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  "body": "\n## Description\n\nLet $Z$ be a [[metric space|0VKD1TP7F3]] and let $A, B \\subseteq Z$ be nonempty.\nThe *Hausdorff distance* $d_H(A, B)$ is the infimum of the numbers $r \\ge 0$ with the following property.\nEach point of $A$ lies within distance $r$ of $B$, and each point of $B$ lies within distance $r$ of $A$.\n\nLet $X$ and $Y$ be nonempty [[compact|0TVJZYB6SH]] metric spaces.\nTheir *Gromov–Hausdorff distance* is\n$$d_{GH}(X, Y) = \\inf d_H(f(X), g(Y)),$$\nwhere the infimum runs over all metric spaces $Z$ and all [[isometric embeddings|02HPP5VCZH]] $f \\colon X \\to Z$ and $g \\colon Y \\to Z$.\n\nThe *Gromov–Hausdorff space* is the set of isometry classes of nonempty compact metric spaces, with the distance $d_{GH}$.\n\n## Remarks\n\nThe function $d_{GH}$ is a metric on isometry classes.\nIn particular, $d_{GH}(X, Y) = 0$ for compact $X$ and $Y$ only when $X$ and $Y$ are isometric.\nThe Gromov–Hausdorff space is a [[complete metric space|0BY52JW6NK]] and is [[separable|0Z2P41SVDP]].\nFinite metric spaces with rational distances form a countable dense subset.\n\nFor a one-point space $P$ and a compact metric space $X$, one has $d_{GH}(P, X) = \\operatorname{diam}(X) / 2$.\n\n## Explanations\n\n### L1 human\n\nThe Gromov–Hausdorff space is a space whose points are themselves shapes, namely compact metric spaces.\nTwo shapes are close when they can be placed in a common space so that each is near the other.\nFor example, a fine finite grid of points is close to the square it sits in.\nThis lets one speak of a sequence of spaces converging to a limit space.\n",
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