Locally compact groups with totally disconnected space of subgroups (Q1755293)

Let \(X\) be a topological space and \(\mathcal{F}(X)\) be the set of closed subsets of \(X\). Define for a compact subset \(K\) and open subsets \(U_1, \ldots, U_n\) of \(X\), the sets of \(X\) defined by: \(\mathcal O_1(K)= \{ F\in \mathcal F(X) ~|~F\cap K= \emptyset\}\) and \(\mathcal O_2(U_i)= \{ F\in \mathcal F(X) ~|~F\cap U_i\not= \emptyset\}, \forall i,\); then the sets: \[ \Omega(K; U_1, \cdots, U_n)= \mathcal O_1(K) \cap \mathcal O_2(U_1) \cap \cdots \cap \mathcal O_2(U_n) \] constitute a basis of the Chabauty-Fell topology. \(\mathcal F(X)\) is quasi-compact, in particular if \( X\) is a locally compact space, then \(\mathcal F(X)\) is Hausdorff. A topological space is called totally disconnected if every connected component consists of a single point. Denote by \(\mathcal K(X)\) the family of compact subsets of \(X\), then \(\mathcal K(X)\) is dense in \(\mathcal F(X)\) with respect to the Chabauty-Fell topology. In this work, the authors prove in a first step that \(X\) is totally disconnected if and only if \(\mathcal F(X)\) (or \(\mathcal K(X)\) is is totally disconnected). An element \(x\) of a locally compact group \( G\) is called elliptic if \(x\) is contained in a compact subgroup of \(G\). A locally compact group \(G\) is elliptic if any element belongs to a compact subgroup. In a second step, the authors study the set \(\mathcal{SUB}(G)\) of closed subgroups of \(G\). It is closed in \(\mathcal{F}(G)\) and compact, the topology inherited is called the Chabauty topology on \(\mathcal{SUB}(G)\). Precisely, if \(G_0\) is the identity component of \(G\), then the authors prove that the Chabauty space \(\mathcal{SUB}(G)\) is totally disconnected if and only if \(G\) is totally disconnected or elliptic and \(G_0\) is central.