Filter (mathematics)

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

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

If G is a subset of X then the family

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

is a filter, the principal filter on G.

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

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

form a filter, the neighbourhood filter of x.

Filter bases

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

Ultrafilters

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

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