Polish group

In mathematics, a Polish group is a topological group that is also a Polish space, in other words homeomorphic to a separable complete metric space.

Examples

• 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.

Properties

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.

References

• Kechris, A. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics. 156. Springer. ISBN 0-387-94374-9.