Suppose $X\neq \emptyset$ is a set. Let $\tau_1, \tau_2$ be Hausdorff topologies on $X$ with the property that the partially ordered sets $(\tau_1,\subseteq)$ and $(\tau_2,\subseteq)$ are order-isomorphic.

Does this imply that $(X,\tau_1)\cong (X,\tau_2)$ as topological spaces?

If Hausdorffness is dropped, then there are easy counterexamples, for instance $X = \omega$ and $\tau_1 = \{\emptyset,\{0\},\omega\}, \tau_2 = \{\emptyset, \omega\setminus\{0\},\omega\}$.