Topologies induced by group actions

TL;DR

This paper explores canonical topologies induced by group actions on groups and rings, analyzing their properties and limitations, especially in the context of Polish groups and structures.

Contribution

It introduces the finest topologies making group or ring actions continuous and examines their properties within Polish structures.

Findings

The finest topology can be non-Hausdorff for certain Polish group actions.

Introduces canonical topologies induced by group actions on algebraic structures.

Studies the limitations of these topologies in the context of Polish groups.

Abstract

We introduce some canonical topologies induced by actions of topological groups on groups and rings. For $H$ being a group [or a ring] and $G$ a topological group acting on $H$ as automorphisms, we describe the finest group [ring] topology on $H$ under which the action of $G$ on $H$ is continuous. We also study the introduced topologies in the context of Polish structures. In particular, we prove that there may be no Hausdorff topology on a group $H$ under which a given action of a Polish group on $H$ is continuous.