# Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell topology construction, which itself derives from the Vietoris topology concept." So, to understand the one, one could be served by understanding the other two. Unfortunately topology and functional analysis are not among my strong suits, so I am looking to grasp the moral purpose or design behind these topologies as well as see how they are connected to each other.

Perhaps some motivation is necessary in order to see that the Chabauty topology is interesting in the first place. For $({\Bbb R},+)$, the proper closed subgroups are the cyclic ones $\alpha\Bbb Z$, $\alpha\in\Bbb R^{+}$. The space of closed subgroups ${\rm Cha}(\Bbb R)$ is then a kind of moduli space of discrete lattices. (It seems people use the notation ${\cal C}(G)$ for this topological space, but I want to distinguish it from the various spaces of continuous maps or the group ${\rm C}^*$-algebra.) The space ${\rm Cha}(\Bbb C)$ is homeomorphic to the $4$-sphere, and the subspace comprised of lattices of $\Bbb C$ is homeomorphic to the product of an open interval with the complement of the trefoil knot in the $3$-sphere.

Questions.

• What is the main idea behind these three topologies respectively; what kind of information are they intended to capture? What do they say about the original space or topological group?
• In what way does the Fell topology derive from the Vietoris topology, and in what way does the Chabauty topology derive from the Fell topology?

Some definitions, more for my sake than anyone else's.

Let $X$ be a topological space. Denote by $K(X)$ the collection of compact subsets of $X$. For open subsets $U\subseteq X$ define $U^+=\{ K\in K(X):K\cap U\ne\varnothing\}$, $U^-=\{K\in K(X):K\subseteq U\}$. Then the Vietoris topology on ${\cal P}(X)\setminus\{\varnothing\}$ has as subbase all the $U^+$ and $U^-$ as $U$ ranges over the open subsets of $U\subseteq X$.

A complex inner product space $H$ is a Hilbert space if it is closed with respect to the metric induced by the inner product. The norm of a linear operator between normed vector spaces is defined by $\|A\|:={\rm sup}\|Av\|/\|v\|$ over $v\ne0$. A ${\rm C}^*$-algebra is an algebra of continuous linear endomorphisms of a Hilbert space which is closed in the norm topology and closed under taking adjoints. A ${}^*$-homomorphism $A\to B$ of ${\rm C}^*$-algebras is an algebra homomorphism which is bounded as a linear operator and commutes with taking adjoints. The group ${\rm C}^*$-algebra of a topological group $G$ is the completion of $C_c(G)$ [the algebra of $\Bbb C$-valued continuous functions from $G$ with compact support] with respect to the norm $\|f\|_{{\rm C}^*}:=\sup_\pi\|\pi(f)\}$ where $\pi$ is taken over ${}^*$-representations of $C_c(G)$ on Hilbert spaces. The Fell topology is the topology on the dual space of ${\rm C}^{*}(G)$.

Let ${\rm Cha}(G)$ denote the collection of closed subgroups of a locally compact topological group $G$; the Chabauty topology it is endowed with admits as neighborhood basis the sets $${\cal V}_{K,U}(C)=\{D\in{\rm Cha}(G):D\cap K\subset CU,~C\cap K\subset DU\}$$ as $C$ ranges over ${\rm Cha}(G)$, $K$ ranges over compact subsets of $G$ and $U$ ranges over open neighborhoods of $G$'s identity element.

• Pierre de la Harpe has a nice survey article about the Chabauty space, see arXiv:0807.2030. – user56706 Feb 28 '14 at 23:46
• This is a really good question -- I wish I knew the answer... – Chill2Macht Jun 9 '16 at 17:16
• I think the right setting for such a question is that of closed subsets of a set, rather than specifying to closed subgroups (which is of course an important motivation). – YCor May 1 at 15:02