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  "body": "\n## Definition\n\nA [[topological space|0CQHFPGJ45]] $X$ is *completely normal* when every subset of $X$, with the [[subspace topology|0WS9P12FQR]], is a [[normal space|0NSR9V8TED]].\nA completely normal space that is also [[T1|0NW6CH38YR]] is called a $T_5$ space.\n\n## Remarks\n\nEquivalently, $X$ is completely normal if and only if the following holds.\nFor all $A, B \\subseteq X$ with $A \\cap \\overline{B} = \\varnothing$ and $\\overline{A} \\cap B = \\varnothing$, there are disjoint [[open sets|02BH3MCBQB]] $U \\supseteq A$ and $V \\supseteq B$.\nIt is enough to check that every open subspace is normal.\n\nExample: every metric space $(X, d)$ is completely normal.\nEach subspace is again a metric space, and every metric space is [[normal|0AD0BZ49DW]].\n\nNon-example: the [[Tychonoff plank|D440JPY3QG]] is compact Hausdorff, hence normal, but it is not completely normal.\nIts subspace obtained by removing one corner point is not normal.\n\nEvery [[perfectly normal space|VK3ZMD1BCP]] is completely normal.\n",
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