The Hausdorff topology as a moduli space

In 1914, F. Hausdorff defined a metric on the set of closed subsets of a metric space $X$. This metric induces a topology on the set $H$ of compact subsets of $X$, called the Hausdorff topology. We show that the topological space $H$ represents the functor on the category of sequential topological spaces taking $T$ to the set of closed subspaces $Z$ of $T \times X$ for which the projection $\pi_1 : Z \to T$ is open and proper. In particular, the Hausdorff topology on $H$ depends on the metric space $X$ only through the underlying topological space of $X$. The Hausdorff space $H$ provides an analog of the Hilbert scheme in topology. As an example application, we explore a certain quotient construction, called the Hausdorff quotient, which is the analog of the Hilbert quotient in algebraic geometry.