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  "body": "\n## Statement\n\nLet $X$ be a [[paracompact|8R29G6NG9Q]] [[Hausdorff space|0S1FJ8VEX9]].\nThen $X$ is [[normal|0NSR9V8TED]].\nThat is, for any two disjoint closed sets $A, B \\subseteq X$, there are disjoint open sets $U \\supseteq A$ and $V \\supseteq B$.\n\n## Remarks\n\nThe proof has two steps of the same shape.\nFirst, $X$ is [[regular|03HMJASQBQ]]: for a closed set $C$ and a point $x \\notin C$, cover $C$ by open sets whose closures miss $x$, and add $X \\setminus C$.\nTake a [[locally finite|DZ297MMGV8]] open refinement.\nThe union of the closures of the members that meet $C$ is closed, by local finiteness, and it misses $x$.\nThe second step repeats this argument with a closed set in place of the point $x$, using regularity.\n\nExample at work: every [[compact|0TVJZYB6SH]] Hausdorff space is paracompact, so it is normal.\n\nThe Hausdorff hypothesis cannot be dropped.\nTake $X = \\set{a, b, c}$ with open sets $\\emptyset$, $\\set{a}$, $\\set{a, b}$, $\\set{a, c}$ and $X$.\nIt is finite, so paracompact.\nThe closed sets $\\set{b}$ and $\\set{c}$ are disjoint, but every open set containing either of them contains $a$.\n\n## Explanations\n\n### L1 human\n\nA normal space is one where two disjoint closed sets can be put in disjoint open boxes.\nDieudonné's theorem says that paracompactness plus the Hausdorff condition already gives this.\nIt extends the classical fact that compact Hausdorff spaces are normal.\n\n### L2 human\n\nNormality is what makes [[Urysohn's lemma|01EY3VR7ZD]] work.\nSo on a paracompact Hausdorff space, there are many continuous real functions.\nThis is the first step toward partitions of unity subordinate to an open cover.\nIn fact, a Hausdorff space is paracompact if and only if every open cover has a subordinate partition of unity.\n",
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