# Well-ordered set

Note: This page exists only to contain a simpler, easier view of Well-ordering - until all the concepts can be united anyway.

## Definition

A set [ilmath]A[/ilmath] with an linear ordering [ilmath]<\subseteq A\times A[/ilmath] where if [ilmath](a,b)\in<[/ilmath] we write [ilmath]a<b[/ilmath] is said to be well ordered[1] if:

• Every nonempty subset of [ilmath]A[/ilmath] has a least element

That is to say that:

• [ilmath]\forall X\in\mathcal{P}(A)\exists p\in X\forall x\in X[p=p\vee p<x][/ilmath]

Or more simply:

• [ilmath]\forall X\in\mathcal{P}(A)\exists p\in X\forall x\in X[p\le x][/ilmath][Note 1]

## Notes

1. Recall that for every linear ordering [ilmath]>[/ilmath] there exists a corresponding partial ordering [ilmath]\ge[/ilmath] and for every [ilmath]\ge[/ilmath] there exists a corresponding [ilmath]>[/ilmath]

## References

1. Topology - James R. Munkres - Second Edition