Lemma 99.30.3. Let $\mathcal{X}$ be an algebraic stack whose diagonal is quasi-compact (for example if $\mathcal{X}$ is quasi-separated). Then there is an open covering $|\mathcal{X}| = \bigcup U_ i$ with $U_ i$ spectral. In particular $|\mathcal{X}|$ is a sober topological space.

Proof. Immediate consequence of Lemma 99.30.1. $\square$

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