Filter (mathematics)

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In set theory, a filter is a family of subsets of a given set which has properties generalising those of neighbourhood in topology.

Formally, a filter on a set X is a subset ℱ of the power set 𝒫X with the properties:

1. X∈ℱ;
2. ∅∉ℱ;
3. A,B∈ℱ⇒A∩B∈ℱ;
4. A∈ℱ and A⊆B⇒B∈ℱ.

If G is a subset of X then the family

⟨G⟩={A⊆X:G⊆A}

is a filter, the principal filter on G.

In a topological space (X,𝒯), the neighbourhoods of a point x

𝒩x={N⊆X:∃U∈𝒯,x∈u⊆N}

form a filter, the neighbourhood filter of x.

### Filter bases[edit]

A base ℬ for the filter ℱ is a non-empty collection of non-empty sets such that the family of subsets of X containing some element of ℬ is precisely the filter ℱ.

## Ultrafilters[edit]

An ultrafilter is a maximal filter: that is, a filter on a set which is not properly contained in any other filter on the set. Equivalently, it is a filter ℱ with the property that for any subset A⊆X either A∈ℱ or the complement X∖A∈ℱ.

The principal filter on a singleton set {x}, namely, all subsets of X containing x, is an ultrafilter.