totally disconnected orbit-stabilizer theorem
Solution 1:
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 .