The Stacks project

Lemma 67.8.5. Let $S$ be a scheme. Let $X$ be a quasi-compact, reasonable algebraic space over $S$. There exist an integer $n$ and open subspaces

\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]

such that each $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) is a scheme.

Proof. Immediate consequence of Lemma 67.8.4. $\square$

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