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  "body": "\n## Summary\n\nA Polish group is a [[topological group|92700QAB5Y]] whose topology is a [[Polish space|584GXXY73M]], that is, separable and completely metrizable.\nPolish groups are the natural setting for descriptive set theory of group actions.\nEvery second-countable locally compact Hausdorff group is Polish, and so is the group of all permutations of $\\N$.\nAnother example is the unitary group of a separable Hilbert space with the strong operator topology.\nTheir theory combines the Baire category theorem with the group structure, for example through automatic continuity results.\n\n## Explanations\n\n### L1 human\n\nA Polish group is a group that has a good notion of distance, with no holes, and a countable set of points that comes close to everything.\nThe real line under addition is one, and so is the circle.\nA larger example is the group of all rearrangements of the natural numbers, where two rearrangements are close when they agree on a long initial list of numbers.\nThese groups are big, but their topology is tame enough for strong theorems.\n",
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