# ultrafilter

Let $X$ be a set.

## Definitions

A collection $\mathcal{U}$ of subsets of $X$ is an ultrafilter if $\mathcal{U}$ is a filter, and whenever $A\subseteq X$ then either $A\in\mathcal{U}$ or $X\setminus A\in\mathcal{U}$.

Equivalently, an ultrafilter on $X$ is a maximal (http://planetmath.org/MaximalElement) filter on $X$.

More generally, an ultrafilter of a lattice (http://planetmath.org/LatticeFilter) is a maximal proper filter of the lattice. This is indeed a generalization, as an ultrafilter on $X$ can then be defined as an ultrafilter of the power set $\mathcal{P}(X)$.

## Types of ultrafilter

For any $x\in X$ the set $\{A\subseteq X\mid x\in A\}$ is an ultrafilter on $X$. An ultrafilter formed in this way is called a fixed ultrafilter, or a principal ultrafilter, or a trivial ultrafilter. Any other ultrafilter on $X$ is called a free ultrafilter, or a non-principal ultrafilter. An ultrafilter on a finite set is necessarily fixed. On any infinite set there are free ultrafilters (in great abundance (http://planetmath.org/NumberOfUltrafilters)), but their existence depends on the Axiom of Choice, and so none can be explicitly constructed.

An ultrafilter $\mathcal{U}$ on $X$ is called a uniform ultrafilter if every member of $\mathcal{U}$ has the same cardinality. (An ultrafilter on a singleton is uniform, but this is a degenerate case and is often excluded. All other uniform ultrafilters are free.)

 Title ultrafilter Canonical name Ultrafilter Date of creation 2013-03-22 12:13:59 Last modified on 2013-03-22 12:13:59 Owner yark (2760) Last modified by yark (2760) Numerical id 11 Author yark (2760) Entry type Definition Classification msc 54A20 Related topic Filter Related topic Ultranet Related topic EveryBoundedSequenceHasLimitAlongAnUltrafilter Related topic LatticeFilter Defines fixed ultrafilter Defines principal ultrafilter Defines trivial ultrafilter Defines free ultrafilter Defines non-principal ultrafilter Defines nonprincipal ultrafilter Defines uniform ultrafilter