We first study separated morphisms in an algebraic site C . Denote by f ' the restriction of f to V . Suppose U is an open affine neighborhood of x contained in U '. X U is affine. Suppose f is separated.
V is a closed special morphism.
Y is a morphism. The other direction is obvious. Suppose X is an object. The proof is complete. Any object X whose underlying space is locally noetherian is quasi-separated.