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

1. ${\displaystyle X\in {ℱ};}$
2. ${\displaystyle \mathrm{\varnothing }\in ̸{ℱ};}$
3. ${\displaystyle A,B\in {ℱ}\Rightarrow A\cap B\in {ℱ};}$
4. ${\displaystyle A\in {ℱ}\text{ and }A\subseteq B\Rightarrow B\in {ℱ}.}$

If G is a subset of X then the family

${\displaystyle ⟨G⟩=\{A\subseteq X:G\subseteq A\}}$

is a filter, the principal filter on G.

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

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

form a filter, the neighbourhood filter of x.

Filter bases[edit]

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

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

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