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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).