Lemma 5.16.5. Let $X$ be a Noetherian topological space. Let $E \subset X$ be a subset. The following are equivalent:

1. $E$ is open in $X$, and

2. for every irreducible closed subset $Y$ of $X$ the intersection $E \cap Y$ is either empty or contains a nonempty open of $Y$.

Proof. This follows formally from Lemmas 5.16.3 and 5.16.4. $\square$

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