Definition 72.3.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A Zariski covering of $X$ is a family of morphisms $\{ f_ i : X_ i \to X\} _{i \in I}$ of algebraic spaces over $S$ such that each $f_ i$ is an open immersion and such that

$|X| = \bigcup \nolimits _{i \in I} |f_ i|(|X_ i|),$

i.e., the morphisms are jointly surjective.

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