Universally closed morphism

Remarks

A closed immersion $Z \to Y$ is universally closed. Its base change along any $Y' \to Y$ is again a closed immersion, and a closed immersion is a homeomorphism onto a closed subset, so it is a closed map. For a field $k$, the morphism $\mathbb{A}^1_k \to \operatorname{Spec} k$ is closed but not universally closed. Its base change along $\mathbb{A}^1_k \to \operatorname{Spec} k$ is the projection $\mathbb{A}^2_k \to \mathbb{A}^1_k$, which sends the closed set $V(xy - 1)$ onto $\mathbb{A}^1_k \setminus \set{0}$, a set that is not closed. Universally closed morphisms are stable under composition and base change.