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I wonder for what kind of topological spaces $X$ the following assertion is true: given any $x, y \in X$, there is some homeomorphism $f: X \to X$ such that $f(x) = y$. I have just learnt that all topological groups satisfying the condition above; and I think that there might be other spaces possessing this property, maybe some closed connected surfaces in $\mathbb{R}^n$ (not sure about that). Can anyone give me some examples?

zyy
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    Such topological spaces are called homogeneous. In group theory terminology it means that the group of homeomorphisms $X\to X$ acts transitively on $X$. As for examples, it can be shown that all connected manifolds are homogeneous. See here: https://math.stackexchange.com/questions/89721/are-all-connected-manifolds-homogeneous – Mark Aug 09 '24 at 14:40
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    Duplicata: https://math.stackexchange.com/questions/25326/homogeneous-topological-spaces – Gleberson Antunes Aug 09 '24 at 14:41

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