According to Wikipedia
there are two common ways to define **algebraic spaces**:
they can be defined as either quotients of schemes by étale
equivalence relations,
or as sheaves on a big étale site that are locally isomorphic
to schemes.
Namely:

I) a la Knutson:

An algebraic space $X$ comprises a scheme $U$ and a closed subscheme $R \subset U \times U$ satisfying the following two conditions:

- $R$ is an equivalence relation as a subset $U \times U$;
- the two projections $P_i: R \to U$ onto each factor are étale.

Knutson adds an extra condition that the diagonal map is quasi-compact.

II) as a sheaf:

An algebraic space $\mathfrak {X}$ can be defined as a sheaf of sets $$\mathfrak {X}:(\operatorname{Sch}/S)^{\text{op}}_{\text{ét}} \to \operatorname{Sets}$$ such that

- There is a surjective étale morphism $h_X \to \mathfrak {X}$;
- the diagonal morphism $\Delta _{{\mathfrak {X}}/S}: \mathfrak {X} \to \mathfrak {X} \times \mathfrak {X}$ is representable and quasicompact (thanks to David's careful remark).

(Rmk: in II)1. we identified a scheme $X$ with its image $h_X$ wrt the Yoneda embedding $X \to \operatorname{Hom}(X,{-})$.)

Two questions:

About construction I). Wikipedia moreover says that if $R$ is the

**trivial equivalence**over each connected conponent of $U$ (i.e. for all $x,y \in U$ lying in same component then $xRy$ iff $x=y$) then the so defined algebraic space is a scheme in the usual sense. Why?Where I can find a proof/ reason that the constructions I) and II) are indeed equivalent?