Gromov–Hausdorff convergence

Remarks

Example: let $X_n = \set{k/n : k = 0, 1, \ldots, n}$ and $X = [0, 1]$, both with the distance $\abs{s - t}$. Both lie in $\R$, and each point of $[0, 1]$ is within $\tfrac{1}{2n}$ of $X_n$. So $d_{GH}(X_n, X) \le \tfrac{1}{2n} \to 0$, and $X_n$ converges to $[0, 1]$.

Non-example: let $X_n$ be the round sphere of radius $n$ with its great-circle distance. One has $\abs{\operatorname{diam} Y - \operatorname{diam} Z} \le 2\, d_{GH}(Y, Z)$ for compact $Y$ and $Z$. The diameters $\pi n$ tend to infinity, so $(X_n)$ converges to no compact space.

The limit is unique up to isometry, because $d_{GH}$ is a metric on isometry classes. For spaces that are not compact, such as complete Riemannian manifolds, one uses pointed Gromov–Hausdorff convergence. There a base point is fixed in each space, and one asks for convergence of the closed balls of each radius $r$ around the base points.