Lemma 66.5.5. 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
Comments (0)