No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit

TL;DR

This paper proves that the set of homeomorphisms cannot be embedded into a compact metric space with a composition operation that extends the usual composition and remains continuous in the limit.

Contribution

It establishes a fundamental limitation on embedding the automorphisms of a topological space into a compact metric space with a compatible composition structure.

Findings

Homeomorphisms cannot be embedded into a compact metric space with continuous composition.

Dissimilarity measures like Hausdorff and Gromov-Hausdorff share a common infimum-based definition.

The main theorem rules out certain natural extensions of automorphism groups into compact metric spaces.

Abstract

The Hausdorff distance, the Gromov-Hausdorff, the Fr\'echet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $\inf_\rho F(\rho)$ where $F$ is a suitable functional and $\rho$ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space $\mathcal{K}$, in such a way that the composition in $\mathcal{K}$ (extending the composition of homeomorphisms) passes to the limit and, at the same time, $\mathcal{K}$ is compact.