# maximality principle

Let $S$ be a collection^{} of sets. If, for each chain $C\subseteq S$, there exists an $X\in S$ such that every element of $C$ is a subset of $X$, then $S$ contains a maximal element^{}. This is known as the *maximality principle*.

The maximality principle is equivalent^{} to the axiom of choice^{}.

Title | maximality principle |

Canonical name | MaximalityPrinciple |

Date of creation | 2013-03-22 12:26:18 |

Last modified on | 2013-03-22 12:26:18 |

Owner | akrowne (2) |

Last modified by | akrowne (2) |

Numerical id | 9 |

Author | akrowne (2) |

Entry type | Theorem |

Classification | msc 03E30 |

Classification | msc 03E25 |

Synonym | maximal principle |

Related topic | ZornsLemma |

Related topic | AxiomOfChoice |

Related topic | WellOrderingPrinciple |

Related topic | TukeysLemma |

Related topic | ZermelosPostulate |

Related topic | HaudorffsMaximumPrinciple |