Does every algebraic space have a Nisnevich cover by a scheme?

(Assume that the algebraic space is quasi-separated, quasi-compact and over a scheme $S$.)

**Background**:

Every algebraic space has an etale cover by a scheme by definition: an algebraic space is a sheaf $$ F : (Sch/S)_{fppf} \to (Set) $$ with representable diagonal $\Delta : F \to F \times F$ and an etale, surjective morphism $U \to F$ from a sheaf associated to an $S$-scheme $U$.

A Nisnevich cover in this context is a surjective, etale morphism $f : U \to X$ such that for all $x \in X$, there exists a $u \in U$ with $f(u) = x$ such that the induced morphism $k(x) \to k(u)$ is an isomorphism.

defined.) So with a gigantic disjoint union you get a huge Nisnevich scheme cover. If $X$ is also quasi-compact then it has a dense open subscheme $V$ (also proved by Knutson) and by noetherian induction you get a quasi-compact Nisnevich scheme cover applying the previous Knutson theorem at the finitely many generic points of $X-V$. $\endgroup$ – user30180 Jun 8 '13 at 3:29