The Baire property and precompact duality

Abstract

We prove that if $G$ is a totally bounded abelian group \st\ its dual group ${\hat{G}}_p$ 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, Domínguez 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${\hat{I}}^2$enhofer and Dikranjan.}