Lemma 29.7.8. Let X be a reduced scheme and let U \subset X be an open subscheme. Then the following are equivalent
U is topologically dense in X,
the scheme theoretic closure of U in X is X, and
U is scheme theoretically dense in X.
Lemma 29.7.8. Let X be a reduced scheme and let U \subset X be an open subscheme. Then the following are equivalent
U is topologically dense in X,
the scheme theoretic closure of U in X is X, and
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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