number of ultrafilters


Let X be a set. The number ( of ultrafilters ( on X is |X| if X is finite (, and 22|X| if X is infiniteMathworldPlanetmath (


If X is finite then each ultrafilter on X is principal, and so there are exactly |X| ultrafilters. In the rest of the proof we will assume that X is infinite.

Let F be the set of all finite subsets of X, and let Φ be the set of all finite subsets of F.

For each AX define BA={(f,ϕ)F×ΦAfϕ}, and BA=(F×Φ)BA. For each 𝒮𝒫(X) define 𝒮={BAA𝒮}{BAA𝒮}.

Let 𝒮𝒫(X), and suppose A1,,Am𝒮 and D1,,Dn𝒫(X)𝒮, so that we have BA1,,BAm,BD1,,BDn𝒮. For i{1,,m} and j{1,,n} let ai,j be such that either ai,jAiDj or ai,jDjAi. This is always possible, since AiDj. Let f={ai,j1im, 1jn}, and put ϕ={Aif1im}. Then (f,ϕ)BAi, for i=1,,m. If for some j{1,,n} we have Djfϕ , then Djf=Aif for some i{1,,m}, which is impossible, as ai,j is in one of these sets but not the other. So Djfϕ, and therefore (f,ϕ)BDj. So (f,ϕ)BA1BAmBD1BDn. This shows that any finite subset of 𝒮 has nonempty intersectionDlmfMathworldPlanetmath, and therefore 𝒮 can be extended to an ultrafilter 𝒰𝒮.

Suppose ,𝒮𝒫(X) are distinct. Then, relabelling if necessary, 𝒮 is nonempty. Let A𝒮. Then BA𝒰, but BA𝒰𝒮 since BA𝒰𝒮. So 𝒰 and 𝒰𝒮 are distinct for distinct and 𝒮. Therefore {𝒰𝒮𝒮𝒫(X)} is a set of 22|X| ultrafilters on F×Φ. But |F×Φ|=|X|, so F×Φ has the same number of ultrafilters as X. So there are at least 22|X| ultrafilters on X, and there cannot be more than 22|X| as each ultrafilter is an element of 𝒫(𝒫(X)). ∎


The number of topologies on an infinite set X is 22|X|.


Let X be an infinite set. By the theorem, there are 22|X| ultrafilters on X. If 𝒰 is an ultrafilter on X, then 𝒰{} is a topology on X. So there are at least 22|X| topologies on X, and there cannot be more than 22|X| as each topology is an element of 𝒫(𝒫(X)). ∎

Title number of ultrafilters
Canonical name NumberOfUltrafilters
Date of creation 2013-03-22 15:51:49
Last modified on 2013-03-22 15:51:49
Owner yark (2760)
Last modified by yark (2760)
Numerical id 14
Author yark (2760)
Entry type Theorem
Classification msc 03E99