Editing Filter (mathematics) (section)

== Examples ==
See the image at the top of this article for a simple example of filters on the finite poset&nbsp;{{Math|{{mathcal|P}}({1, 2, 3, 4})}}.

Partially order {{Math|{{mathbb|R}} &rarr; {{mathbb|R}}}}, the space of real-valued functions on {{Math|{{mathbb|R}}}}, by pointwise comparison.  Then the set of functions "large at infinity,"<math display="block">\left\{f:\lim_{x\to\pm\infty}{f(x)}=\infty\right\}\text{,}</math>is a filter on {{Math|{{mathbb|R}} &rarr; {{mathbb|R}}}}.  One can generalize this construction quite far by [[Compactification (mathematics)|compactifying]] the domain and [[Completion (order theory)|completing]] the codomain: if {{Mvar|X}} is a set with distinguished subset&nbsp;{{Mvar|S}} and {{Mvar|Y}} is a poset with distinguished element&nbsp;{{Mvar|m}}, then {{Math|{{brace|''f'' : ''f''&thinsp;{{pipe}}<sub>''S''</sub> &geq; ''m''}}}} is a filter in {{Math|''X'' &rarr; ''Y''}}.

The set {{Math|{{brace|{{brace|''k'' : ''k'' &geq; ''N''}} : ''N'' &isin; {{mathbb|N}}}}}} is a filter in {{Math|{{mathcal|P}}({{mathbb|N}})}}.  More generally, if {{Mvar|D}} is any [[directed set]], then<math display="block">\{\{k:k\geq N\}:N\in D\}</math>is a filter in {{Math|{{mathcal|P}}(''D'')}}, called the tail filter.  Likewise any [[Net (topology)|net]]&nbsp;{{Math|{{brace|''x''<sub>&alpha;</sub>}}<sub>&alpha;&isin;&Alpha;</sub>}} generates the eventuality filter {{Math|{{brace|{{brace|''x''<sub>&beta;</sub> : &alpha; &leq; &beta;}} : &alpha; &isin; &Alpha;}}}}.  A tail filter is the eventuality filter for {{Math|''x''<sub>&alpha;</sub> {{=}} &alpha;}}.

The [[Fréchet filter]] on an infinite set&nbsp;{{Mvar|X}} is<math display="block">\{A:X\setminus A\text{ finite}\}\text{.}</math>If {{Math|(''X'', &mu;)}} is a [[measure space]], then the collection {{Math|{{brace|''A'' : &mu;(''X'' &setminus; ''A'') {{=}} 0}}}} is a filter.   If {{Math|&mu;(''X'') {{=}} &infin;}}, then {{Math|{{brace|''A'' : &mu;(''X'' &setminus; ''A'') < &infin;}}}} is also a filter; the Fréchet filter is the case where {{Math|&mu;}} is [[counting measure]].

Given an ordinal&nbsp;{{Mvar|a}}, a subset of {{Mvar|a}} is called a [[Club set|club]] if it is closed in the [[order topology]] of {{Mvar|a}} but has net-theoretic limit {{Mvar|a}}.  The clubs of {{Mvar|a}} form a filter: the [[club filter]],&nbsp;{{Math|♣(''a'')}}.

The previous construction generalizes as follows: any club&nbsp;{{Mvar|C}} is also a collection of dense subsets (in the [[ordinal topology]]) of {{Mvar|a}}, and {{Math|♣(''a'')}} meets each element of {{Mvar|C}}.  Replacing {{Mvar|C}} with an arbitrary collection&nbsp;{{Mvar|C&#771;}} of [[Dense set (order)|dense sets]], there "typically" exists a filter meeting each element of {{Mvar|C&#771;}}, called a [[generic filter]].  For countable {{Mvar|C&#771;}}, the [[Rasiowa–Sikorski lemma]] implies that such a filter must exist; for "small" [[Uncountable set|uncountable]] {{Mvar|C&#771;}}, the existence of such a filter can be [[Forcing (mathematics)|forced]] through [[Martin's axiom]].

Let {{Math|''P''}} denote the set of [[Partial Order|partial orders]] of [[Universe (mathematics)|limited cardinality]], [[Modulo (mathematics)|modulo]] [[Isomorphism (algebra)|isomorphism]].  Partially order {{Mvar|P}} by: 
:{{Math|''A'' &leq; ''B''}} if there exists a strictly increasing {{Math|''f'' : ''A'' &rarr; ''B''}}.  
Then the subset of [[Atom (order theory)|non-atomic]] partial orders forms a filter.  Likewise, if {{Mvar|I}} is the set of [[injective module]]s over some given [[commutative ring]], of limited cardinality, modulo isomorphism, then a partial order on {{Mvar|I}} is: 
:{{Math|''A'' &leq; ''B''}} if there exists an [[injective function|injective]] [[module homomorphism|linear map]] {{Math|''f'' : ''A'' &rarr; ''B''}}.<ref>{{Cite journal |last=Bumby |first=R. T. |date=1965-12-01 |title=Modules which are isomorphic to submodules of each other |url=https://doi.org/10.1007/BF01220018 |journal=Archiv der Mathematik |language=en |volume=16 |issue=1 |pages=184–185 |doi=10.1007/BF01220018 |issn=1420-8938}}</ref>  
Given any infinite cardinal&nbsp;{{Math|&kappa;}}, the modules in {{Mvar|I}} that cannot be generated by fewer than {{Math|&kappa;}} elements form a filter.

Every [[uniform structure]] on a set&nbsp;{{Mvar|X}} is a filter on {{Math|''X'' &times; ''X''}}.