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:

  1. X \in \mathcal{F} ; \,
  2. \empty \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 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}.

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

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