Open and universally open morphisms of schemes

Let $f:X\to S$ be a morphism of schemes; write $\mathrm{\mid }f\mathrm{\mid }:\mathrm{\mid }X\mathrm{\mid }\to \mathrm{\mid }S\mathrm{\mid }$ for its underlying continuous map of topological spaces (Morphisms of schemes).

The morphism $f$ is open if $\mathrm{\mid }f\mathrm{\mid }$ is an open map: for every open subset $U\subseteq \mathrm{\mid }X\mathrm{\mid }$ the image $\mathrm{\mid }f\mathrm{\mid }(U)$ is open in $\mathrm{\mid }S\mathrm{\mid }$ (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

The morphism $f$ is universally open if for every $S$-scheme $T$, that is, every morphism of schemes $T\to S$ (Schemes and morphisms over a base), the base-changed projection $f_T:X_T=X{\times }_{S}T⟶T$ is open, where the base change is as in Base change of objects, morphisms and properties.

The quantifier in the second clause ranges over all $S$-schemes $T$; no finiteness, quasi-compactness or separatedness hypothesis is part of the definition. Universal openness implies openness by taking $T=S$. A morphism whose source is empty is open, and a morphism whose target is empty has empty source and is open.