I'm a little stuck with this problem and I'd appriciate any advice or hint you could give me:
Let $(X,\leq)$, $(Y,\preccurlyeq)$ totally oredered sets and $\psi: X \rightarrow Y$ a function. Then:
$\psi$ is a homeomorphism if and only if for all $a,b \in X$ such that $a \leq b$ then $\psi(a) \preccurlyeq \psi(b)$.
i.e., $\psi$ is a homeomorphism if and only if $\psi$ is order-preserving.
So far in this direction $\Leftarrow$ I was able to prove that $\psi$ is continuous and injective. But I really have any idea about how to proceed in the other direction. Or if it's not true I haven't found any counterexample.
I really appreciate all the help you can give me. Thanks in advance.