View source for Filter (mathematics)

{{short description|In mathematics, a special subset of a partially ordered set}}
[[File:Filter vs ultrafilter.svg|thumb|The power set lattice of the set {{Math|{{brace|1, 2, 3, 4}}}}, with [[Upper set|upper set]] {{Math|&uparrow;{{brace|1, 4}}}} colored dark green. This upper set is a {{em|filter}}, and even a {{em|principal filter}}. It is not an {{em|ultrafilter}}, because including also the light green elements extends it to the larger nontrivial filter {{Math|&uparrow;{{brace|1}}}}. Since the latter cannot be extended further, {{Math|&uparrow;{{brace|1}}}} is an ultrafilter.]] 
In [[Mathematics|mathematics]], a '''filter''' or '''order filter''' is a special [[Subset|subset]] of a [[Partially ordered set|partially ordered set]] (poset), describing "large" or "eventual" elements. Filters appear in [[Order theory|order]] and lattice theory, but also [[Topology|topology]], whence they originate. The notion [[Duality (order theory)|dual]] to a filter is an [[Ideal (order theory)|order ideal]].

Special cases of filters include [[Ultrafilter|ultrafilter]]s, which are filters that cannot be enlarged, and describe nonconstructive techniques in [[Mathematical logic|mathematical logic]].

[[Filter (set theory)|Filters on sets]] were introduced by [[Biography:Henri Cartan|Henri Cartan]] in 1937.  Nicolas Bourbaki, in their book ''Topologie Générale'', popularized filters as an alternative to [[Biography:E. H. Moore|E. H. Moore]] and [[Biography:Herman L. Smith|Herman L. Smith]]'s 1922 notion of a net; order filters generalize this notion from the specific case of a [[Power set|power set]] under inclusion to arbitrary [[Partially ordered set|partially ordered set]]s.   Nevertheless, the [[Filter (set theory)|theory of power-set filters]] retains interest in its own right, in part for substantial [[Filters in topology|applications in topology]].

==Motivation==

Fix a [[Partially ordered set|partially ordered set (poset)]]&nbsp;{{Mvar|P}}.  Intuitively, a filter&nbsp;{{Mvar|F}}  is a subset of {{Mvar|P}} whose members are elements large enough to satisfy some criterion.{{sfn|Koutras|Moyzes|Nomikos|Tsaprounis|2021|p=}} For instance, if {{Math|''x'' &isin; ''P''}}, then the set of elements above {{Mvar|x}} is a filter, called the principal filter at {{Mvar|x}}. (If {{Mvar|x}} and {{Mvar|y}} are [[Comparability|incomparable]] elements of {{Mvar|P}}, then neither the principal filter at {{Mvar|x}} nor {{Mvar|y}} is contained in the other.)

Similarly, a filter on a set&nbsp;{{Mvar|S}} contains those subsets that are sufficiently large to contain some given {{em|thing}}. For example, if {{Mvar|S}} is the [[Real line|real line]] and {{Math|''x'' &isin; ''S''}}, then the family of sets including {{Mvar|x}} in their [[Interior (topology)|interior]] is a filter, called the neighborhood filter at {{Mvar|x}}. The {{em|thing}} in this case is slightly larger than {{Mvar|x}}, but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the [[Filter (mathematics)#Definition|definition below]]: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider {{Mvar|F}} as a "locating scheme" to find {{Mvar|x}}.  In this interpretation, one searches in some space&nbsp;{{Mvar|X}}, and expects {{Mvar|F}} to describe those subsets of {{Mvar|X}} that contain the goal. The goal must be located somewhere; thus the [[Empty set|empty set]]&nbsp;{{Math|&emptyset;}} can never be in {{Mvar|F}}.  And if two subsets both contain the goal, then should "zoom in" to their common region.

An ultrafilter describes a "perfect locating scheme" where each scheme component gives new information (either "look here" or "look elsewhere").  Compactness is the property that "every search is fruitful," or, to put it another way, "every locating scheme ends in a search result."

A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space.<ref>{{cite arXiv|last1=Igarashi|first1=Ayumi|last2=Zwicker|first2=William S.|date=16 February 2021|title=Fair division of graphs and of tangled cakes|class=math.CO|eprint=2102.08560}}</ref>  This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.

