Lemma 29.7.9. Let $X$ be a scheme and let $U \subset X$ be a reduced open subscheme. Then the following are equivalent

1. the scheme theoretic closure of $U$ in $X$ is $X$, and

2. $U$ is scheme theoretically dense in $X$.

If this holds then $X$ is a reduced scheme.

Proof. This follows from Lemma 29.7.7 and the fact that the scheme theoretic closure of $U$ in $X$ is reduced by Lemma 29.6.7. $\square$

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