Polish Group Topologies

Summary

In what circumstances does a given group carry a Polish group topology? (A metric, separable, complete topology is called Polish, and a topology is a group if multiplication and inverse are continuous.) Two natural interpretations of this question can be considered.

1. We consider a group equipped only with its algebraic structure. This algebraic structure may or may not be compatible with the possibility of defining a Polish group topology on the group.

2. The group is endowed with a σ-algebra Σ of subsets, and the multiplication (regarded as function of two variables) and inverse operations axe assumed to be measurable with respect to Σ, that is, preimages of sets from Σ are in the product σ-algebra Σ × Σ in case of multiplication and in Σ in case of inverse. Now, we would like to investigate the possibility of putting a Polish group topology on the group whose Borel sets coincide with Σ.

Problem 2 is more “canonical” than 1 in the following sense. A group can carry many Polish group topologies compatible with its algebraic structure. Take for example the reals, R, with their natural topology and R^w with the product topology. Both these groups are topological groups with their respective Polish topologies. Now, R and R and R^ω are isomorphic as groups (both of them are linear spaces over the rationals, Q, with bases of cardinality continuum) but the topologies are clearly different (R is locally compact while R^w is not).