Lemma 66.29.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E \subset |Y|$ be a subset. If $E$ is étale locally constructible in $Y$, then $f^{-1}(E)$ is étale locally constructible in $X$.

Proof. Choose a scheme $V$ and a surjective étale morphism $\varphi : V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to X$ is surjective étale and the inverse image of $f^{-1}(E)$ in $U$ is the inverse image of $\varphi ^{-1}(E)$ by $U \to V$. Thus the lemma follows from the case of schemes for $U \to V$ (Morphisms, Lemma 29.22.1) and the definition (Properties of Spaces, Definition 65.8.2). $\square$

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