Is every homeomorphism between topological spaces an order isomorphism (for orders of inclusion $\subseteq$ of sets)?
Are homeomorphisms order-isomorphisms?
2
$\begingroup$
general-topology
order-theory
-
0@BrianM.Scott Thank you for pointing that out! In retrospect you may ignore my previous comment since your first comment answers the question whether OP wants open sets or all sets. : ) – 2013-02-28
1 Answers
11
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.