Successors of topologies of connected locally compact groups
Streszczenie
Let $G$ be a group and $\sigma , \tau $ be topological group topologies on $G$. We say that $\sigma $ is a successor of $\tau $ if $\sigma $ is strictly finer than $\tau $ and there is no group topology properly between them. In this note, we explore the existence of successor topologies in topological groups, particularly focusing on non-abelian connected locally compact groups. Our main contributions are twofold: for a connected locally compact group $(G, \tau )$, we show that (1) if $(G, \tau )$ is compact, then $\tau $ has a precompact successor if and only if there exists a discontinuous homomorphism from $G$ into a simple connected compact group with dense image, and (2) if $G$ is solvable, then $\tau $ has no successors. The result (1) implies that the topology on a connected compact Lie group does not have a successor. Our work relies on the previous characterization of locally compact group topologies on abelian groups having successors.
