# Hausdorff’s maximum principle

Theorem
Let $X$ be a partially ordered set^{}. Then there exists a maximal totally
ordered^{} subset of $X$.

The Hausdorff^{}’s maximum principle is one of the many theorems equivalent^{}
to the
axiom of choice^{} (http://planetmath.org/AxiomOfChoice).
The below proof uses Zorn’s lemma, which
is also equivalent to the
.

###### Proof.

Let $S$ be the set of all totally ordered subsets of $X$. $S$ is not empty, since the empty set^{} is an element of $S$. Partial order^{} $S$ by inclusion. Let $\tau $ be a chain (of elements) in $S$. Being each totally ordered, the union of all these elements of $\tau $ is again a totally ordered subset of $X$, and hence an element of $S$, as is easily verified. This shows that $S$, ordered by inclusion, is inductive. The result now follows from Zorn’s lemma.
∎

Title | Hausdorff’s maximum principle |

Canonical name | HausdorffsMaximumPrinciple |

Date of creation | 2013-03-22 13:04:42 |

Last modified on | 2013-03-22 13:04:42 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 03E25 |

Synonym | maximum principle |

Synonym | Hausdorff maximality theorem |

Related topic | ZornsLemma |

Related topic | AxiomOfChoice |

Related topic | ZermelosWellOrderingTheorem |

Related topic | ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple |

Related topic | EveryVectorSpaceHasABasis |

Related topic | MaximalityPrinciple |