I'm studing Andreas Gathmann's notes on algebraic geometry (pdf here: https://agag-gathmann.math.rptu.de/de/alggeom.php). In chapter 4 (about Morphisms) he was using the universal property of products to explain why the product $X\times Y$ of affine varieties $X,Y$ does not carry the product topology. I don't get it.
He defined the Zariski topology on an affine variety $X$ as the topology where every closed subset is exactly an affine subvariety of $X$, i.e. subset of the form $V(S)$ for some $S\subset A(X)$ ($A(X)$ is the coordinate ring and $V(S)$ the zero locus of $S$). The universal property of products yields that giving a morphism from an affine variety to $X\times Y$ is the same as giving a morphism to each of the factors $X$ and $Y$.
Thanks for your help.