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  "body": "\n## Definition\n\nA *Polish group* is a [[topological group|92700QAB5Y]] $G$ whose underlying topological space is a [[Polish space|584GXXY73M]].\nThat is, the topology of $G$ is separable and is induced by some complete metric.\n\n## Remarks\n\nExample: the [[symmetric group|0515A6D2Y1]] $S_\\infty$ of all bijections $\\N \\to \\N$, with the topology of pointwise convergence.\nIt is a $G_\\delta$ subset of the Polish space $\\N^\\N$, hence Polish, and composition and inversion are continuous for this topology.\nNon-example: $(\\Q, +)$ with the topology from $\\R$.\nIt is separable and metrizable, but it is countable without isolated points.\nSo it is a countable union of nowhere dense singletons and is not a [[Baire space|0ZH5AJ2ED7]]; hence no complete metric induces its topology.\nThe metric need not be invariant: a Polish group always has a compatible left-invariant metric, but this metric may fail to be complete.\n",
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