The Stacks project

Lemma 99.9.12. Let $\mathcal{X}$ be an algebraic stack. The rule $\mathcal{U} \mapsto |\mathcal{U}|$ defines an inclusion preserving bijection between open substacks of $\mathcal{X}$ and open subsets of $|\mathcal{X}|$.

Proof. Choose a presentation $[U/R] \to \mathcal{X}$, see Algebraic Stacks, Lemma 93.16.2. By Lemma 99.9.11 we see that open substacks correspond to $R$-invariant open subschemes of $U$. On the other hand Lemmas 99.4.5 and 99.4.7 guarantee these correspond bijectively to open subsets of $|\mathcal{X}|$. $\square$

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