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 there for all $x,y\in X$ with $x\neq y$ a clopen (closed and open) set containing $x$ but not $y$?



Homeomorphic open sets and total disconnectedness
According to this paper, [Spaces of diversity two](https://doi.org/10.1016/0166-8641(95)00073-9), the paper [Spaces of diversity one](https://zbmath.org/?q=an%3A0724.54020) by Franklin and Rajagopalan contains examples of various spaces with just one open set, also one that is not totally disconnected. Unfortunately the latter paper does not appear to be available on-line and the review does not mention Hausdorffness, but the last sentence suggests that the Hausdorff property is present.


Addendum: the Franklin-Rajagopalan paper is [on-line but behind a paywall](http://journals.thescientificindian.com/jrms/1990-99).


