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

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