Pontryagin duality in the class of precompact Abelian groups and the Baire property

TL;DR

This paper explores a broad class of reflexive, precompact Abelian groups with the Baire property, revealing their duals have only finite compact subsets and showing how many non-reflexive groups relate to these via quotients.

Contribution

It introduces a new class of reflexive precompact Abelian groups characterized by the Baire property and open refinement condition, and demonstrates their relation to non-reflexive groups through quotients.

Findings

All compact subsets of the dual groups are finite.

Many non-reflexive precompact Abelian groups are quotients of reflexive ones.

Existence of dense subgroups with contrasting reflexivity and metrizability properties.

Abstract

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups $G$ determined by three requirements. They must have the Baire property, satisfy the \textit{open refinement condition}, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group $G^\wedge$ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group $G$ of weight $\geq 2^\om$, we find proper dense subgroups $H_1$ and $H_2$ of $G$ such that $H_1$ is reflexive and pseudocompact, while $H_2$ is non-reflexive and almost metrizable.