Closed-set criterion for continuity ($f\text{ continuous}⟺f^{-1}(F)\text{ closed for every closed }F$)

A function $f:X\to Y$ between topological spaces is continuous exactly when the inverse image of every closed set in $Y$ is closed in $X$. The defining open set criterion is equivalent, because $f^{-1}(Y\setminus A)=X\setminus f^{-1}(A)$. This criterion concerns inverse images; a continuous map need not be a closed map.