A topological space $X$ is called Noetherian if closed subsets satisfy the descending chain condition, equivalently, the open subsets satisfy the ascending chain condition.
Let $A$ and $B$ be two topological space such that $B$ is Noetherian. If $f:A\rightarrow B$ is a continuous function, is there any condition $T$ on $f$ such that any of the following to be true:
1)if $f$ satisfies $T$, then $A$ is Noetherian.
2) if $A$ if Noetherian, then $f$ satisfies $T$.
3) $f$ satisfies $T$ if and only is $A$ if Noetherian.