Lemma 65.5.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Every point of $|X|$ has a fundamental system of open quasi-compact neighbourhoods. In particular $|X|$ is locally quasi-compact in the sense of Topology, Definition 5.13.1.
Proof. This follows formally from the fact that there exists a scheme $U$ and a surjective, open, continuous map $U \to |X|$ of topological spaces. To be a bit more precise, if $u \in U$ maps to $x \in |X|$, then the images of the affine neighbourhoods of $u$ will give a fundamental system of quasi-compact open neighbourhoods of $x$. $\square$
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