Characterized subgroups of topological abelian groups

TL;DR

This paper investigates the properties and classifications of characterized subgroups in topological abelian groups, extending known results, and introduces autochacaracterized groups, providing a comprehensive understanding of their structure.

Contribution

It extends the theory of characterized subgroups beyond compact groups, introduces autochacaracterized groups, and describes characterized subgroups in discrete abelian groups.

Findings

Characterized subgroups are extended to general topological abelian groups.

Autochacaracterized groups are exactly the non-compact locally compact abelian groups.

Complete description of characterized subgroups in discrete abelian groups.

Abstract

A subgroup $H$ of a topological abelian group $X$ is said to be characterized by a sequence $\mathbf v =(v_n)$ of characters of $X$ if $H=\{x\in X:v_n(x)\to 0\ \text{in}\ \mathbb T\}$. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of autochacaracterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proved to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.