On the compact space of closed subgroups of locally compact groups
For a locally compact group \(G\), denote by \(\mathcal{C} (G)\) the hyperspace of all closed subgroups of \(G\) equipped with the Chabauty topology [\textit{C. Chabauty}, Bull. Soc. Math. Fr. 78, 143--151 (1950; Zbl 0039.04101)]. It is a well known fact that \(\mathcal{C} (G)\) endowed with the Chabauty topology is compact for a locally compact group \(G\). The Main Theorem states that the functor \(\mathcal{C}: G \mapsto \mathcal{C}(G)\) preserves strong projective limits on the category of locally compact topological groups and proper morphisms. For the case \(G\) compact, the analogous theorem was proved in [\textit{S. Fisher} and \textit{P. Gartside}, On the space of subgroups of a compact group. I. Topology Appl. 156, No. 5, 862--871 (2009; Zbl 1169.22003)].NEWLINENEWLINEAs an application, certain conditions under which the hyperspace \(\mathcal{C}(G)\) is connected are established: namely, the main result is used to reduce the study of connectedness of the hyperspace \(\mathcal{C}(G)\) of an arbitrary locally compact pro-Lie group \(G\) to that of Lie groups (Theorem 4.7), from which the well-known theorem by Protasov and Tsybenko about disconnectedness of \(\mathcal{C}(G)\) in case of a compact group \(G\) follows [\textit{I. V. Protasov} and \textit{Yu. V. Tsybenko}, Ukr. Mat. Zh. 35, No. 3, 382--385 (1983; Zbl 0517.22004)].
scientific article; zbMATH DE number 6365979