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 neighbourhoods in topology.

Formally, a filter on a set X is a subset $\mathcal{F}$ of the power set $\mathcal{P}X$ with the properties:

$X \in \mathcal{F} ; \,$
$\empty \not\in \mathcal{F} ; \,$
$A,B \in \mathcal{F} \Rightarrow A \cap B \in \mathcal{F} ; \,$
$A \in \mathcal{F} \mbox{ and } A \subseteq B \Rightarrow B \in \mathcal{F} . \,$

If G is a nonempty subset of X then the family

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

is a filter, the principal filter generated by 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.

[edit] 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 $\mathcal{F}$ .

[edit] 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 $X \setminus A \in \mathcal{F}$ .