Clarifying an interpretation of algebraic spaces From several lecture notes and some posts, people claim that while schemes are constructed by gluing affine schemes over the Zariski topology, algebraic spaces are constructed by gluing affine schemes over the étale topology, which I do not really understand. Could someone explain this point carefully? Examples?
 A: If I remember correctly, this goes roughly as follows. Consider the category $\mathcal C=\operatorname{Rings}^{op}$, first endowed with the Zariski topology. You can consider sheaves on this site that are locally covered by representable sheaves. Such sheaves form a category equivalent to the category of schemes. 
As you can guess, if you now consider $\mathcal C$ endowed with the étale topology, you will get a category equivalent to the category of algebraic spaces.
ps : I am looking for a reference. The best I found by now : 
Commutative rings to algebraic spaces in one jump?
Erratum : as nfdc23 points out, some condition on the diagonal is missing. The correct definition that I copy from Chris Schommer-Pries answer here 
Quasi-separatedness for Algebraic Spaces
is the following
Definition: An algebraic space over $S$ is a functor $X : (Sch/S)^{op} \to S_{et}$ such
that


*

*$X$ is a sheaf on the big  étale topology on S,

*$\Delta : X \to X \times_S X$ is representable, and 

*there exists an $S$-scheme $U \to S$ and a surjective  étale morphism $U \to X$.


This is Definition 5.1.10 in Olsson's book Algebraic Spaces and Stacks
https://bookstore.ams.org/coll-62/ .
In remark 5.1.11 he remarks that Knutson's definition includes the fact that $\Delta$ is quasi-compact.
The same definition and more information can be found in the stacks project :
see https://stacks.math.columbia.edu/tag/025Y and https://stacks.math.columbia.edu/tag/076M .
