If $(X,\tau)$ is a $T\_2$-space such that [all non-empty open sets are homeomorphic](https://mathoverflow.net/questions/300253/t-2-spaces-where-all-non-empty-open-sets-are-homeomorphic) (with the subspace topology) to $X$, is $(X,\tau)$ necessarily [homogeneous](https://en.wikipedia.org/wiki/Homogeneous_space)?



Homeomorphic open sets and homogeneity
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](https://ac.els-cdn.com/0166864195000739/1-s2.0-0166864195000739-main.pdf?_tid=d1869c44-f529-44e0-b3c5-16a13d46acf0&acdnat=1527187377_4747fcdffc586d139b7c540a9c28e0f8), 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).


