Lemma 98.9.11. 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 92.16.2. By Lemma 98.9.10 we see that open substacks correspond to $R$-invariant open subschemes of $U$. On the other hand Lemmas 98.4.5 and 98.4.7 guarantee these correspond bijectively to open subsets of $|\mathcal{X}|$. $\square$

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