==Definition==
{{anchor|Filter on a partially ordered set|directed downward|order filter}}
A subset&nbsp;{{Mvar|F}} of a partially ordered set&nbsp;{{Math|(''P'', &leq;)}} is a '''filter''' or '''dual ideal''' if the following are satisfied:

; Nontriviality: The set {{Mvar|F}} is [[Empty set|non-empty]].
; [[Directed set|Downward directed]]: For every {{Math|''x'', ''y'' &isin; ''F''}}, there is some {{Math|''z'' &isin; ''F''}} such that {{Math|''z'' &leq; ''x''}} and {{Math|''z'' &leq; ''y''}}.  
; [[Upper set|Upward closure]]: For every {{Math|''x'' &isin; ''F''}} and {{Math|''p'' &isin; ''P''}}, the condition {{Math|''x'' &leq; ''p''}} implies {{Math|''p'' &isin; ''F''}}.  
If, additionally, {{Math|''F'' &NotEqual; ''P''}}, then {{Mvar|F}} is said to be a '''proper filter'''. Authors in [[Set theory|set theory]] and [[Mathematical logic|mathematical logic]] often require all filters to be proper; this article will ''eschew'' that convention.{{sfn|Dugundji|1966|pp=211-213}}  An [[Ultrafilter|ultrafilter]] is a filter contained in no other proper filter.

=== Filter bases ===
{{anchor|Filter base|Prefilter|Filter subbase}}
A subset&nbsp;{{Mvar|S}} of {{Mvar|F}} is a '''base''' or '''basis''' for {{Mvar|F}} if the [[Upper set|upper set]] generated by {{Mvar|S}} (i.e., the smallest upwards-closed set containing {{Mvar|S}}) is all of {{Mvar|F}}.  Every filter is a base for itself.

Moreover, if {{Math|''B'' &subseteq; ''P''}} is nonempty and downward directed, then {{Mvar|B}} generates an upper set&nbsp;{{Mvar|F}} that is a filter (for which {{Mvar|B}} is a base).  Such sets are called '''prefilters''', as well as the aforementioned '''filter base/basis''', and {{Mvar|F}} is said to be '''generated''' or '''spanned''' by {{Mvar|B}}.  A prefilter is proper if and only if it generates a proper filter.

Given {{Math|''p'' &isin; ''P''}}, the set {{Math|{{brace|''x'' : ''p'' &leq; ''x''}}}} is the smallest filter containing {{Math|''p''}}, and sometimes written {{Math|&uparrow; ''p''}}.  Such a filter is called a '''principal filter'''; {{Math|''p''}} is said to be the '''principal element''' of {{Mvar|F}}, or generate {{Mvar|F}}.

==== Refinement ====
Suppose {{Mvar|B}} and {{Mvar|C}} are two prefilters on {{Mvar|P}}, and, for each {{Mvar|''c'' &isin; ''C''}}, there is a {{Math|''b'' &isin; ''B''}}, such that {{Math|''b'' &leq; ''c''}}.  Then we say that {{Mvar|B}} is '''{{visible anchor|finer}}''' than (or '''refines''') {{Mvar|C}}; likewise, {{Mvar|C}} is '''coarser''' than (or '''coarsens''') {{Mvar|B}}.  Refinement is a [[Preorder|preorder]] on the set of prefilters.  In fact, if {{Mvar|C}} also refines {{Mvar|B}}, then {{Mvar|B}} and {{Mvar|C}} are called '''equivalent''', for they generate the same filter.  Thus passage from prefilter to filter is an instance of passing from a preordering to associated partial ordering.

== Special cases ==
Historically, filters generalized to [[Lattice (order)|order-theoretic lattice]]s before arbitrary partial orders.  In the case of lattices, downward direction can be written as closure under finite meets: for all {{Math|''x'', ''y'' &isin; ''F''}}, one has {{Math|''x'' &and; ''y'' &isin; ''F''}}.<ref>{{cite book |last1=Davey |first1=B. A. |title=Introduction to Lattices and Order |title-link=Introduction to Lattices and Order |last2=Priestley |first2=H. A. |publisher=Cambridge University Press |year=1990 |series=Cambridge Mathematical Textbooks |page=184}}</ref>

=== Linear filters ===
A linear (ultra)filter is an (ultra)filter on the [[Lattice (order)|lattice]] of vector subspaces of a given [[Vector space|vector space]], ordered by inclusion.  Explicitly, a linear filter on a vector space&nbsp;{{Mvar|X}} is a family&nbsp;{{Math|{{mathcal|B}}}} of vector subspaces of {{Mvar|X}} such that if {{Math|''A'', ''B'' &isin; {{mathcal|B}}}} and {{Mvar|C}} is a vector subspace of {{Mvar|X}} that contains {{Mvar|A}}, then {{Math|''A'' &cap; ''B'' &isin; {{mathcal|B}}}} and {{Math|''C'' &isin; {{mathcal|B}}}}.{{sfn|Bergman|Hrushovski|1998|p=}}

A linear filter is proper if it does not contain {{Math|{{brace|0}}}}.{{sfn|Bergman|Hrushovski|1998|p=}}

