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

–*homogeneous* if

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 .)

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