  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.
