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

The Hausdorff distance, the Gromov–Hausdorff, the Fréchet and the natural pseudo-distance are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as infρF(ρ)infρF(ρ) where FF is a suitable functional and ρρ 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 KK, in such a way that the composition in KK (extending the composition of homeomorphisms) passes to the limit and, at the same time, KK is compact.