Homogeneous spaces generalized – and an open problem

Homogeneous spaces are topological spaces $(X,\tau)$ that locally look everywhere the same. Put in mathematical terms, $(X,\tau)$ is homogeneous if for any $x,y\in X$ there is an isomorphism $\varphi:X\to X$ such that $\varphi(x)=y$.

We can generalize this by not restricting ourselves to a pair of points $(x,y)$ in the following way:

Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$homogeneous if

1. $|X| \geq \kappa$, and
2. whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X$ such that $\varphi|_A = \psi$.

A natural question is now to ask whether for cardinals $\alpha<\beta$ we have
that there is a space that is $\alpha$-homogeneous, but not $\beta$-homgeneous.

Joel David Hamkins gave this wonderful partial answer, pointing out that the disjoint union of 2 circles is 1-homogeneous but not 2-homogeneous; and moreover that $\mathbb{R}^2$ is
2-homogeneous but not 3-homogeneous.

Interestingly, neither Joel nor another mathematician, Andreas Blass, who wrote several comments on Joel’s post, could figure out how to continue from there – which gives rise to the following

Open problem: For integers $n\geq 5$, are there spaces that are $n$-homogeneous, but not $(n+1)$-homogeneous?

(The question has a positive answer given in Joel’s post for cardinals $\kappa\geq\aleph_0$.) 