A topometric Effros theorem

TL;DR

This paper characterizes when the closure of a group orbit under a Polish group action is co-meagre in a topometric space, providing necessary and sufficient conditions that complete previous criteria.

Contribution

It introduces a complete criterion for co-meagerness of orbit closures in topometric spaces under Polish group actions, extending earlier work.

Findings

Provides necessary and sufficient conditions for co-meagerness of orbit closures.

Characterizes existence of points with co-meagre orbit closures.

Completes previous criteria in the literature.

Abstract

Given a continuous and isometric action of a Polish group $G$ on an adequate Polish topometric space $(X,\tau,\rho)$ and $x \in X$, we find a necessary and sufficient condition for $\overline{Gx}^\rho$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.