Polish group are topological groups which are complete and separable, i.e., have a countable dense subset and are completely metrizable. Polish groups are homeomorphic to separable complete metric spaces.
Use this tag when asking a question regarding these groups.
A Polish group is a topological group $G$ that is also a Polish space, i.e., homeomorphic to a separable complete metric space.
A remarkable fact about Polish groups is that Baire-measurable (i.e., the preimage of any open set has the property of Baire) homomorphisms between them are automatically continuous. The group of homeomorphisms of the Hilbert cube $[0,1]^N$ is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it.
Standard Examples of Polish Groups include
- All finite-dimensional Lie groups with a countable number of components are Polish groups.
- The unitary group of a separable Hilbert space (with the strong topology) is a Polish group.
- The group of homeomorphisms of a compact metric space is a Polish group.
- The product of a countable number of Polish groups is a Polish group.
- The group of isometries of a separable complete metric space is a Polish group.
Source: https://en.wikipedia.org/wiki/Polish_space#Polish_groups