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  "body": "\n## Statement\n\nLet $X$ be a [[metrizable space|09KDHB82SG]].\nThen $X$ is [[normal|0AJ646N339]].\n\n## Remarks\n\nProof: choose a metric $d$ for the topology. For disjoint closed sets $A$ and $B$, let $U = \\{x : d(x, A) \u003c d(x, B)\\}$ and $V = \\{x : d(x, B) \u003c d(x, A)\\}$. Here $d(x, A)$ is the [[distance to a set|07TXYYVYZR]].\nThe sets $U$ and $V$ are open, because $x \\mapsto d(x, A)$ is continuous. They are disjoint, and $A \\subseteq U$ because $d(x, A) = 0 \u003c d(x, B)$ for $x \\in A$.\nSingle points are closed in a metric space. Thus each metrizable space is normal and T1, and also [[Hausdorff|0P9DCDES73]].\n\n## Explanations\n\n### L1 human\n\nIn a metric space, you separate two disjoint closed sets by the points that are nearer to one set than to the other.\n",
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