Chabauty topology

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In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G.

The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. It is also linked to the Hausdorff topology for closed subsets of metric spaces.

This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space. More precisely, it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept.

• Claude Chabauty, Limite d'ensembles et géométrie des nombres. Bulletin de la Société Mathématique de France, 78 (1950), p. 143-151