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$X$ is a smooth manifold (Hausdorff, locally Euclid) with an action by a group $G$. I have the following questions.

  1. Quotient space $X/G$ has a quotient topology. Does $X/G$ have Hausdorff property?
  2. Let $\pi : X \to X/G$ be a quotient map. Is $\pi$ locally homeomorphic?
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minerva
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    Perhaps this answers your question? https://math.stackexchange.com/a/496652/26501 (look at the first displayed statement) Note that you are missing two hypotheses: the action should be smooth, and it should be free. – Lee Mosher Jul 31 '20 at 13:48
  • I'm thinking about a diffeomorphisms group of X. In this case, the action seems smooth and meets the hypothesis of the theorem. – minerva Jul 31 '20 at 14:35
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    You also have to assume that the action is proper, otherwise, even if the action is free and smooth, the quotient need not be Hausdorff: Think about the additive group of rational numbers acting on the real line by translations. – Moishe Kohan Aug 09 '20 at 17:45

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