Locally minimal topological groups

TL;DR

This paper investigates locally minimal topological groups, exploring their properties, connections to NSS and GTG sets, and characterizing when such groups admit finer topologies, with a focus on abelian groups.

Contribution

It introduces the concept of locally GTG groups, establishes their relation to UFSS and NSS groups, and provides constructions and characterizations of these groups in the abelian setting.

Findings

Locally minimal NSS groups are often metrizable under certain conditions.

A topological abelian group is UFSS iff it is locally minimal, locally GTG, and NSS.

Bounded abelian groups admit non-discrete locally minimal and GTG topologies iff their size is at least continuum.

Abstract

A Hausdorff topological group $(G,\tau)$ is called locally minimal if there exists a neighborhood $U$ of 0 in $\tau$ such that $U$ fails to be a neighborhood of zero in any Hausdorff group topology on $G$ which is strictly coarser than $\tau.$ Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudo--convex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. We also prove that a bounded abelian group $G$ admits a non-discrete locally minimal and locally GTG group topology iff $|G|\geq \cont$.