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.