# well ordered set

A *well-ordered* set is a totally ordered set^{} in which every nonempty subset has a least member.

An example of well-ordered set is the set of positive integers with the standard order relation $$, because any nonempty subset of it has least member. However, ${\mathbb{R}}^{+}$ (the positive reals) is not a well-ordered set with the usual order, because $$ is a nonempty subset but it doesn’t contain a least number.

A well-ordering of a set $X$ is the result of defining a binary relation^{} $\le $ on $X$ to itself in such a way that $X$ becomes well-ordered with respect to $\le $.

Title | well ordered set |

Canonical name | WellOrderedSet |

Date of creation | 2013-03-22 11:47:22 |

Last modified on | 2013-03-22 11:47:22 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 16 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 03E25 |

Classification | msc 06A05 |

Classification | msc 81T17 |

Classification | msc 81T13 |

Classification | msc 81T75 |

Classification | msc 81T45 |

Classification | msc 81T10 |

Classification | msc 81T05 |

Classification | msc 42-02 |

Classification | msc 55R15 |

Classification | msc 47D03 |

Classification | msc 55U35 |

Classification | msc 55U40 |

Classification | msc 47D08 |

Classification | msc 55-02 |

Classification | msc 18-00 |

Synonym | well-ordered |

Synonym | well-ordered set |

Related topic | WellOrderingPrinciple |

Related topic | NaturalNumbersAreWellOrdered |

Defines | well-ordering |