Topological groups with \(\sigma\)-compact space of subgroups (Q1063700)

The paper is a contribution to the theory of locally compact groups. The space \({\mathcal L}(G)\) of all closed subgroups (of a locally compact group G) endowed with the E-topology is investigated. The author has formulated the hypothesis that \({\mathcal L}(G)\) is \(\sigma\)-compact if and only if G is \(\sigma\)-compact and the set of all non-compact elements from \({\mathcal L}(G)\) is at most countable. It is proved that the hypothesis is true for any metrizable locally compact group G. The author raised the following question: Does there exist a non-compact locally compact group having no non-compact closed metrizable subgroups?