The Stacks project

Lemma 29.7.6. Let $X$ be a scheme. If $U$, $V$ are scheme theoretically dense open subschemes of $X$, then so is $U \cap V$.

Proof. Let $W \subset X$ be any open. Consider the map $\mathcal{O}_ X(W) \to \mathcal{O}_ X(W \cap V) \to \mathcal{O}_ X(W \cap V \cap U)$. By Lemma 29.7.5 both maps are injective. Hence the composite is injective. Hence by Lemma 29.7.5 $U \cap V$ is scheme theoretically dense in $X$. $\square$

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