Gromov–Hausdorff space

Description

Let $Z$ be a metric space and let $A, B \subseteq Z$ be nonempty. The Hausdorff distance $d_H(A, B)$ is the infimum of the numbers $r \ge 0$ with the following property. Each point of $A$ lies within distance $r$ of $B$, and each point of $B$ lies within distance $r$ of $A$.

Let $X$ and $Y$ be nonempty compact metric spaces. Their Gromov–Hausdorff distance is $$d_{GH}(X, Y) = \inf d_H(f(X), g(Y)),$$ where the infimum runs over all metric spaces $Z$ and all isometric embeddings $f \colon X \to Z$ and $g \colon Y \to Z$.

The Gromov–Hausdorff space is the set of isometry classes of nonempty compact metric spaces, with the distance $d_{GH}$.