Suppose $X$ and $Y$ are two varieties and $f: X \to Y$ is a dominant morphism (i.e. $\overline{f(X)} = Y$) between them. Prove that for any nonempty open subset $U \subset X$, the restriction $f|_U : U \to Y$ is dominant.
I have observed that many textbooks takes this for granted. For example, they sometimes substitute $X$ and $Y$ with appropriate open subsets of them without explaining why they could do this.
The only thing I know $\overline{U} = X$ and $\overline{f(X)} = Y$, but how to continue?
Any suggestion is welcomed and appreciated.
Most of what is going on is purely topological.
Recall the following fact, which is true in general for any continuous map $f:X \rightarrow Y$ between topological space (it is in fact equivalent to continuity I believe): for any subset $S \subset X$ , one has $f(\overline{S})\subset \overline{f(S)}$.
If $U$ is a dense set in $X$, then $f(U)$ is dense in $\overline{f(X)}$. Indeed, we compute $$\overline{f(X)}=\overline{f(\overline{U})}\subset \overline{f(U)}$$ hence $\overline{f(X)} \subset \overline{f(U)}$.
In particular, a dominant continuous map between topological spaces is still dominant when you restrict to a dense subset From this you see that in you initial question all you need is open sets being dense. This is always the case for open sets of irreducible varieties (I can tell you why if you need but try to do it as an exercise first).