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

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    @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

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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.