Lemma 109.26.1. Nonexistence quasi-compact opens of affines:

There exist an affine scheme $S$ and affine open $U \subset S$ such that there is no quasi-compact open $V \subset S$ with $U \cap V = \emptyset $ and $U \cup V$ dense in $S$.

There exists an affine scheme $S$ and a closed point $s \in S$ such that $S \setminus \{ s\} $ does not contain a quasi-compact dense open.

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