Closure and closed map

Question

Suppose $f:A\to B$ is continuous between topological spaces. If $f$ is an open map, then $f^{-1}$ commutes with the interior operator, i.e. for every subset $U$ of $A$ it is $$f^{-1}(\operatorname{int} (U)) = \operatorname{int} (f^{-1}(U))$$ Is it the dual true, i.e. if a continuous functions $f:A\to B$ is closed (and not necessarily open) then $f^{-1}$ commutes with the closure operator? Meaning that $$f^{-1}(\overline{U}) = \overline{f^{-1}(U)}$$ If no, is it true if we assume that both $A$ and $B$ are compact and Hausdorff?