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A topological space $X$ is called homogeneous, if for every two points $x,y \in X$ there exists a homeomorphism $\phi : X \rightarrow X$ s.t. $\phi(x) = y$.

It is not hard to prove that all connected 2-manifolds are homogeneous. The proof basically comes down to the fact that if $D$ is the open disk in $\mathbb{R}^2$ then for every $x,y \in D$ there exists a homeomorphism $\phi : \bar{D} \rightarrow \bar{D}$ such that $\phi(x) = y$, and $\phi \vert_{\partial D}$ is the identity.

Is it true that a general connected manifold is homogeneous?

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    If the manifold has dimension 2 or larger you can go further and send any collection of $n$ points to any other collection of $n$ points via a homeomorphism. Terminology for this is that $Homeo(M)$ acts on $M$ $n$-transitively for all $n$.2011-12-08

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Yes, any connected topological manifold $X$ of arbitrary dimension $n$ is homogeneous .

1) The crucial lemma is that given two points $a,b\in \mathbb B^{\circ}$ in the interior of a closed ball $\mathbb B \subset \mathbb R^n$, there exists a homeomorphism $f: \mathbb B\to \mathbb B$ which is the identity on $\partial \mathbb B$ and such that $f(a)=b$.

2) It then follows that if you fix any point $x_0 \in X$, then the set of points $y\in X$ that can be written $y=F(x_0)$ for some homeomorphism $F:X\to X$ is both open and closed, hence is equal to $X$.
Hence $X$ is homogeneous.
(By the way, an obvious modification of the proof shows that the analogous result is also true for a differential manifold: its diffeomorphisms act transitively on the manifold)

Edit: a fishy image
Let me give a physical model which might help visualize the lemma in 1) ( a totally rigorous and amazingly crisp proof is given in t.b.'s great comment).
Imagine you have a spherical fishbowl completely filled with water and a goldfish sitting somewhere in it.
The lemma says that you can send the goldfish to any preassigned place in the bowl by skilfully (!) shaking the bowl.

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    Dear Georges, Thank you for your kind words and enthusiasm. I like that proof, too. I don't remember where I learned it -- probably in my differential geometry course quite$a$while back. Best wishes, Theo.2011-12-09