# 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 $(\mathbbmss{Z}^{+},<)$, because any nonempty subset of it has least member. However, $\mathbbmss{R}^{+}$ (the positive reals) is not a well-ordered set with the usual order, because $(0,1)=\{x:0 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 $\leq$ on $X$ to itself in such a way that $X$ becomes well-ordered with respect to $\leq$.

 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