Chabauty topology
Encyclopedia
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
G.
The intuitive idea may be seen in the case of the set of all lattices
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
in a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
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 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.