Lemma 115.16.7. Let $S$ be a scheme. Let $X$ be a very reasonable algebraic space over $S$. There exists a set of schemes $U_ i$ and morphisms $U_ i \to X$ such that

each $U_ i$ is a quasi-compact scheme,

each $U_ i \to X$ is étale,

both projections $U_ i \times _ X U_ i \to U_ i$ are quasi-compact, and

the morphism $\coprod U_ i \to X$ is surjective (and étale).

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