On Glimm's Theorem for almost Hausdorff G-spaces

TL;DR

This paper generalizes Glimm's theorem to a broader class of group actions on topological spaces, providing a new characterization of well-behaved orbits in almost Hausdorff G-spaces.

Contribution

It extends Glimm's theorem from second countable groups to hereditary Lindel"of locally compact groups acting on second countable spaces.

Findings

Characterization of well-behaved orbits in almost Hausdorff G-spaces.

Reformulation of Glimm's theorem for broader group classes.

Generalization from second countable to hereditary Lindel"of locally compact groups.

Abstract

We establish a characterization of the well-behaved orbits of a totally Baire $G$-space of a hereditary Lindel\"of locally compact group under a mild assumption of Hausdorffness. Furthermore we give a reformulation of the proof of Glimm's theorem generalizing the assertion from second countable to hereditary Lindel\"of locally compact groups acting on second countable spaces.