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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is an *Urysohn space* when the following holds.\nFor all $x, y \\in X$ with $x \\ne y$, there are [[neighbourhoods|0ZCY0Z80B6]] $U$ of $x$ and $V$ of $y$ such that $\\overline{U} \\cap \\overline{V} = \\varnothing$.\n\n## Remarks\n\nEquivalently, any two distinct points have disjoint [[closed|0ZCK8DQYN1]] neighbourhoods.\nEvery Urysohn space is [[Hausdorff|0S1FJ8VEX9]], since the interiors of $\\overline{U}$ and $\\overline{V}$ are disjoint open sets around $x$ and $y$.\nEvery [[regular|03HMJASQBQ]] Hausdorff space is Urysohn.\n\nExample: $\\R$ is Urysohn.\nFor $x \u003c y$, put $r = (y - x)/3$; the closures $[x - r, x + r]$ and $[y - r, y + r]$ of the open intervals of radius $r$ are disjoint.\n\nNon-example: an infinite set with the [[cofinite topology|0B906ZBTMZ]] is not Urysohn.\nAny two nonempty open sets in it meet, so even the Hausdorff condition fails.\n\nNotation: $T_{2\\frac12}$.\nA space in which distinct points are separated by a continuous function $f \\colon X \\to [0, 1]$ is called completely Hausdorff; some authors call that an Urysohn space.\n",
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