Successors of topologies of connected locally compact groups

Let G be a group and σ,τ be topological group topologies on G. We say that σ is a successor of τ if σ is strictly finer than τ 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,τ), we show that (1) if (G,τ) is compact, then τ 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 τ 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.