The Hausdorff topology as a moduli space

TL;DR

This paper explores the Hausdorff topology on compact subsets of a metric space, showing it as a moduli space representing a specific functor and drawing analogies to algebraic geometry's Hilbert scheme.

Contribution

It establishes that the Hausdorff space functions as a moduli space for certain closed subspaces, depending only on the topology of the underlying space, and introduces the Hausdorff quotient as an analog of the Hilbert quotient.

Findings

Hausdorff space represents a functor on sequential topological spaces.

The topology depends only on the underlying topological space, not the metric.

Introduction of the Hausdorff quotient as an algebraic geometry analog.

Abstract

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.