We working in the following with Knutson's definition of an algebraic space
(ie via equivalence relation; there is also another equivalent def via
sheaves but let us work here with the following one):



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

1. $R$ is an equivalence relation as a subset $U \times U$;
2. 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.

A couple of notes on used notations: the equivalence realtion $R
\subset U \times R$ is considered as 
categoretical equivalence relation (also called "internal relation"),
that means that for all 
$T \in (Sch)$ the set $Hom(T,R) \subset Hom(T, U \times U)=
Hom(T,U) \times Hom(T,U)$ is the equivalence relation in usual sense.

**Question**: How one can see that an "usual" scheme $U$ is an 
algebraic space in the sense above? Assume wlog $U$ affine. The
crucial task is to find an equivalence relation $R \subset U \times U$
corresponding to $U$ such that projections $p_i: R \to U$ are etale.

The most natural choice seems to me the image with respect the
diagonal map $\Delta: U \to U \times U$, ie $R:= \Delta(U)$.
$\Delta$ is always an immersion and thus
$\Delta(U)$ is always a locally closed subscheme of $U \times U$.

If we take this choice for $R$, why $p_i: R \to U$ are etale?
Or is it conventional to take another choice for $R$? eg the
closure of the image? if yes, why?