If tuples are defined as nested ordered pairs, it can be identified with . If a tuple is defined as a function on that takes its value at "i" to be the "i"th element of the tuple, then the Cartesian product "X"1×...×"X"n is the set of functions  The Cartesian square of a set "X" is the Cartesian product . An example is the 2-dimensional plane where R is the set of real numbers: R2 is the set of all points where "x" and "y" are real numbers (see the Cartesian coordinate system). The "n"-ary Cartesian power of a set "X" can be defined as  An example of this is , with R again the set of real numbers, and more generally R"n". The "n"-ary cartesian power of a set "X" is isomorphic to the space of functions from an "n"-element set to "X". As a special case, the 0-ary cartesian power of "X" may be taken to be a singleton set, corresponding to the empty function with codomain "X". It is possible to define the Cartesian product of an arbitrary (possibly infinite) indexed family of sets. If "I" is any index set, and formula_23 is a family of sets indexed by "I", then the Cartesian product of the sets in formula_23 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 "Xi" 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. For each "j" in "I", the function  defined by formula_27 is called the "j"th projection map. Cartesian power is a Cartesian product where all the factors "Xi" are the same set "X". In this case,  is the set of all functions from "I" to "X", and is frequently denoted "XI". This case is important in the study of cardinal exponentiation. An important special case is when the index set is formula_29, 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.