 Closedness of projection from projective space

Projection $X\times {\mathbb{P}}^n\to X$ sends closed subsets to closed subsets, with geometric points or schemes understood. On an affine base, choose finitely many homogeneous equations for a closed subset. Its fiber is nonempty exactly when every graded multiplication map for a projective fiber fails to be surjective, by the Projective Nullstellensatz. The image is thus an intersection of closed rank-defect loci. An affine open cover proves the general case. This is the universally closed part of the proper morphism property of projective space.