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

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.