Is it true that the set of zeros of a regular function on a quasiprojective variety $X$ is closed in $X$?
I have doubt about this because by definition, a closed subset of $X$ is the intersection of $X$ and a closed subset of projective space. A closed subset of the projective space is by definition the zeros of a system of homogeneous polynomials. However, regular functions on a quasiprojective variety are not only polynomials.
Thanks in advance.