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.

Over the real numbers, the closed equation $y_0^2=t{y}_1^2$ in ${\mathbb{A}}^1\times {\mathbb{P}}^1$ projects on real points to $\{t\ge 0\}$, which is not Zariski closed. Projective properness concerns scheme or geometric images, not the image of rational points over an arbitrary field. Similarly, $y_0^2+y_1^2$ has no real projective zero, but its homogeneous ideal contains no power of the irrelevant ideal of projective space, since it has a complex projective zero.