mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Chabauty topology is a certain

topological structure In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

introduced in 1950 by

Claude Chabauty Claude Chabauty (born May 4, 1910 in Oran, died June 2, 1990 in Dieulefit) was a French mathematician. Career He was admitted in 1929 to the École normale supérieure (Paris), École normale supérieure in Paris. In 1938 he obtained his docto ...

, on the set of all

closed subgroup In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...

s of a

locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...

''G''. The intuitive idea may be seen in the case of the set of all lattices in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

''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. This topology can be derived from the

Vietoris topology In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

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.

References

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