---
tag: 9C1WSBNQX9
kind: concept
name: Polish group
msc: [22A05, 54H11, 54H05]
specializes: [7GKCGQ7215]
sources:
  - url: https://en.wikipedia.org/wiki/Polish_space
---

## Summary

A Polish group is a [[topological group|92700QAB5Y]] whose topology is a [[Polish space|584GXXY73M]], that is, separable and completely metrizable.
Polish groups are the natural setting for descriptive set theory of group actions.
Every second-countable locally compact Hausdorff group is Polish, and so is the group of all permutations of $\N$.
Another example is the unitary group of a separable Hilbert space with the strong operator topology.
Their theory combines the Baire category theorem with the group structure, for example through automatic continuity results.

## Explanations

### L1 human

A 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.
The real line under addition is one, and so is the circle.
A 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.
These groups are big, but their topology is tame enough for strong theorems.