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, ${ℝ}^{+}$ (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