Lemma 67.25.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have

where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes étale over $X$ or over all affine schemes étale over $X$.

Lemma 67.25.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We have

\[ X_{\text{ft-pts}} = \bigcup \nolimits _{\varphi : U \to X\text{ étale }} |\varphi |(U_0) \]

where $U_0$ is the set of closed points of $U$. Here we may let $U$ range over all schemes étale over $X$ or over all affine schemes étale over $X$.

**Proof.**
Immediate from Lemma 67.25.1.
$\square$

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