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  "body": "\n## Description\n\nThe *Urysohn universal space* $\\mathbb{U}$ is a [[complete|0BY52JW6NK]] [[separable|0Z2P41SVDP]] [[metric space|0VKD1TP7F3]] with two properties.\n\n1. Universality: every separable metric space is [[isometric|02HPP5VCZH]] to a subset of $\\mathbb{U}$.\n2. Ultrahomogeneity: every isometry between two finite subsets of $\\mathbb{U}$ extends to an isometry of $\\mathbb{U}$ onto itself.\n\nUp to isometry, there is exactly one complete separable metric space with both properties.\n\n## Remarks\n\nAn equivalent characterisation uses a one-point extension property.\nLet $A \\subseteq \\mathbb{U}$ be finite, and let $A \\cup \\set{x}$ carry a metric extending that of $A$.\nThen there is a point $u \\in \\mathbb{U}$ with $d(u, a) = d(x, a)$ for all $a \\in A$.\nA complete separable metric space with this property is isometric to $\\mathbb{U}$.\n\nThe space $C[0, 1]$ with the sup norm is also universal for separable metric spaces, but it is not ultrahomogeneous.\n\n## Explanations\n\n### L1 human\n\nThe Urysohn space is one separable metric space that contains a perfect copy of every other one.\nIt is also as symmetric as possible: any finite configuration of points can be moved onto any other congruent configuration by a symmetry of the whole space.\nIt is built step by step, adding at each stage every point that a finite metric extension asks for.\n",
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