The geometric topology

We shall define a topology on the set of closed subsets of a topological space. We shall thereby derive topologies for both the space of complete hyperbolic manifolds and the space of geodesic laminations (see Section I.4.1 (Geodesic Laminations)). This topology was first considered by Chabauty (1950) as a topology on the space of closed subgroups of a locally compact topological group, and later by Harvey (1977) with specific reference to Fuchsian groups. See also Michael, (1951).

Definition. Given a topological space X, the Chabauty topology on C(X) (the set of all closed subsets of X) has a sub-basis given by sets of the following form:

1. (1) O1(K) = {A |A∩K = Ø} where K is compact;

2. (2) O2(U) = {A |A∩U ≠ Ø} where U is open.

If X is compact and metrizable, the Chabauty topology agrees with the topology induced by the Hausdorff metric. The Chabauty topology has the following nice topological properties.

Proposition: Properties of Chabauty topology. Let X be an arbitrary topological space (no particular assumptions), then:

1. (1) C(X) the set of closed subsets of X with the Chabauty topology is compact.

2. (2) If X is Hausdorff, locally compact and second countable, C(X) is separable and metrizable.

proof. (1) By Alexander's Sub-base theorem (Rudin, 1973, p. 368), we need only show that every covering by sub-basis elements has a finite sub-covering.