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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *perfectly normal* when the following two conditions hold.\n\n1. $X$ is a [[normal space|0NSR9V8TED]].\n2. Every [[closed set|0ZCK8DQYN1]] of $X$ is a [[G-delta set|V1TBNP8R3S]], that is, a countable intersection of [[open sets|02BH3MCBQB]].\n\n## Remarks\n\nEquivalently, for all disjoint closed sets $A, B \\subseteq X$ there is a [[continuous function|0X3WGX95FE]] $f \\colon X \\to [0, 1]$ with $f^{-1}(0) = A$ and $f^{-1}(1) = B$.\nAnother equivalent form: every closed set is $f^{-1}(0)$ for some continuous $f \\colon X \\to \\R$.\nA perfectly normal space is [[completely normal|9HBDQ2W9CF]].\nA perfectly normal $T_1$ space is called a $T_6$ space.\n\nExample: every metric space $(X, d)$ is perfectly normal.\nIt is normal, and a closed set $A$ equals $\\bigcap_{n \\ge 1} \\set{x : d(x, A) \u003c 1/n}$, an intersection of countably many open sets.\n\nNon-example: the space $[0, \\omega_1]$ of countable ordinals together with $\\omega_1$, with the order topology, is compact Hausdorff and so normal.\nThe closed set $\\set{\\omega_1}$ is not a G-delta set: any countably many neighbourhoods $(\\alpha_n, \\omega_1]$ contain $(\\sup_n \\alpha_n, \\omega_1]$, which has more than one point.\nSo clause 2 fails.\n",
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