The Stacks project

Lemma 27.2.3. Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$.

Proof. If $U$ and $V$ are quasi-compact open then $U \cap V = \Delta ^{-1}(U \times V)$, where $\Delta : X \to X \times X$ is the diagonal. As $X$ is quasi-separated we see that $\Delta $ is quasi-compact. Hence we see that $U \cap V$ is quasi-compact as $U \times V$ is quasi-compact (details omitted; use Schemes, Lemma 25.17.4 to see $U \times V$ is a finite union of affines). The other assertions follow from the first and Topology, Lemma 5.27.1. $\square$


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