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# Ultrafilter

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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 and that free ultrafilters exist, but the proofs involve the axiom of choice in the form of Zorn's lemma, and therefore 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.

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 DA = { U in G : A in U }. Then the set of all DA 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.

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Last edited August 16, 2001 5:02 am (diff)
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