I'd like to know if there are any interesting theorems/facts about the image $f(U)$ of an open set $U$ under a continuous mapping $f$.
Is there maybe a characterization of sets that are such images?
Or maybe something can be said about $f^{-1}(f(U))$?
EDIT: As per Mike Earnest's answer I'd like to modify the characterization part of this question. Given two fixed topological spaces $(X,\tau_X)$ and $(Y,\tau_Y)$, is there a characterization of subsets of $Y$ that are images of some open set of $X$ under some continuous mapping?