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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA family of subsets of $X$ is *$\\sigma$-locally finite* if it is a union of countably many [[locally finite families|DZ297MMGV8]].\nThen $X$ is [[metrizable|091GJ3FYD7]] if and only if $X$ is [[regular|03HMJASQBQ]], [[Hausdorff|0S1FJ8VEX9]] and has a $\\sigma$-locally finite base.\n\n## Remarks\n\nFor a metric space, [[Stone's theorem|7QFKD5JXH6]] gives, for each $n$, a locally finite open refinement $\\mathcal{B}_n$ of the cover by balls of radius $1/n$.\nThe union of the $\\mathcal{B}_n$ is a $\\sigma$-locally finite base.\nFor the converse one shows that $X$ is normal and builds countably many locally finite families of continuous functions to $[0,1]$.\nThese functions embed $X$ into a metrizable product space.\nA countable base is $\\sigma$-locally finite, since each one-element family is locally finite.\nSo the [[Urysohn metrization theorem|HBFHDH4EJX]] is a special case.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives an exact test for when a topology comes from a distance.\nCountably many basic open sets are no longer needed.\nIt is enough that the basic open sets come in countably many layers, and that near each point only finitely many sets of each layer are present.\n\n### L2 human\n\nTogether with the [[Bing metrization theorem|RGCZ2P9QRM]], this is the general solution of the metrization problem.\nIts proof uses [[Urysohn's lemma|01EY3VR7ZD]] to build the continuous functions of the embedding.\nA useful consequence is that a paracompact Hausdorff space that is locally metrizable is metrizable.\n",
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