Chabauty topology

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. It is closely related to the Fell topology on the set of all closed subets of G and to the Hausdorff distance.

Intuitively, two closed subgroups of G are close in the Chabauty topology if, within any compact subset of G, every point of one subgroup is close to some point of the other, and vice versa. For instance, if ${\displaystyle (\alpha _{n})}$ is a sequence of positive real numbers, then the sequence of lattices ${\displaystyle \alpha _{n}\mathbb {Z} }$ in the additive group ${\displaystyle \mathbb {R} }$ converges to

• ${\displaystyle \alpha \mathbb {Z} }$ if ${\displaystyle \alpha _{n}\to \alpha }$ with ${\displaystyle \alpha \neq 0}$,
• ${\displaystyle \mathbb {R} }$ if ${\displaystyle \alpha _{n}\to 0}$,
• the trivial subgroup ${\displaystyle \{0\}}$ if ${\displaystyle \alpha _{n}\to \infty }$.

Chabauty's original motivation was to study limit groups of lattices in ${\displaystyle \mathbb {R} ^{n}}$.