Ultrafilter

In mathematics, an ultrafilter is a maximal filter. Equivalently, an ultrafilter F on a set S is a filter on S with the additional property that for every subset A of S, either A is in F or S \\ A is in F.

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter on S consists of all sets containing a particular point of S. Any ultrafilter which is not principal is called a free (or non-principal) ultrafilter.

One can show that every filter is contained in an ultrafilter (see Ultrafilter Lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma, so explicit examples of free ultrafilters cannot be given. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter on a finite set is principal.

Another way of looking at ultrafilters is to define a function m on the power set of S by setting m(A) = 1 if A is contained in F and m(A) = 0 otherwise. Then m is a finitely additive measure on S, and every property of elements of S is either true almost everywhere or false almost everywhere.

Ultrafilters are useful in topology, especially in relation to compact Hausdorff spaces. Every ultrafilter on a compact Hausdorff space converges to exactly one point.

The set G of all ultrafilters on a set S can be topologized in a natural way. For any subset A of S, let D_A = { U in G : A in U }. Then the set of all D_A is a base for a compact Hausdorff topology on G. The resulting topological space is the Stone-Čech compactification of a discrete space of cardinality |S|.

Ultrafilters are also used in the construction of hyperreal numbers.

Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.