Topologies on families of closed subsets

In 1914, Felix Hausdorff introduced a metric, called the Hausdorff metric on the set of closed subspaces, Hausdorff space, of a metric space. This metric and the corresponding topology are the main objects of the study. In algebraic geometry, there is an analogous object called the Hilbert scheme, whose points correspond to closed subschemes of a projective variety X. We give a modular interpretation of the Hausdorff space analogous to the one for the Hilbert scheme. The Hilbert scheme is used in various structures in algebraic geometry; using the Hausdorff space we can mimic these constructions in topology. For example, when an algebraic group acts on a projective variety, one can form a quotient X //Hilb G called the Hilbert quotient. We consider the analogous Hausdorff quotient X //Haus G associated to a topological group acting on a metric space. We note that this quotient has some desirable properties: When X is compact X //Haus G is compact and when G is also compact X //Haus G is the usual quotient X / G. In addition to the Hausdorff topology, one can also topologize the set of closed subspaces of X by considering the compact open topology on the set of continuous maps from X to the Sierpinski space. Although the resulting Hilbert space is less well-behaved than the Hausdorff space, it admits a nice modular interpretation when X is locally compact.