Let $f:X \to Y$ be a continuous, closed map between topological spaces. Let $E,F$ be closed subsets of $X$. Is $f(E-F)$ the difference of closed subsets of $Y$?
I am interested in this because, as applied to the Zariski topology, a positive answer would imply a positive answer to the question Image of quasiprojective variety under closed map.
(Note also that for a general continuous map $g$, it is not necessarily true that $g(E-F)=g(E)-g(F)$, which would be the obvious first thing to try).
UPDATE: A counter-example has been found - see Image of quasiprojective variety under closed map.
