Closed-set criterion for continuity
= Closed-set criterion for continuity
{title2=$f\text{ continuous}\iff f^{-1}(F)\text{ closed for every closed }F$}
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>.