well ordered set


A well-ordered set is a totally ordered setMathworldPlanetmath 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, + (the positive reals) is not a well-ordered set with the usual order, because (0,1)={x:0<x<1} 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 relationMathworldPlanetmath on X to itself in such a way that X becomes well-ordered with respect to .

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