Chabauty topology

We begin by defining a topology on the set of all closed subset of G. This is given by defining a neighbourhood basis for any closed subset X of G. Elements of the neighbourhood basis are given by V(X;C,U) = \{ Y \subset G \text { closed } \mid Y\cap C \subseteq UX \text{ and } X\cap C \subseteq UY \}, where C is any compact subset of G and U is any open neighbourhood U of the identity. The topology determined by this neighbourhood basis is the same as the Fell topology, and the set \mathrm{Subgp}(G) of closed subgroups of G is a closed subset in this topology. The inherited topology is called the Chabauty topology, and \mathrm{Subgp}(G) with this topology is called the Chabauty space. ==Examples==

The Chabauty space of the group \mathbb{R} is homeomorphic to the closed interval [0,\infty] via the map \alpha \mapsto \begin{cases} \mathbb{R}, & \text{if } \alpha = 0,\\ \alpha\mathbb{Z}, & \text{if } 0 The Chabauty space of \mathbb{R}^2 is homemorphic to a 4-sphere. The Chabauty space of \mathbb{R}^n for n>2 becomes more complicated. ==Relation to other topologies==

The definition Chabauty topology can be used to define a uniform structure on the set \mathrm{\mathcal{F}(G)} of all closed subsets of G. Namely, the sets \mathcal{V}(C,U) = \{ (X,Y) \in \mathcal{F}(G)^2 \mid X \cap C \subseteq UY \text{ and } Y\cap C \subseteq UX \}, define a set of entourages for \mathcal{F}(G), where C and U vary over the compact subsets of G and the open neighbourhoods of the identity, respectively. The induced topology of this uniform structure is the Chabauty topology. If the topology of G is first countable, then G can be endowed with a left-invariant metric which induces the topology. In this case, a series of closed subgroups converges in the Chabauty topology if and only if their intersections with any compact subset converge with respect to the Hausdorff distance. == References ==