Polish group

Summary

A Polish group is a topological group whose topology is a Polish space, 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.