Lemma 69.16.3. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$ such that $|X|$ has finitely many irreducible components.
There exists a surjective finite morphism $f : Y \to X$ of finite presentation where $Y$ is a scheme such that $f$ is finite étale over a quasi-compact dense open $U \subset X$,
given a surjective étale morphism $V \to X$ we may choose $Y \to X$ such that for every $y \in Y$ there is an open neighbourhood $W \subset Y$ such that $W \to X$ factors through $V$.