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
\kappahomogeneous 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.)


About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
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