Lemma 66.17.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U \subset X$ be an open subspace. The following are equivalent

for every étale morphism $\varphi : V \to X$ (of algebraic spaces) the scheme theoretic closure of $\varphi ^{-1}(U)$ in $V$ is equal to $V$,

there exists a scheme $V$ and a surjective étale morphism $\varphi : V \to X$ such that the scheme theoretic closure of $\varphi ^{-1}(U)$ in $V$ is equal to $V$,

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