When is $f(\operatorname{int}A)=\operatorname{int}f(A)$ We have that $f:X\rightarrow Y$ is an open map iff $f(\operatorname{int} A)\subseteq \operatorname{int} f(A)$. My question is: when does it become an equality? Obviously $f$ has to be an open map. The equality will hold if $f$ is a homeomorphism, but can that condition be weakened?

$f:\mathbb R \rightarrow {a}\ $ is an open map.  Let $A$ be the set of rationals.
$\text{int} f(A) = {a}$;  $f(\text{int} A)$ is empty.
Though $f$ is an open map, $\text{int} f(A)$ is not a subset of $f(\text{int} A)$.
