Chabauty topology

Definition
Suppose $$G$$ is a locally compact topological group. The Chabauty topology on $$G$$ is a topology defined as follows:


 * The underlying set of the topological space is the set of all closed subgroups of $$G$$, which we denote $$\mathcal{C}(G)$$ here.
 * The basis is given as follows. For each compact subset $$K$$ of $$G$$, non-empty open subset $$U$$ of $$G$$ containing the identity element, and compact subgroup $$C$$ of $$G$$, define:

$$\mathcal{V}_{K,U}(C) := \{ D \in \mathcal{C}(G) \mid D \cap K \subseteq CU \mbox{ and } C \cap K \subseteq DU \}$$.

The basis is the collection of all such subsets. The local basis at any point $$C \in \mathcal{C}(G)$$ is given by the set of all basis sets for that particular $$C$$.