Lemma 100.4.9 (04Y9)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 100: Properties of Algebraic Stacks
Section 100.4: Points of algebraic stacks
Lemma 100.4.9 (cite)

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Lemma 100.4.9. Let $\mathcal{X}$ be an algebraic stack. Every point of $|\mathcal{X}|$ has a fundamental system of quasi-compact open neighbourhoods. In particular $|\mathcal{X}|$ 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 $U \to |\mathcal{X}|$ of topological spaces. Namely, if $U \to \mathcal{X}$ is surjective and smooth, then Lemma 100.4.7 guarantees that $|U| \to |\mathcal{X}|$ is continuous, surjective, and open. $\square$

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