A NOTE ON THE OPEN MAPPINGS OF LOCALLY COMPACT GROUPS

Abstract Let G be a locally compact topological group and $\mathcal {L}(G)$ the space of all its closed subgroups endowed with the Vietoris topology. Let $\mathcal {L}_c(G)$ be the subspace of all compact subgroups of G . Any continuous morphism $\varphi \colon G\to H$ between locally compact groups G and H functorially induces a continuous map $\varphi _*\colon \mathcal {L}_c(G)\to \mathcal {L}_c(H)$ given by $\varphi _*(L)=\varphi (L)$ . The main problem addressed in this paper is that of determining the relationship between the openness of $\varphi $ and the openness of $\varphi _*$ . For example, we show that if G is locally compact with compact identity component and H is locally compact and totally disconnected, then $\varphi $ is open if and only if $\varphi _*$ is open.