# 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

$$f:J\to \bigcup _{j\in J}{A}_{j}$$ |

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

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

$${\pi}_{i}:\prod _{j\in J}{A}_{j}\to {A}_{i}$$ |

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 |