generalized Cartesian product

Given any family of sets $\{A_{j}\}_{j\in J}$ indexed by an index set $J$ , the generalized Cartesian product

\prod_{j\in J}A_{j}

is the set of all functions

f\colon J\to\bigcup_{j\in J}A_{j}

such that $f(j)\in A_{j}$ for all $j\in J$ .

For each $i\in J$ , the projection map

\pi_{i}\colon\prod_{j\in J}A_{j}\to A_{i}

is the function defined by

\pi_{i}(f):=f(i).

The generalized Cartesian product is the product (http://planetmath.org/CategoricalDirectProduct) in the category of sets.

The axiom of choice is the statement that the generalized Cartesian product of nonempty sets is nonempty. The generalized Cartesian product is usually called the Cartesian product.

Title	generalized Cartesian product
Canonical name	GeneralizedCartesianProduct
Date of creation	2013-03-22 11:49:02
Last modified on	2013-03-22 11:49:02
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	15
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 03E20
Related topic	CartesianProduct
Related topic	ProductTopology
Related topic	AxiomOfChoice
Related topic	OrderedTuplet
Related topic	FunctorCategory2
Defines	projection map