Coset spaces of metrizable groups

TL;DR

This paper characterizes when coset spaces of topological groups are metrizable or Polish, and shows how actions of certain groups can be extended to Polish groups, enhancing understanding of the structure of these spaces.

Contribution

It provides new characterizations of coset spaces of metrizable and Polish groups and demonstrates how group actions can be extended to Polish groups.

Findings

Characterization of coset spaces of metrizable and Polish groups.

Existence of Polish extensions for separable metrizable coset spaces.

Construction of groups with controlled weight and cardinality for given actions.

Abstract

We characterize coset spaces of topological groups which are coset spaces of (separable) metrizable groups and complete metrizable (Polish) groups. Besides, 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 then $X$ has a Polish extension which is a coset space of a Polish group.