Filter (mathematics)

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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 {\displaystyle {\mathcal {F}}} of the power set  P X {\displaystyle {\mathcal {P}}X} with the properties:

1. X ∈ F ; {\displaystyle X\in {\mathcal {F}};\,}
2. ∅ ∉ F ; {\displaystyle \emptyset \not \in {\mathcal {F}};\,}
3. A , B ∈ F ⇒ A ∩ B ∈ F ; {\displaystyle A,B\in {\mathcal {F}}\Rightarrow A\cap B\in {\mathcal {F}};\,}
4. A ∈ F and A ⊆ B ⇒ B ∈ F . {\displaystyle A\in {\mathcal {F}}{\mbox{ and }}A\subseteq B\Rightarrow B\in {\mathcal {F}}.\,}

If G is a subset of X then the family

⟨ G ⟩ = { A ⊆ X : G ⊆ A } {\displaystyle \langle G\rangle =\\{A\subseteq X:G\subseteq A\\}\,}

is a filter, the principal filter on G.

In a topological space ( X , T ) {\displaystyle (X,{\mathcal {T}})} , the neighbourhoods of a point x

N x = { N ⊆ X : ∃ U ∈ T , x ∈ u ⊆ N } {\displaystyle {\mathcal {N}}_{x}=\\{N\subseteq X:\exists U\in {\mathcal {T}},x\in u\subseteq N\\}\,}

form a filter, the neighbourhood filter of x.

### Filter bases[edit]

A base B {\displaystyle {\mathcal {B}}} for the filter  F {\displaystyle {\mathcal {F}}} is a non-empty collection of non-empty sets such that the family of subsets of X containing some element of  B {\displaystyle {\mathcal {B}}} is precisely the filter  F {\displaystyle {\mathcal {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 {\displaystyle {\mathcal {F}}} with the property that for any subset  A ⊆ X {\displaystyle A\subseteq X} either  A ∈ F {\displaystyle A\in {\mathcal {F}}} or the complement X ∖ A ∈ F {\displaystyle X\setminus A\in {\mathcal {F}}} .

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