Non-isomorphic $T_0$-spaces with order-isomorphic topologies

Question

Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?

Answer 1

Yes. For the first space, topologize $\mathbb R$ by taking the intervals $(-\infty,r)$, $r\in \mathbb R$, for the nonempty proper open subsets. For the second space, take the subspace with underlying set $\mathbb Q$.