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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nAssume that $X$ is [[second-countable|03PPNCM9GG]], [[regular|03HMJASQBQ]] and [[Hausdorff|0S1FJ8VEX9]].\nThen $X$ is [[metrizable|091GJ3FYD7]]: there is a metric on $X$ whose open sets are exactly the open sets of $X$.\n\n## Remarks\n\nA second-countable regular space is normal, so [[Urysohn's lemma|01EY3VR7ZD]] applies to it.\nWith a countable base one picks countably many continuous functions $f_n \\colon X \\to [0, 1]$ that separate points from closed sets.\nThe map $x \\mapsto (f_n(x))_n$ is then an embedding of $X$ into the [[Hilbert cube|JKMMJ8RFC6]] $[0,1]^{\\N}$, which is metrizable.\nFor example, every subspace of $\\R^n$ satisfies the hypotheses, and so does every compact Hausdorff space with a countable base.\nSecond countability cannot be dropped: the [[Sorgenfrey line|Y4T3GVK8GV]] is regular and Hausdorff but not metrizable.\nRegularity cannot be dropped either: there are second-countable Hausdorff spaces that are not regular, and a metrizable space is always regular.\nThe converse holds only in part: a metrizable space is regular and Hausdorff, and it is second-countable exactly when it is separable.\n\n## Explanations\n\n### L1 human\n\nA metric is a way to measure distances, and many spaces are given only by their open sets.\nThe theorem says when distances can be put back in.\nIt is enough that the space has a countable family of basic open sets and that points and closed sets can be kept apart by open sets.\nSo a large class of spaces that look abstract can be treated as metric spaces.\n\n### L2 human\n\nThe theorem is the first answer to the metrization problem: which topological spaces come from a metric.\nThe proof is an embedding into the Hilbert cube, a standard method that also shows each such space is a subspace of one fixed compact metric space.\nIt characterizes separable metrizable spaces as the second-countable regular Hausdorff spaces.\nThe [[Nagata–Smirnov metrization theorem|KB11T6389G]] and the [[Bing metrization theorem|RGCZ2P9QRM]] remove the countability condition and give a full characterization.\n",
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