Filter (mathematics)

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:


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

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.

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 generated by a singleton set {x}, namely, all subsets of X containing x, is an ultrafilter.