If $(X,\tau)$ is a $T_2$-space such that all non-empty open sets are homeomorphic (with the subspace topology) to $X$, is $(X,\tau)$ necessarily homogeneous?
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asked May 24, 2018 at 6:45
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1$\begingroup$ Don't forget to link preceding relevant questions of yours like mathoverflow.net/questions/300253/… $\endgroup$– Pietro MajerCommented May 24, 2018 at 7:44
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$\begingroup$ Related: In the Cantor set all clopen sets are homeomorphic (I believe) and it is not homogeneous. $\endgroup$– HenrikRüpingCommented May 24, 2018 at 8:17
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3$\begingroup$ @HenrikRüping Why do say that the Cantor set is not homogeneous? It is, after all, the product of countably many $2$-point discrete spaces. Isn't a product of homogeneous spaces homogeneous? $\endgroup$– bofCommented May 24, 2018 at 8:28
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1$\begingroup$ @bof: Thanks of course it is homogeneous.There are points which are endpoints of some removed intervals and points which are not. I was mistakenly thinking that a homeomorphism can't map one of the first kind to one of the second kind. However that idea at least shows that there are no order preserving (thinking of the Cantor set as a subset of the real line) homeomorphisms that map a point of the first kind to one of the second kind. $\endgroup$– HenrikRüpingCommented May 24, 2018 at 8:34
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1$\begingroup$ @RamirodelaVega And if $X$ is compact and has exactly two non-homeomorphic "types", $X$ is homogeneous. $\endgroup$– Henno BrandsmaCommented May 25, 2018 at 9:34
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1 Answer
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For an infinite Hausdorff space the diversity of a space is the number of homeomorphism types of non-empty open sets, so if all non-empty open sets are homeomorphic, the space is said to be of diversity one.
According to this paper, reference [11]:
S.P Franklin and M. Rajagopolan, spaces of diversity one, Ramanujan Math. Soc. 5 (1990), 7-31
has an example such that a space of diversity one need not be homogeneous. I have no access to this paper, so I couldn't look to see and describe the example. Maybe someone else can.
But it's not hard to prove that a zero-dimensional, first countable space of diversity one is homogeneous (it's also in that paper, but the proof idea is classical); examples of such spaces are $\mathbb{Q}$ and $\mathbb{P}$ (the irrationals). Any compact space of diversity two is homogeneous (reference in the linked paper) and a compact metric space of diversity two (diversity one is impossible for a compact space) is homeomorphic to the Cantor set. (the two types being clopen = Cantor set, and open, non-clopen = Cantor set minus a point).
answered May 24, 2018 at 18:50