Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in V\big\}$.


We endow $[X]^2$ with the topology $[\tau]^2$, which is generated by $\{[U,V]: U,V\in \tau\land U\cap V =\emptyset\}$.


Is there a non-discrete Hausdorff space $X$ with more than 3 points such that $X\cong [X]^2$?



Hausdorff space $X$ with $X\cong [X]^2$
It is easy to see that the operation $X \mapsto [X]^2$ preserves the properties: countable, second-countable, regular, no isolated points. Hence $\mathbb{Q} \cong [\mathbb{Q}]^2 $.


