Lemma 29.7.3. Let $X$ be a scheme. Let $U \subset X$ be an open subscheme. If the inclusion morphism $U \to X$ is quasi-compact, then $U$ is scheme theoretically dense in $X$ if and only if the scheme theoretic closure of $U$ in $X$ is $X$.

**Proof.**
Follows from Lemma 29.6.3 part (3).
$\square$

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