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  "body": "\n## Definition\n\nLet $X_1, X_2, \\ldots$ and $X$ be nonempty [[compact|0TVJZYB6SH]] [[metric spaces|0VKD1TP7F3]].\nLet $d_{GH}$ be the Gromov–Hausdorff distance, the metric of the [[Gromov–Hausdorff space|ETRF2FHAEE]].\nThe sequence $(X_n)$ *converges to $X$ in the Gromov–Hausdorff sense* if $d_{GH}(X_n, X) \\to 0$ as $n \\to \\infty$.\n\n## Remarks\n\nExample: let $X_n = \\set{k/n : k = 0, 1, \\ldots, n}$ and $X = [0, 1]$, both with the distance $\\abs{s - t}$.\nBoth lie in $\\R$, and each point of $[0, 1]$ is within $\\tfrac{1}{2n}$ of $X_n$.\nSo $d_{GH}(X_n, X) \\le \\tfrac{1}{2n} \\to 0$, and $X_n$ converges to $[0, 1]$.\n\nNon-example: let $X_n$ be the round sphere of radius $n$ with its great-circle distance.\nOne has $\\abs{\\operatorname{diam} Y - \\operatorname{diam} Z} \\le 2\\, d_{GH}(Y, Z)$ for compact $Y$ and $Z$.\nThe diameters $\\pi n$ tend to infinity, so $(X_n)$ converges to no compact space.\n\nThe limit is unique up to isometry, because $d_{GH}$ is a metric on isometry classes.\nFor spaces that are not compact, such as complete Riemannian manifolds, one uses *pointed* Gromov–Hausdorff convergence.\nThere 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.\n",
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