Universally closed morphism

A morphism is universally closed when every base change is a closed map on underlying topological spaces.

Among quasi-compact morphisms of schemes, universal closedness is equivalent to the existence part of the valuative criterion: for every valuation ring $D$ with fraction field $K$, every compatible pair $\mathrm{Spec}K\to X$, $\mathrm{Spec}D\to S$ extends to $\mathrm{Spec}D\to X$. All valuation rings must be tested without further hypotheses. A universally closed morphism is automatically a quasi-compact morphism.

Glue two copies of ${\mathbb{P}}_k^1$ by the identity away from one chosen point. The resulting finite-type scheme is universally closed over $k$: after any base change, a closed subset meets each projective-line chart in a closed subset whose image is closed, and the total image is their finite union. The two copies of the chosen point cannot be separated, so the structure morphism is not separated and therefore is not proper.