every filter is contained in an ultrafilter

Let X be a set and be a filter on X. Then there exists an ultrafilterMathworldPlanetmath 𝒰 on X which is finer than .

An importance consequnce of this theorem is the existence of free ultrafilters on infinite setsMathworldPlanetmath. According to the theorem, there must exist an ultrafilter which is finer than the cofinite filter. Since the cofinite filter is free, every filter finer than it must also be free, and hence there exists a free ultafilter.

Also note that this theorem requires the axiom of choiceMathworldPlanetmath.

Title every filter is contained in an ultrafilter
Canonical name EveryFilterIsContainedInAnUltrafilter
Date of creation 2013-03-22 14:41:41
Last modified on 2013-03-22 14:41:41
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Theorem
Classification msc 54A20
Related topic LindenbaumsLemma