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  "body": "\n## Statement\n\nLet $X$ be a [[compact space|0QPHW703PV]] that is [[Hausdorff|0MXB4CKHFW]].\nThen $X$ is [[normal|0AJ646N339]].\n\n## Remarks\n\nProof sketch: closed subsets of $X$ are compact, by [[closed subsets of a compact set are compact|066HW6D9A7]]. In a Hausdorff space, a point and a disjoint compact set have disjoint open neighbourhoods. A second use of compactness separates two disjoint compact sets.\nWith [[Urysohn's lemma|01EY3VR7ZD]], this shows that compact Hausdorff spaces have many continuous real functions.\n\n## Explanations\n\n### L1 human\n\nIn a compact Hausdorff space, you first separate points by open sets, and then compactness lets you combine finitely many of these choices.\nThe result separates any two disjoint closed sets.\n",
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