Definitions

A collection $\mathcal{U}$ of subsets of 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 is a maximal filter on .

More generally, an ultrafilter of a lattice is a maximal proper filter of the lattice. This is indeed a generalization, as an ultrafilter on 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 . An ultrafilter formed in this way is called a fixed ultrafilter, or a principal ultrafilter, or a trivial ultrafilter. Any other ultrafilter on 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), but their existence depends on the Axiom of Choice, and so none can be explicitly constructed.

An ultrafilter $\mathcal{U}$ on 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.)

"ultrafilter" is owned by yark. [ full author list (2) | owner history (1) ]

(view preamble)

Also defines:

fixed ultrafilter, principal ultrafilter, trivial ultrafilter, free ultrafilter, non-principal ultrafilter, nonprincipal ultrafilter, uniform ultrafilter

Attachments:: every filter is contained in an ultrafilter (Theorem) by rspuzio
alternative characterization of ultrafilter (Theorem) by yark
number of ultrafilters (Theorem) by yark

Log in to rate this entry.
(view current ratings)

Cross-references: singleton, cardinality, axiom of choice, infinite set, finite set, power set, lattice, filter, subsets, collection
There are 12 references to this entry.

This is version 7 of ultrafilter, born on 2002-01-24, modified 2006-11-17.
Object id is 1611, canonical name is Ultrafilter.
Accessed 7237 times total.

Classification:

AMS MSC:

54A20 (General topology :: Generalities :: Convergence in general topology )

Pending Errata and Addenda

None.[ View all 3 ]

Discussion

forum policy

No messages.

Interact