Lemma 109.62.1. There exists a reduced scheme $X$ and a schematically dense open $U \subset X$ such that some weakly associated point $x \in X$ is not in $U$.

**Proof.**
In the first example we have $\mathfrak p \not\in U$ by construction. In Gabber's examples the schemes $\mathop{\mathrm{Spec}}(R)$ or $\mathop{\mathrm{Spec}}(R')$ are reduced.
$\square$

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