Homeomorphic open sets and homogeneity 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?
 A: 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).
