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?
Best Answer
One of very interesting as well as important theorem about continuous image of open sets is Invariance of domain theorem which states,
Above thorem is very useful while studying topological manifolds.