Is the preimage of a variety a variety, or is the preimage of an affine variety an affine variety?
More precisely, Let $X,Y$ prevarieties, $\varphi: X \to Y$ a morphism. given a (affine) variety $Z \subset Y$, is $\varphi^{-1}(Z)$ a (affine) variety?
My definition is the following, an affine variety is a topological space $X$ with a sheaf of $k$ algebras that is isomorphic a to an algebraic set, a prevariety is just something locally isomorphic to an affine variety and a variety is a prevariety satisfying the "Hausdorff" propriety.
EDIT: what can you say if $X$ is a variety?