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Definition 73.4.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. An étale 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 étale and such that

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

i.e., the morphisms are jointly surjective.

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