# 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