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So I'm aware that the orbit-stabilizer theorem does not hold for arbitrary spaces with a transitive action by a topological group, but I wonder if it works in the following situation.

Let $G$ be a totally disconnected, locally compact Hausdorff topological group and $X$ a topological space satisfying the same conditions (I would call such things $\ell$-groups and $\ell$-spaces respectively). If $G$ acts transitively on $X$ and $x \in X$ is any point there is an obvious $G$-equivariant continuous bijection $G/G_x \to X$, where $G_x$ denotes the stabilizer of $x$ in $G$. Can we conclude, in this situation, that $G/G_x \to X$ is a homeomorphism? If not, what further conditions do we need to impose? Notice that this is true if $G/G_x$ is compact, since a continuous bijection of compact Hausdorff spaces is a homeomorphism.

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    Well, I don't have much intuition about totally disconnected groups and spaces, but one general criterion that may (or may not) be useful is if the action is *proper*, i.e. the map $G \times X \to X \times X$, $(g,x) \mapsto (gx,x)$ is proper in the sense that pre-images of compact sets are compact then the map $G/G_x \to Gx$ is a homeomorphism, see [this MO-thread](http://mathoverflow.net/questions/55726/properly-discontinuous-action/56490#56490) for some basics on proper actions and references.2011-06-23

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This is true for G a locally compact, Hausdorff topological group, and X locally compact, Hausdorff, with a countable local basis. This "apocryphal lemma" appears many places, but is easily misplaced.

I reproduced the usual argument in an appendix in the "Solenoids" class notes on my modular forms course page, here .

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    Assuming $G$ is locally compact second countable and $X$ is Polish, a refinement of this is proved as Theorem 2.1.14 in *Ergodic Theory and Semisimple Groups* by Robert J. Zimmer. The references given there are [Glimm](http://dx.doi.org/10.1090/S0002-9947-1961-0136681-X) (locally compact case) and [Effros](http://www.jstor.org/stable/1970381). @Pete, maybe you're interested.2011-06-23