Correspondence as a graph of a multifunction Suppose I'd like to say that a projection of $R\subset X\times Y$ on $X$ is the whole $X$. That is, $R$ is a graph of a certain multifunction, or equivalently it is a left-total relation. I do remember seeing somewhere the term correspondence being used exactly for such purposes. Is such terminology commonly used in set theory? Could you advise some classical text books where it is used?
 A: It is standard terminology in mathematical economics. See for example, Aliprantis & Border 2007 Infinite Dimensional Analysis: A Hitchhiker's Guide. Terminology varies as to whether any subset of $X\times Y$ is a correspondence or whether the projectiont to $X$ has to be surjective (a "nonempty-valued correspondence"). Also, some people define it as a relation and some as a function with sets as values. Mathemtical economics and related fields such as optimization theory are some of the main users of the concept.
The term originated probably with Nicolas Bourbaki and was imported into economics by Gerard Debreu. In Bourbaki's book on sets, correspondences are treated in Chapter II §3, where a correspondence is defined as a triple of sets $(G,A,B)$ with $G\subseteq A\times B$.
A: Multivalued function is exactly the thing in the question. "Left-total relation" is not nearly as commonly used, although many readers might be able to guess that it means the same thing.  In some situations, multivalued function has more specific meanings such as a differential 1-form, algebraic function or branched covering, but these are compatible (or can be made so) with the set-theoretic idea of a multifunction.

term correspondence ... commonly used in set theory?

It has been standard in mathematics for a very long time, to use correspondence to mean a subset of $A \times B$, and the formal definition of inverses and composition of correspondences.  Subset can be generalized to a context-specific notion of sub-object (such as subvariety, submanifold, subspace, etc) of the product, maybe with additional properties such as being closed, and not necessarily the same thing as sub-objects in the category from which the $A$ and $B$'s are taken.
Common special cases are:


*

*one-to-one correspondence (bijections) 

*many-to-one correspondences (functions)

*one-to-many correspondences (multivalued functions) 

*correspondences in algebraic geometry

