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For compact metric spaces $X$ and $Y$, the Gromov-Hausdorff distance iswhere the infimum runs over all metric spaces $Z$ and all isometric embeddings $\phi :X\to Z$ and $\psi :Y\to Z$. It measures how closely the two spaces can be placed inside one ambient metric space.

The Gromov-Hausdorff topology on isometry classes of compact metric spaces is the topology induced by the Gromov-Hausdorff distance.