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?