TITLE: Hausdorff space $X$ with $X\cong [X]^2$
QUESTION [5 upvotes]: 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 $X\cong [X]^2$?

REPLY [4 votes]: Yet another example: $X:=\{−3,−2,−1\}∪\mathbb{Q}_+$ with the standard Euclidean topology. $\mathbb{Q}_+$ denotes the positive rationals.
I post it on an explicit request in the comments, but this is in fact a combination of the answer by Ramiro de la Vega, the finite example banned by Dominic van der Zypen and the fact that a product of a finite set by the rationals is homeomorphic to the rationals.