number of ultrafilters
Theorem.
Let X be a set. The number (http://planetmath.org/CardinalNumber) of ultrafilters (http://planetmath.org/Ultrafilter) on X is |X| if X is finite (http://planetmath.org/Finite), and 22|X| if X is infinite (http://planetmath.org/Infinite).
Proof.
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 A⊆X define BA={(f,ϕ)∈F×Φ∣A∩f∈ϕ}, and B∁A=(F×Φ)∖BA. For each 𝒮⊆𝒫(X) define ℬ𝒮={BA∣A∈𝒮}∪{B∁A∣A∉𝒮}.
Let 𝒮⊆𝒫(X), and suppose A1,…,Am∈𝒮 and D1,…,Dn∈𝒫(X)∖𝒮, so that we have BA1,…,BAm,B∁D1,…,B∁Dn∈ℬ𝒮. For i∈{1,…,m} and j∈{1,…,n} let ai,j be such that either ai,j∈Ai∖Dj or ai,j∈Dj∖Ai. This is always possible, since Ai≠Dj. Let f={ai,j∣1≤i≤m, 1≤j≤n}, and put ϕ={Ai∩f∣1≤i≤m}. Then (f,ϕ)∈BAi, for i=1,…,m. If for some j∈{1,…,n} we have Dj∩f∈ϕ , then Dj∩f=Ai∩f for some i∈{1,…,m}, which is impossible, as ai,j is in one of these sets but not the other. So Dj∩f∉ϕ, and therefore (f,ϕ)∈B∁Dj. So (f,ϕ)∈BA1∩⋯∩BAm∩B∁D1∩⋯∩B∁Dn. This shows that any finite subset of ℬ𝒮 has nonempty intersection, and therefore ℬ𝒮 can be extended to an ultrafilter 𝒰𝒮.
Suppose ℛ,𝒮⊆𝒫(X) are distinct. Then, relabelling if necessary, ℛ∖𝒮 is nonempty. Let A∈ℛ∖𝒮. Then BA∈𝒰ℛ, but BA∉𝒰𝒮 since B∁A∈𝒰𝒮. So 𝒰ℛ and 𝒰𝒮 are distinct for distinct ℛ and 𝒮. Therefore {𝒰𝒮∣𝒮⊆𝒫(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)). ∎
Corollary.
The number of topologies on an infinite set X is 22|X|.
Proof.
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 |