# Homeomorphic open sets and total disconnectedness

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 $y$?

• It might be worth looking at "The Toronto Problem" which is about Hausdorff spaces which are homeomorphic to each full-cardinality subspace. – Daron Jun 6 '18 at 20:25
• @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. – Will Brian Nov 1 '18 at 12:49