Let $X$ be a set and let $<_1,<_2$ be order relations on $X$.
Let $T_1,T_2$ be the topologies induced on $X$ respectively.
If $(X,T_1)$ is homeomorphic to $(X,T_2)$, does that imply that $(X,<_1)$ and $(X,<_2)$ are order isomorphic?
And a derived philisophical question: The other way around is easy to prove, so if this holds this means that, in some sense, homeomorphism between order topologies is equivalent to order isomorphism. What does that mean?