Homogeneous spaces are topological spaces that locally look everywhere the same. Put in mathematical terms, is homogeneous if for any there is an isomorphism
such that .
We can generalize this by not restricting ourselves to a pair of points in the following way:
Let be a cardinal and let be a topological space. We say that is
1. , and
2. whenever are subsets with and is a bijective map, then there is a homeomorphism such that .
A natural question is now to ask whether for cardinals we have
that there is a space that is -homogeneous, but not -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 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 , are there spaces that are -homogeneous, but not -homogeneous?
(The question has a positive answer given in Joel’s post for cardinals .)