applies these to function spaces and precompact groups, resolving an open question in the field.

Findings

Many completeness properties coincide in metric spaces and precompact groups.

Weakly pseudocompact precompact groups are pseudocompact.

New proofs are provided even for classical cases like G = real line.

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 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. Not only the results but also the proofs themselves are novel even in the classical case when G is the real line. A space X is weakly pseudocompact if it is G_delta-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 Tkachenko.