Lemma 29.8.2. Let $f : X \to S$ be a morphism of schemes. If every generic point of every irreducible component of $S$ is in the image of $f$, then $f$ is dominant.
Proof. This is a topological fact which follows directly from the fact that the topological space underlying a scheme is sober, see Schemes, Lemma 26.11.1, and that every point of $S$ is contained in an irreducible component of $S$, see Topology, Lemma 5.8.3. $\square$
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