A topological version of the Bergman property

TL;DR

This paper introduces a topological analogue of the Bergman property for Polish groups, explores its reformulations, and verifies it for various groups of isometries and homeomorphisms, including the rational Urysohn space.

Contribution

It defines property (OB) for topological groups, establishes its reformulations, and proves it for several important groups, expanding understanding of group actions in topology.

Findings

Property (OB) characterized for Polish groups

Verified property (OB) for groups of isometries and homeomorphisms

Proved the isometry group of the rational Urysohn space is Bergman

Abstract

A topological group G is defined to have property (OB) if any G-action by isometries on a metric space, which is separately continuous, has bounded orbits. We study this topological analogue of the socalled Bergman property in the context of Polish groups, where we show it to have several interesting reformulations and consequences. We subsequently apply the results obtained in order to verify property (OB) for a number of groups of isometries and homeomorphism groups of compact metric spaces. We also give a proof that the isometry group of the rational Urysohn metric space of diameter 1 is Bergman.