Characterizing sequences for precompact group topologies

Abstract

A precompact group topology $τ$ 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 $τ$ 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 $(G,τ)$ is $ss$-characterized iff it is countable. For every metrizable precompact group topology $τ$ on a countably infinite abelian group $G$ there exists a group topology $η$ such that $η$ is strictly finer than $τ$ and the groups $(G,τ)$ and $(G,η)$ 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.