What is known about the continuous image of an open set?

Question

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?

Answer 1

One of very interesting as well as important theorem about continuous image of open sets is Invariance of domain theorem which states,

If $U \subset \mathbb R^n$ be a domain and $f:U \to \mathbb R^n$ be an injective continuous function then $f$ is an open map.

Above thorem is very useful while studying topological manifolds.

Answer 2

As to your characterization question, every set is the image of an open set under some mapping.

Namely, given a topological space $(X,\tau)$, and $A\subset X$, then consider the identity map from $(X,\text{discrete})$ to $(X,\tau)$. This is continuous since every subset of $X$ is open in the discrete topology. Furthermore, $f(A)=A$, so $A$ in our original space is the image of the open set $A$ in the discrete space.

Answer 3

If the open set in question is $\sigma$-compact (i.e., a union of countably many compact sets) then the image is also $\sigma$-compact, as the image of each compact set is compact. This implies in particular that the image is measurable.