The notion for convergence of metric spaces that is a generalization of Hausdorff convergence.
The Gromov–Hausdorff distance measures how far two compact metric spaces are from being isometric. If $X$ and $Y$ are two compact metric spaces, then $d_{GH}(X, Y)$ is defined to be the infimum of all numbers $d_H(f(X), g(Y))$ for all metric spaces $M$ and all isometric embeddings $f: X \rightarrow M$ and $g: Y \rightarrow M$. Here, $d_H$ denotes Hausdorff distance between subsets in $M$.
The Gromov–Hausdorff distance turns the set of all isometry classes of compact metric spaces into a metric space, called Gromov–Hausdorff space, and it therefore defines a notion of convergence for sequences of compact metric spaces, called Gromov–Hausdorff convergence. A metric space to which such a sequence converges is called the Gromov–Hausdorff limit of the sequence.