Suppose $G$ is a topological group that acts on a connected topological space $X$. Show that if this action is transitive (and continuous), then so is the action of the identity component of the group.
$G$ acts transitively on connected space, then so does identity component
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1It's entirely possible that the author has, in a preface or something, a statement along the lines of "Every topological space considered will be Hausdorff...". But barring that, I'm in agreement that the counterexample works. Then again, I'm biased and am often wrong... ;-). – 2012-05-04
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Once you assume the space $X$ is a connected homogeneous space with the group acting on it as $G$, the quotient map $\pi:G\to G/G_x$, where $G_x$ is a stabilizer, is an open map and $G/G_x\to X$ is a homeomorphism. This tells us that $g\mapsto gx$ is an open map. Now it follows that (in particular) $G_0$, the identity component, acts transitively on $X$ (since X is connected). This is what Morris assumed in (the exercise) his book I think.
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0$G_0$ is a connected component and hence clopen – 2018-04-29