Ultrafilter on a set

In the mathematical field of set theory, an ultrafilter on a set ${\displaystyle X}$ is a maximal filter on the set ${\displaystyle X.}$ In other words, it is a collection of subsets of ${\displaystyle X}$ that satisfies the definition of a filter on ${\displaystyle X}$ and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of ${\displaystyle X}$ that is also a filter. (In the above, by definition a filter on a set does not contain the empty set.) Equivalently, an ultrafilter on the set ${\displaystyle X}$ can also be characterized as a filter on ${\displaystyle X}$ with the property that for every subset ${\displaystyle A}$ of ${\displaystyle X}$ either ${\displaystyle A}$ or its complement ${\displaystyle X\setminus A}$ belongs to the ultrafilter.

Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set ${\displaystyle \wp (X)}$ and the partial order is subset inclusion ${\displaystyle \,\subseteq .}$ This article deals specifically with ultrafilters on a set and does not cover the more general notion.

There are two types of ultrafilter on a set. A principal ultrafilter on ${\displaystyle X}$ is the collection of all subsets of ${\displaystyle X}$ that contain a fixed element ${\displaystyle x\in X}$. The ultrafilters that are not principal are the free ultrafilters. The existence of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exists models of ZF where every ultrafilter on a set is principal.

Ultrafilters have many applications in set theory, model theory, and topology.^: 186 Usually, only free ultrafilters lead to non-trivial constructions. For example, an ultraproduct modulo a principal ultrafilter is always isomorphic to one of the factors, while an ultraproduct modulo a free ultrafilter usually has more complex structures.