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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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