proof of Tukey’s lemma

Let S be a set and F a set of subsets of S such that F is of finite character. By Zorn’s lemma, it is enough to show that F is inductive. For that, it will be enough to show that if (Fi)iI is a family of elements of F which is totally orderedPlanetmathPlanetmath by inclusion, then the union U of the Fi is an element of F as well (since U is an upper bound on the family (Fi)). So, let K be a finite subset of U. Each element of U is in Fi for some iI. Since K is finite and the Fi are totally ordered by inclusion, there is some jI such that all elements of K are in Fj. That is, KFj. Since F is of finite character, we get KF, QED.

Title proof of Tukey’s lemma
Canonical name ProofOfTukeysLemma
Date of creation 2013-03-22 13:54:58
Last modified on 2013-03-22 13:54:58
Owner Koro (127)
Last modified by Koro (127)
Numerical id 4
Author Koro (127)
Entry type Proof
Classification msc 03E25