Lemma 66.4.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $U$ be a scheme and let $\varphi : U \to X$ be an étale morphism. The following are equivalent:

$x$ is in the image of $|U| \to |X|$, and setting $R = U \times _ X U$ the fibres of both

\[ |U| \longrightarrow |X| \quad \text{and}\quad |R| \longrightarrow |X| \]over $x$ are finite,

there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ with $k$ a field in the equivalence class of $x$, and the fibre product $\mathop{\mathrm{Spec}}(k) \times _ X U$ is a finite nonempty scheme over $k$.

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