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 $X$.
Best Answer
Googling
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.