We use the Zariskii topology. Let $\phi:\mathbb C^n\rightarrow \mathbb C^m$ be a polynomial map and let $S\subset \mathbb C^n$ be a locally closed set (that is, $S$ is the intersection of an open set and a closed set). Assume that $\phi$ induces an isomorphism of $S$ onto a nonempty open subset.
${\bf Question:}$ Is it true that $S$ is irreducible, that is if $S=(C_1\cap S)\cup(C_2\cap S)$ with closed $C_1$ and $C_2$, then $C_1\cap S=S$ or $C_2\cap S=S$.
${\bf Solution (attempt):}$ By definition, a constructible set is a finite union of locally closed sets.
Chevalley’s theorem states that the image of a constructible set under a polynomial map is constructible.
Thus, if we assume that $S=(C_1\cap S)\cup(C_2\cap S)$, then both $\phi(C_1\cap S)$ and $\phi(C_2\cap S)$ are constructible.
Since the open subset $\phi(S)$ is never a finite union of closed subsets, it follows that at least one of the sets $\phi(C_1\cap S)$, $\phi(C_2\cap S)$ contains an open subset. Then I should probably use the fact that $\phi$ induces an isomorphism, but I do no how to do this.
