Lemma 100.49.2. Let $\mathcal{X}$ be a decent, locally Noetherian algebraic stack. Then $|\mathcal{X}|$ is a sober locally Noetherian topological space.

**Proof.**
By Lemma 100.8.3 the topological space $|\mathcal{X}|$ is locally Noetherian. By Lemma 100.49.1 the topological space $|\mathcal{X}|$ is Kolmogorov. By Lemma 100.8.4 the topological space $|\mathcal{X}|$ is quasi-sober. This finishes the proof, see Topology, Definition 5.8.6.
$\square$

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