Filter (set theory)

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In mathematics, a filter on a set ${\displaystyle X}$ is a family ${\displaystyle {\mathcal {B}}}$ of subsets such that: ^[1]

1. ${\displaystyle X\in {\mathcal {B}}}$ and ${\displaystyle \emptyset \notin {\mathcal {B}}}$
2. if ${\displaystyle A\in {\mathcal {B}}}$ and ${\displaystyle B\in {\mathcal {B}}}$, then ${\displaystyle A\cap B\in {\mathcal {B}}}$
3. If ${\displaystyle A\subset B\subset X}$ and ${\displaystyle A\in {\mathcal {B}}}$, then ${\displaystyle B\in {\mathcal {B}}}$

A filter on a set may be thought of as representing a "collection of large subsets",^[2] one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.

Filters were introduced by Henri Cartan in 1937^[3][4] and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.