# Homogeneous topological spaces

Let $X$ be a topological space.

Call $x,y\in X$ swappable if there is a homeomorphism $\phi\colon X\to X$ with $\phi(x)=y$. This defines an equivalence relation on $X$.

One might call $X$ homogeneous if all pairs of points in $X$ are swappable.

Then, for instance, topological groups are homogeneous, as well as discrete spaces. Also any open ball in $\mathbb R^n$ is homogeneous. On the other hand, I think, the closed ball in any dimension is not homogeneous.

I assume that these notions have already been defined elsewhere. Could you please point me to that?

Are there any interesting properties that follow for $X$ from homogeneity? I think for these spaces the group of homeomorphisms of $X$ will contain a lot of information about $X$.

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Connected manifolds are also homogeneous. –  Ryan Budney Mar 6 '11 at 20:05
@Ryan: wow, really? Then they are extremely "flexible". –  Rasmus Mar 6 '11 at 20:19
Yes, the argument is quite fun. I suggest giving it a try. Let $M$ be a connected manifold, and fix $p \in M$. Consider the set $U = \{ x \in M : f(p)=x, f \text{ a homeomorphism of } M \}$. Check that $U$ is non-empty, open and closed. –  Ryan Budney Mar 6 '11 at 20:22
You can extend your question further. Your notion of homogeneous is that any two points can be swapped by a homeomorphism, which you could also call a "1-transitive action" of the homeomorphism group. You could further ask if any collection of $n$ points in $X$ can be moved to any other collection of $n$ points in $X$ by a homeomorphism. This would be an $n$-transitive action. Connected manifolds of dimension $k \geq 2$ also satisfy this $n$-transitivity condition -- you can't do it for $1$-manifolds because points in a $1$-manifold inherit a (cyclic) ordering from the ambient space. –  Ryan Budney Mar 6 '11 at 20:26
Homogeneous is a completely standard notion, and many papers have been written on it. There are also notions like locally homogeneous, strongly locally homogeneous, countable dense homogeneous etc. Indeed all finite dimensional closed balls are non-homogeneous, as we can only map points of the boundary to other such points, and likewise for interior points (2 classes, in your terminology). I believe your classes are called homogeneity components by some authors. –  Henno Brandsma Mar 7 '11 at 17:47

## 1 Answer

Googling

"topological space is homogeneous"

brings up several articles that use the same terminology, for example this one. It is also the terminology used in the question Why is the Hilbert cube homogeneous?. The Wikipedia article on Perfect space mentions that a homogeneous space is either perfect or discrete. The Wikipedia article on Homogeneous space, which uses a more general definition, may also help.

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I've also seen this referred in the literature as the homeomorphism (diffeomorphism) group acting transitively on a space (smooth manifold). –  Eric O. Korman May 31 '11 at 2:50
@Eric: That is a clear and accurate description based on the usual meaning of "transitive group action", so it makes sense. I suppose "homogeneous" saves a few words (aside from having to define it initially). –  Jonas Meyer May 31 '11 at 3:33