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 Xi . 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 Xi . 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 Xi 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.
Read more about this topic: Cartesian Product
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