left and right euclidean;
A binary relation from a set to a set is called entire if every element of is related to at least one element of . This includes most examples of what the pre-Bourbaki literature calls a (total) multi-valued function (although that term usually implied some continuity or analyticity properties as well). An entire relation is sometimes called total, although that has another meaning in the theory of partial orders; see total relation.
Like any relation, an entire relation can be viewed as a span
The axiom of choice says precisely that every entire relation contains a function. Failing that, the COSHEP axiom may be interpreted to say that, given , there is a single surjection such that every entire relation from contains a relation given by a span whose left leg is . In any case, entire relations may be preferable to functions in some contexts where the axiom of choice fails.