Lemma 67.17.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. If $U$, $V$ are scheme theoretically dense open subspaces of $X$, then so is $U \cap V$.
Proof. Let $W \to X$ be any étale morphism. Consider the map $\mathcal{O}(W) \to \mathcal{O}(W \times _ X V) \to \mathcal{O}(W \times _ X (V \cap U))$. By Lemma 67.17.5 both maps are injective. Hence the composite is injective. Hence by Lemma 67.17.5 $U \cap V$ is scheme theoretically dense in $X$. $\square$
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