mapping

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.