Coset spaces of metrizable groups

Abstract

We characterize topological spaces which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Moreover, it is shown that for a G-space X with a d-open action there is a topological group H of weight and cardinality less than or equal to the weight of X such that H admits a d-open action on X. This is further applied to show that if X is a separable metrizable coset space of some topological group, then X has a Polish extension which is a coset space of a Polish group.