ultrafilter

An ultrafilter on a nonempty set X is a maximal filter on X. A filter on X is a nonempty family F of subsets of X that is closed under supersets and under finite intersections and does not contain the empty set. An ultrafilter U is maximal with respect to inclusion among filters on X. Equivalently, for every subset A of X, either A ∈ U or X \ A ∈ U (and not both, since the empty set cannot lie in a filter).

There are two basic kinds of ultrafilters. Principal ultrafilters are generated by a point x in X: U = {A ⊆ X : x ∈ A}. On finite sets, every ultrafilter is principal. On infinite sets, non-principal (free) ultrafilters exist, but their existence requires some form of the axiom of choice (often proved using Zorn’s lemma).

Ultrafilters have important roles in topology and logic. In topology, they underpin the Stone–Čech compactification βX and provide a notion of convergence: an ultrafilter U on X converges to a point x if every neighborhood of x lies in U. In logic and model theory, ultrafilters are used to form ultraproducts and to develop nonstandard analysis.

Related results include the Ultrafilter Lemma, which states that every filter is contained in an ultrafilter; this lemma is equivalent to the Boolean Prime Ideal Theorem and is weaker than full axiom of choice.