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I'm using following definitions:

Definition 1. A topological space $X$ is (globally) homogenous if for any two points $x,y\in X$ there exists a homeomorphism $f:X\to X$ such that $f(x)=y$.

and the local version:

Definition 2. A topological space $X$ is locally homogenous if for any two points $x,y\in X$ and any two open neighbourhoods $U_x, U_y$ of $x,y$ respectively there are open sub-neighbourhoods $x\in V_x\subseteq U_x$, $y\in V_y\subseteq U_y$ such that there is a homeomorphism $f:V_x\to V_y$ such that $f(x)=y$.

Obviously every manifold is locally homogenous. Also I've done some research and actually connected manifolds are additionally globally homogenous.

Now I'm quite sure that the global version implies the local one. I'm looking for a counterexample that the local one implies global.

One obvious counterexample is the disjoint union $X=S^1\sqcup\mathbb{R}$. However this space is not connected. And the connectedness plays crucial role in showing that $X$ is not globally homogenous (otherwise we would get a homeomorphism $S^1\to\mathbb{R}$).

I'm having trouble in finding a connected counterexample. Or perhaps there is none? Unlikely. Any help?

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  • $\begingroup$ I haven't worked out the details, but I'm guessing the real line with two origins (i.e. $\mathbb R\times\{1,2\}/\sim$ with $(x,1)\sim(x,2)$ for all $x\neq0$) gives a counterexample. $\endgroup$ – Aweygan Apr 27 '18 at 11:02
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Hint: You may want to consider the real line with two origins, i.e., the space $X=\mathbb R\times\{1,2\}/\sim$, where $(x,1)\sim(x,2)$ if $x\neq0$. This is connected, and locally homogeneous, but can we have an automorphism of $X$ taking one of the zeros to anything but itself or the other zero?

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