Characterizing sequences for precompact group topologies

TL;DR

This paper investigates the properties of sequences that characterize precompact group topologies on abelian groups, establishing conditions for when such groups are ss-characterized, and exploring their structural and duality properties.

Contribution

It provides a complete characterization of ss-characterized precompact abelian groups, especially in the metrizable case, and describes their structural and duality features.

Findings

Metrizable precompact abelian groups are ss-characterized iff they are countable.

No infinite pseudocompact abelian group is ss-characterized.

An ss-characterized precompact abelian group is hereditarily disconnected.

Abstract

A precompact group topology $\tau$ on an abelian group $G$ is called {\em single sequence characterized} (for short, {\em ss-characterized}) if there is a sequence $\mathbf{u}= (u_n)$ in $G$ such that $\tau$ is the finest precompact group topology on $G$ making $\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group $(G,\tau)$ is $ss$-characterized iff it is countable. For every metrizable precompact group topology $\tau$ on a countably infinite abelian group $G$ there exists a group topology $\eta$ such that $\eta$ is strictly finer than $\tau$ and the groups $(G,\tau)$ and $(G,\eta)$ have the equal Pontryagin dual groups. We give a complete description of all $ss$-characterized precompact abelian groups modulo countable $ss$-characterized groups from which we derive: (1) No infinite pseudocompact abelian group is $ss$-characterized. (2) An $ss$-characterized precompact abelian group is hereditarily disconnected.