Compact subspaces in the space of subgroups of a topological group (Q1090422)

Let G be a locally compact topological group and \({\mathfrak L}(G)\) be the space of closed subgroups of G endowed with the Vietoris topology. Compact subsets of \({\mathfrak L}(G)\) are studied. It is shown that a subset \({\mathfrak F}\subseteq {\mathfrak L}(G)\) is compact iff the following conditions are fulfilled: (1) \({\mathfrak F}\) is closed in \({\mathfrak L}(G)\); (2) \({\mathfrak F}\) does not contain infinite decreasing chains consisting of non-compact subgroups; (3) each closed subspace \({\mathfrak F}'\subseteq {\mathfrak F}\) contains finitely many maximal (in \({\mathfrak F}')\) non-compact subgroups; (4) if a closed subspace \({\mathfrak F}'\) of \({\mathfrak F}\) does not contain non-compact subgroups, then \(\cup {\mathfrak F}'\) is compact in G. In particular, a subset \({\mathfrak F}\subseteq {\mathfrak L}(G)\) is compact iff \({\mathfrak F}\) is closed and countably compact (Corollary 1), and every compact subspace of n\({\mathfrak K}(G)\) is scattered (Corollary 3), where n\({\mathfrak K}(G)\) is the subspace of \({\mathfrak L}(G)\) consisting of all non- compact subgroups. It is also proved that if G is a group of countable weight and the set of its inductively compact subgroups is closed, then \({\mathfrak L}(G)\) is a k-space (Theorem 3).