Filter (mathematics)

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

1. ${\displaystyle X\in {\mathcal {F}};\,}$
2. ${\displaystyle \emptyset \not \in {\mathcal {F}};\,}$
3. ${\displaystyle A,B\in {\mathcal {F}}\Rightarrow A\cap B\in {\mathcal {F}};\,}$
4. ${\displaystyle A\in {\mathcal {F}}{\mbox{ and }}A\subseteq B\Rightarrow B\in {\mathcal {F}}.\,}$

If G is a subset of X then the family

${\displaystyle \langle G\rangle =\{A\subseteq X:G\subseteq A\}\,}$

is a filter, the principal filter on G.

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

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

form a filter, the neighbourhood filter of x.

Filter bases[edit]

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

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 ${\displaystyle {\mathcal {F}}}$ with the property that for any subset ${\displaystyle A\subseteq X}$ either ${\displaystyle A\in {\mathcal {F}}}$ or the complement ${\displaystyle X\setminus A\in {\mathcal {F}}}$.

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