Complete varieties

For a scheme-theoretic $k$-variety $X$ (integral, separated and finite type over $k$), call $X$ complete when its structure morphism $X\to \mathrm{Spec}k$ is proper. Completeness here is relative to the given $k$-structure. This is a property of the structure morphism and makes no assertion about compactness of $X(k)$ in an unspecified topology.

Unfolding Proper morphisms, $X$ is complete over $k$ exactly when its structure morphism is separated, of finite type, and universally closed. The source convention Scheme-theoretic varieties supplies the meaning of $k$-variety used here. The term concerns algebraic properness over $k$; it does not add a topology to the set of $k$-rational points.