ultrafilters

An ultrafilter on a set X is a maximal among filters on X. A filter on X is a nonempty family F of subsets of X that contains X, is closed under finite intersections, and is closed upward under supersets. An ultrafilter F has the extra property that for every subset A of X, either A is in F or its complement X \ A is in F. This makes ultrafilters a precise notion of “largeness” for subsets of X.

There are two basic types of ultrafilters. A principal ultrafilter at a point x in X consists of all subsets of X that contain x. Such ultrafilters exist on any X. If X is finite, every ultrafilter is principal. On infinite X, there exist nonprincipal (free) ultrafilters, which are not determined by a single point.

Key properties include: an ultrafilter is maximal among proper filters; it is closed under finite intersections and supersets; and it satisfies the dichotomy that every subset or its complement belongs to the ultrafilter. Ultrafilters extend any given filter, and they can be characterized via Zorn’s lemma as guaranteed to exist for any set, a statement known as the Ultrafilter Lemma.

Existence of nonprincipal ultrafilters on infinite sets typically depends on a choice principle equivalent to the Boolean Prime Ideal Theorem. In practice, ultrafilters are central to areas such as topology (Stone–Čech compactification), model theory, nonstandard analysis, and the construction of ultralimits in analysis.

Examples include principal ultrafilters corresponding to points and nonprincipal ultrafilters on N used to define limit processes that behave like generalized limits.