Are projection maps open in the Zariski topology?

Question

The projection map $\pi:U\times V\to U$ is an open map if $U$ and $V$are topological spaces and $U\times V$ has the product topology. Is the same true if $U$ and $V$ are (abstract) varieties with the Zariski topology and $U\times V$ has the product Zariski topology? And is this a particular example of a more general principle in categorical topology?

Answer 1

Flat morphisms which are locally of finite presentation are open; this follows from Chevalley's Theorem on the preservation of constructible sets under finitely presented morphisms and the going-down theorem for flat morphisms. See When is a flat morphism open?

This applies to any projection map $X\times_k Y\to X$ for varieties $X,Y$ over a field $k$.