Continuous maps and frame homomorphisms

Question

Given a continuous map $f : X \rightarrow Y$ between topological spaces, we get a function $\mathcal{O}(f) : \mathcal{O}(Y) \rightarrow \mathcal{O}(X)$ defined by taking preimages. This is always a frame homomorphism, meaning that it preserves finite meets and arbitrary joins.

Question. Is it possible for $\mathcal{O}(f)$ to fail to preserve pseudocomplements?

(By the pseudocomplement of an open set $U$, I mean the interior of its complement.)

Answer 1

Any pseudocomplement is necessarily a regular open set: a set equal to the interior of its complement. As you correctly pointed out in this answer, the preimage of $(0,1)$ under the squaring map ($x\mapsto x^2$ from $\mathbb R$ to $\mathbb R$ with the usual topology) is $(-1,0) \cup (0,1),$ which is not regular.