Is the projection from a Zariski product of Zariski closed sets continuous?

Is the projection from a Zariski product of Zariski closed sets continuous?

Let $X \subseteq \mathbb A^n$ and $Y \subseteq \mathbb A^m$ be Zariski
closed. Then the (Zariski) product $X \times Y \subseteq \mathbb A^{n +
m}$ is closed and there is a projection map $p\colon X \times Y \to X$
which is continuous in the Zariski topology.
Question: When is $p$ an open map?
This is always the case when $X \times Y$ is given the product topology
but here we give it the Zariski product topology and when we do that the
result is not always true. I know it's true when $X$ and $Y$ are
irreducible and the field is algebraically closed. Martin gave a very
short answer to that effect here (and if someone could point me too a more
elementary proof in that case I would appreciate it). The projection
$\mathbb R^2 \to \mathbb R$ onto the first coordinate is not open (look at
the complement of a circle) so even when $X$ and $Y$ are irreducible we
need the field to be algebraically closed. What about when the field is
algebraically closed but $X$ and $Y$ are not necessarily irreducible?