# On convergent sequences in dual groups

## TL;DR

This paper characterizes when dual groups of precompact abelian groups contain nontrivial convergent sequences, especially in torsion groups, and provides examples illustrating these properties and their implications for reflexivity and category.

## Contribution

It offers new characterizations of convergent sequences in dual groups of precompact abelian groups, including torsion groups, and constructs examples with specific topological and duality properties.

## Key findings

- Dual groups of certain precompact abelian groups contain nontrivial convergent sequences.
- In torsion groups, the existence of such sequences relates to the absence of infinite countable quotients.
- Constructed examples show groups with specific measure, category, and duality properties.

## Abstract

We provide some characterizations of precompact abelian groups $G$ whose dual group $G_p^\wedge$ endowed with the pointwise convergence topology on elements of $G$ contains a nontrivial convergent sequence. In the special case of precompact abelian \emph{torsion} groups $G$, we characterize the existence of a nontrivial convergent sequence in $G_p^\wedge$ by the following property of $G$: \emph{No infinite quotient group of $G$ is countable.} Finally, we present an example of a dense subgroup $G$ of the compact metrizable group $\mathbb{Z}(2)^\omega$ such that $G$ is of the first category in itself, has measure zero, but the dual group $G_p^\wedge$ does not contain infinite compact subsets. This complements Theorem 1.6 in [J.E.~Hart and K.~Kunen, Limits in function spaces and compact groups, \textit{Topol. Appl.} \textbf{151} (2005), 157--168]. As a consequence, we obtain an example of a precompact reflexive abelian group which is of the first Baire category.