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  "body": "\n## Statement\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA family $\\mathcal{F}$ of subsets of $X$ is *discrete* if each point of $X$ has a neighbourhood that meets at most one member of $\\mathcal{F}$.\nA family is *$\\sigma$-discrete* if it is a union of countably many discrete families.\nThen $X$ is [[metrizable|091GJ3FYD7]] if and only if $X$ is [[regular|03HMJASQBQ]], [[Hausdorff|0S1FJ8VEX9]] and has a $\\sigma$-discrete base.\n\n## Remarks\n\nA discrete family is locally finite, so one direction follows from the [[Nagata–Smirnov metrization theorem|KB11T6389G]].\nFor the other direction, take a metric space.\nThe cover by balls of radius $1/n$ has a $\\sigma$-discrete open refinement, built with a well-order of the balls.\nThe union over $n$ of these refinements is a $\\sigma$-discrete base.\nIn the real line, the open intervals with rational endpoints form a countable base, and a countable base is $\\sigma$-discrete.\n\n## Explanations\n\n### L1 human\n\nThe theorem is another exact test for when a topology comes from a distance.\nIt asks for basic open sets in countably many layers, where in each layer the sets are spread apart.\nNear each point at most one set of the layer is present.\n\n### L2 human\n\nThe theorems of Bing and of Nagata–Smirnov were found independently and give the same class of spaces.\nBing's form is often handy in proofs, because a discrete family behaves like a disjoint family of sets that are also kept apart from each other.\nIn the same work Bing proved that a collectionwise normal [[Moore space|68MA0RM5P6]] is metrizable.\n",
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