A NOTE ON THE OPEN MAPPINGS OF LOCALLY COMPACT GROUPS

Source abstract

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