# Mapping

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single-valued

A law according to which to every element of a given set has been assigned a completely defined element of another given set ( may coincide with ). Such a relation between the elements and is denoted in the form , or . One also writes and says that the mapping operates from into . The set is called the domain (of definition) of the mapping, while the set is called the range (of values) of the mapping. The mapping is also called a mapping of the set into the set (or onto the set if ). Logically, the concept of a "mapping" coincides with the concept of a function, an operator or a transformation.

A mapping gives rise to a set , which is called the graph of the mapping. On the other hand, a set defines a single-valued mapping having graph if and only if for all one and only one exists such that ; and then .

Two mappings and are said to be equal if their domains of definition coincide and if for each . In this case the ranges of these mappings also coincide. A mapping defined on is constant if there is an such that for every . The mapping defined on a subset of by the equality , , is called the restriction of the mapping to ; this restriction is often denoted by . A mapping defined on a set and satisfying the equality for all is called an extension (or continuation) of the mapping to . If three sets are given, if a mapping with values in is defined on , and a mapping with values in is defined on , then there exists a mapping with domain of definition , taking values in , and defined by the equality . This mapping is called the composite of the mappings and , while and are called component (factor) mappings. The mapping is also called the compound mapping (composite mapping, composed mapping), consisting of the interior mapping and the exterior mapping . The composed mapping is denoted by , where the order of the notation is vital (for functions of a real variable, the term superposition is also used). The concept of a compound mapping can be generalized to any finite number of components of the mapping.

A mapping , defined on and taking values in , gives rise to a new mapping defined on the subsets of and taking subsets of as values. In fact, if , then

The set is called the image of . If , the initial mapping is obtained; thus, is an extension of from the set to the set of all subsets of if a one-element set is identified with the element comprising it. When , a set is called an invariant subset for if , while a point is called a fixed point for if . Invariant sets and fixed points are important in solving functional equations of the form or .

Every mapping gives rise to a mapping defined on the subsets of the set or and taking subsets of the set as values. In fact, for every (or ), the set is denoted by , and is called the complete inverse image (complete pre-image) of . If for each consists of a single element, then is a mapping of elements, is defined on , and takes values in . It is also called the inverse mapping for . The existence of an inverse mapping is equivalent to the solvability of the equation , , for a unique when is given.

If the sets and have certain properties, then interesting classes can be distinguished in the set of all mappings from into . Thus, for partially ordered sets and , the mapping is isotone if implies (cf. Isotone mapping). For complex planes and , the class of holomorphic mappings is naturally selected. For topological spaces and , the class of continuous mappings between these spaces is distinguished naturally; an extended theory of differentiation of mappings (cf. Differentiation of a mapping) has been constructed. For mappings of a scalar argument and, in the most general case, for mappings defined on a measure space, the concept of (weak or strong) measurability can be introduced, and various Lebesgue-type integrals can be constructed (for example, the Bochner integral and the Daniell integral).

A mapping is called a multi-valued mapping if subsets consisting of more than one element are assigned to certain values of . Examples of this type of mappings include multi-sheeted functions of a complex variable, multi-valued mappings of topological spaces, and others.

#### References

 [1] N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French) [2] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) [3] J.L. Kelley, "General topology" , Springer (1975)