Cartesian Product - Infinite Products

Infinite Products

It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If I is any index set, and is a collection of sets indexed by I, then the Cartesian product of the sets in X is defined to be

that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of X_i . Even if each of the is nonempty, the Cartesian product may be empty in general. The axiom of choice postulates that the product is nonempty.

For each j in I, the function

defined by is called the j -th projection map.

An important case is when the index set is, the natural numbers: this Cartesian product is the set of all infinite sequences with the i -th term in its corresponding set X_i . For example, each element of

can be visualized as a vector with countably infinite real-number components. This set is frequently denoted, or .

The special case Cartesian exponentiation occurs when all the factors X_i involved in the product are the same set X. In this case,

is the set of all functions from I to X, and is frequently denoted . This case is important in the study of cardinal exponentiation.

The definition of finite Cartesian products can be seen as a special case of the definition for infinite products. In this interpretation, an n-tuple can be viewed as a function on {1, 2, ..., n} that takes its value at i to be the i-th element of the tuple (in some settings, this is taken as the very definition of an n-tuple).

Nothing in the definition of an infinite Cartesian product implies that the Cartesian product of nonempty sets must itself be nonempty. This assertion is equivalent to the axiom of choice.