Metrizability in a space of subgroups of a Lie group (Q2640711)

Let G be a real Lie group and L(G) be the space of closed subgroups of the topological group G. The space L(G) is equipped with the Vietoris topology. Theorem: If G satisfies the following conditions: 1) the set of compact subgroups of G is closed in L(G); 2) for every noncompact subgroup \(H\subset G\) the group \(H/H_ 0\) is finitely generated. Here \(H_ 0\) is the connected component of the subgroup H, then L(G) admits a complete metric.