In
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 introduced in 1950 by
Claude Chabauty, on the set of all
closed subgroup
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
s 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
''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 point ...
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