stylistic:style_transfer
Infinite Cartesian 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 { X i } i ∈ I {\displaystyle \{X_{i}\}_{i\in I}} is a family of sets indexed by I, then the Cartesian product of the sets in { X i } i ∈ I {\displaystyle \{X_{i}\}_{i\in I}} is defined to be

that is, the set of all functions defined on the index set I such that the value of the function at a particular index i is an element of X. Even if each of the X is nonempty, the Cartesian product may be empty if the axiom of choice, which is equivalent to the statement that every such product is nonempty, is not assumed. ∏ i ∈ I X i {\displaystyle \prod _{i\in I}X_{i}} may also be denoted X {\displaystyle {\mathsf {X}}} i ∈ I X i {\displaystyle {}_{i\in I}X_{i}}.

For each j in I, the function

defined by π j (f) = f (j) {\displaystyle \pi _{j}(f)=f(j)} is called the j -th projection map.

Cartesian power is a Cartesian product where all the factors X are the same set X. In this case,

is the set of all functions from I to X, and is frequently denoted X. This case is important in the study of cardinal exponentiation. An important special case is when the index set is N {\displaystyle \mathbb {N} }, the natural numbers: this Cartesian product is the set of all infinite sequences with the i -th term in its corresponding set X. For example, each element of

can be visualized as a vector with countably infinite real number components. This set is frequently denoted R ω {\displaystyle \mathbb {R} ^{\omega }}, or R N {\displaystyle \mathbb {R} ^{\mathbb {N} }}.