On Glimm's Theorem for almost Hausdorff G-spaces

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.