Coset spaces of metrizable groups
Sergey Antonyan, Natella Antonyan, Konstantin Kozlov

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 -space with a -open action there is a topological group of weight and cardinality less than or equal to the weight of such that admits a -open action on . This is further applied to show that if is a separable metrizable coset space then has a Polish extension which is a coset space of a Polish group.
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