In

mathematics,

mapping (or just

map) is a common

synonym for

function. The

traditional definition: a

mapping from a

set X to a

set Y is a

subset f of the

cartesian product X x Y such that, for each element
x of X, there is exactly one y in Y with (x,y) in f. Then this y is denoted f(x), the

value of the

mapping f at x.

If the objects X and Y are more sophisticated structures than sets, for instance topological spaces,
algebras or manifolds, use of the word mapping *frequently*, but not *always*, connotes that f preserves the relevant structure. Thus a mapping of topological spaces is a continuous function, a mapping of smooth manifolds is a smooth function, a mapping of groups or algebras is a homomorphism.