Well-ordered set
From Maths
Note: This page exists only to contain a simpler, easier view of Well-ordering - until all the concepts can be united anyway.
Contents
[hide]Definition
A set A with an linear ordering <⊆A×A where if (a,b)∈< we write a<b is said to be well ordered[1] if:
- Every nonempty subset of A has a least element
That is to say that:
- ∀X∈P(A)∃p∈X∀x∈X[p=p∨p<x]
Or more simply:
- ∀X∈P(A)∃p∈X∀x∈X[p≤x][Note 1]
Notes
- Jump up ↑ Recall that for every linear ordering > there exists a corresponding partial ordering ≥ and for every ≥ there exists a corresponding >
References
- Jump up ↑ Topology - James R. Munkres - Second Edition
