Lemma 101.50.4. Let \mathcal{X} be a decent algebraic stack such that \mathcal{I}_\mathcal {X} \to \mathcal{X} is quasi-compact. There are canonical bijections between the following sets:
the set of points of \mathcal{X}, i.e., |\mathcal{X}|,
the set of irreducible closed subsets of |\mathcal{X}|,
the set of integral closed substacks of \mathcal{X}.
The bijection from (1) to (2) sends x to \overline{\{ x\} }. The bijection from (3) to (2) sends \mathcal{Z} to |\mathcal{Z}|.
Proof.
Our map defines a bijection between (1) and (2) as |\mathcal{X}| is sober by Proposition 101.49.3. Given T \subset |\mathcal{X}| closed and irreducible, there is a unique reduced closed substack \mathcal{Z} \subset \mathcal{X} such that |\mathcal{Z}| = T, namely, \mathcal{Z} is the reduced induced subspace structure on T, see Properties of Stacks, Definition 100.10.4. Then \mathcal{Z} is an integral algebraic stack because it is decent (Lemma 101.48.3), the morphism \mathcal{I}_\mathcal {Z} \to \mathcal{Z} is quasi-compact (as the base change of \mathcal{I}_\mathcal {X} \to \mathcal{X}, see Lemma 101.5.6), \mathcal{Z} is reduced, and |\mathcal{Z}| is irreducible.
\square
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