Do products and/or coproducts exist in the category of closed algebraic sets?

Question

I was recently reading a note by Mel Hochster where he introduces the category of closed algebraic sets over an algebraically closed field $K$ a ways down on page 6.

Out of curiosity, do products and/or coproducts exist in the category of closed algebraic sets?

Answer 1

Yes, they do. Note, by the theorem at the bottom of page 7, that your category is dual to the category of reduced finitely generated $k$-algebras, so (if they exist) products and coproducts in fg reduced $k$-algebras are taken to coproducts and products in closed algebraic sets over $k$. Showing products and coproducts in fg reduced $k$-algebras exist should be easier, if only because its a category you are more familiar with. Just in case, here's the final answer:

Coproducts in fg reduced $k$-algebras are tensor products over $k$, and products are the normal cartesian products. These translate into product of topological spaces and disjoint union for the underlying topological spaces in your category of closed algebraic sets.