  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.
