The Stacks project

Lemma 29.7.8. Let $X$ be a reduced scheme and let $U \subset X$ be an open subscheme. Then the following are equivalent

  1. $U$ is topologically dense in $X$,

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

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

Proof. This follows from Lemma 29.7.7 and the fact that a closed subscheme $Z$ of $X$ whose underlying topological space equals $X$ must be equal to $X$ as a scheme. $\square$

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