Are homeomorphisms order-isomorphisms?

Question

Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?

Answer 1

Every bijection $f \colon X\to Y$ induces an order-isomorphism between $(\mathcal P(X),\subseteq)$ and $(\mathcal P(Y),\subseteq)$.

This follows easily from the following two observations:

• $A\subseteq B$ $\Rightarrow$ $f[A]\subseteq f[B]$ for any map $f$
• $f^{-1}[f[A]]=A$, if $f$ is a bijection.