Lemma 66.6.2. Let S be a scheme. Let X be an algebraic space over S. There exists a Zariski covering X = \bigcup X_ i such that each algebraic space X_ i has a surjective étale covering by an affine scheme. We may in addition assume each X_ i maps into an affine open of S.
Proof. By Lemma 66.6.1 we can find a surjective étale morphism U = \coprod U_ i \to X, with U_ i affine and mapping into an affine open of S. Let X_ i \subset X be the open subspace of X such that U_ i \to X factors through an étale surjective morphism U_ i \to X_ i, see Lemma 66.4.10. Since U = \bigcup U_ i we see that X = \bigcup X_ i. As U_ i \to X_ i is surjective it follows that X_ i \to S maps into an affine open of S. \square
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