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If $(X,\tau)$ is a $T_2$-space such that 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 $y$?

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    $\begingroup$ It might be worth looking at "The Toronto Problem" which is about Hausdorff spaces which are homeomorphic to each full-cardinality subspace. $\endgroup$ – Daron Jun 6 '18 at 20:25
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    $\begingroup$ @Daron: That's a great problem, but not directly relevant to this question. A Toronto space (if it exists) has uncountably many different kinds of open sets up to homeomorphism. This is because Toronto spaces are scattered, with uncountable scattered height, and they contain open sets of any given height. $\endgroup$ – Will Brian Nov 1 '18 at 12:49
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According to this paper, Spaces of diversity two, the paper Spaces of diversity one 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.

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