The Baire property and precompact duality

TL;DR

This paper establishes a link between the Baire property of dual groups and the finiteness of compact subsets in totally bounded abelian groups, solving open problems in topological group theory.

Contribution

It proves that if the dual group with finite-open topology is Baire, then the original group has only finite compact subsets, addressing open questions in the field.

Findings

Dual group with finite-open topology being Baire implies finiteness of compact subsets.

Provides an example of a group that is g-dense but not g-barrelled.

Solves open problems by Chasco, Domínguez, Tkachenko, Außenhofer, and Dikranjan.

Abstract

We prove that if $G$ is a totally bounded abelian group \st\ its dual group $\widehat{G}_p$ equipped with the finite-open topology is a Baire group, then every compact subset of $G$ must be finite. This solves an open question by Chasco, Dom\'inguez and Tkachenko. {Among other consequences, we obtain an example of a group that is $g$-dense in its completion but is not $g$-barrelled. This solves a question proposed by Au${\beta}$enhofer and Dikranjan.}