Hausdorff’s maximum principle
The Hausdorff’s maximum principle is one of the many theorems equivalent to the axiom of choice (http://planetmath.org/AxiomOfChoice). The below proof uses Zorn’s lemma, which is also equivalent to the .
Let be the set of all totally ordered subsets of . is not empty, since the empty set is an element of . Partial order by inclusion. Let be a chain (of elements) in . Being each totally ordered, the union of all these elements of is again a totally ordered subset of , and hence an element of , as is easily verified. This shows that , ordered by inclusion, is inductive. The result now follows from Zorn’s lemma. ∎
|Title||Hausdorff’s maximum principle|
|Date of creation||2013-03-22 13:04:42|
|Last modified on||2013-03-22 13:04:42|
|Last modified by||CWoo (3771)|
|Synonym||Hausdorff maximality theorem|