What are examples of non-homeomorphic connected $T\_2$-spaces $(X\_i,\tau\_i)$ for $i=1,2$ such that the posets $(\tau\_1, \subseteq)$ and $(\tau\_2,\subseteq)$ are order-isomorphic?



Non-homeomorphic connected $T_2$-spaces with isomorphic topology poset
There aren't any. Hausdorff spaces are [sober spaces](https://ncatlab.org/nlab/show/sober+topological+space). If $X, Y$ are sober, then every frame map $\mathcal{O}(Y) \to \mathcal{O}(X)$, i.e., every poset map between their topologies that preserves finite meets and arbitrary joins, arises from a uniquely determined continuous map $X \to Y$. It follows that a poset isomorphism $\mathcal{O}(X) \cong \mathcal{O}(Y)$, being a frame isomorphism, arises from a homeomorphism between the spaces.


(Just to give slightly more detail: for a sober space $X$, the points of $X$ are in natural bijection with frame maps $\mathcal{O}(X) \to \mathcal{O}(1)$ where the codomain is the topology on a one-point space. Thus a frame map $\phi: \mathcal{O}(Y) \to \mathcal{O}(X)$ induces, via composition with frame maps $\mathcal{O}(X) \to \mathcal{O}(1)$, a function $f: X \to Y$, and is itself of the form $\phi(V) = f^{-1}(V)$.)