=== Filters on a set; subbases ===
{{Main|Filter (set theory)}}
Given a set&nbsp;{{Mvar|S}}, the [[Power set|power set]]&nbsp;{{Math|{{mathcal|P}}(''S'')}} is [[Partially ordered set|partially ordered]] by set inclusion; filters on this poset are often just called "filters on {{Mvar|S}}," in an abuse of terminology.  For such posets, downward direction and upward closure reduce to:{{sfn|Dugundji|1966|pp=211-213}}
; Closure under finite intersections: If {{Math|''A'', ''B'' &isin; ''F''}}, then so too is {{Math|''A'' &cap; ''B'' &isin; ''F''}}.  
; Isotony:{{sfn|Dolecki|Mynard| 2016|pp=27-29}} If {{Math|''A'' &isin; ''F''}} and {{Math|''A'' &subseteq; ''B'' &subseteq; ''S''}}, then {{Math|''B'' &isin; ''F''}}.

A '''proper<ref>{{cite book |last1=Goldblatt |first1=R |url=https://archive.org/stream/springer_10.1007-978-1-4612-0615-6/10.1007-978-1-4612-0615-6#page/n31/mode/2up/search/proper+filter |title=Lectures on the Hyperreals: an Introduction to Nonstandard Analysis |page=32}}</ref>/non-degenerate{{sfn|Narici|Beckenstein|2011|pp=2-7}}''' filter is one that does not contain {{Math|&emptyset;}}, and these three conditions (including non-degeneracy) are [[Biography:Henri Cartan|Henri Cartan]]'s original definition of a filter.{{sfn|Cartan|1937a|p=}}{{sfn|Cartan|1937b|p=}}   It is common&nbsp;— ''though not universal''&nbsp;— to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.

Prefilters on a set are proper if and only if they do not contain {{Math|&emptyset;}} either.

For every subset&nbsp;{{Mvar|T}} of {{Math|{{mathcal|P}}(''S'')}}, there is a smallest filter&nbsp;{{Mvar|F}} containing {{Mvar|T}}.  As with prefilters, {{Mvar|T}} is said to generate or span {{Mvar|F}}; a base for {{Mvar|F}} is the set&nbsp;{{Mvar|U}} of all finite intersections of {{Mvar|T}}.  The set {{Mvar|T}} is said to be a '''filter subbase''' when {{Mvar|F}} (and thus {{Mvar|U}}) is proper.

Proper filters on sets have the [[Finite intersection property|finite intersection property]].

If {{Math|''S'' {{=}} &emptyset;}}, then {{Mvar|S}} admits only the improper filter {{Math|{{brace|&emptyset;}}}}.

==== Free filters ====
A filter is said to be a '''free''' if the intersection of its members is empty.  A proper principal filter is not free.

Since the intersection of any finite number of members of a filter is also a member, no proper filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members.  But a nonprincipal filter on an infinite set is not necessarily free: a filter is free if and only if it includes the [[Fréchet filter]] (see {{Slink||Examples}}).

== 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 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|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&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|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|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|order topology]] of {{Mvar|a}} but has net-theoretic limit {{Mvar|a}}.  The clubs of {{Mvar|a}} form a filter: the [[Club filter|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 sets, there "typically" exists a filter meeting each element of {{Mvar|C&#771;}}, called a [[Generic filter|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 orders of [[Universe (mathematics)|limited cardinality]], [[Modulo (mathematics)|modulo]] 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|injective module]]s over some given [[Commutative ring|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''}}.

== Relationship to ideals ==
{{Main|Ideal (order theory)}}
The [[Duality (mathematics)|dual notion]] to a filter&nbsp;— that is, the concept obtained by reversing all {{Math|&leq;}} and exchanging {{Math|&and;}} with {{Math|&or;}}&nbsp;— is an order ideal.  Because of this duality, any question of filters can be mechanically translated to a question about ideals and vice-versa; in particular, a '''prime''' or '''maximal''' filter is a filter whose corresponding ideal is (respectively) prime or maximal.

A filter is an ultrafilter if and only if the corresponding ideal is minimal.

==In model theory==
For every filter&nbsp;{{Mvar|F}} on a set&nbsp;{{Mvar|S}}, the set function defined by<math display=block>m(A) = \begin{cases}
1 & \text{if }A \in F \\
0 & \text{if }S \smallsetminus A \in F \\
\text{is undefined} & \text{otherwise}
\end{cases}</math>is finitely additive&nbsp;— a "[[Measure (mathematics)|measure]]," if that term is construed rather loosely.  Moreover, the measures so constructed are defined everywhere if {{mvar|F}} is an [[Ultrafilter|ultrafilter]].  Therefore, the statement<math display="block">\left\{\,x \in S : \varphi(x)\,\right\} \in F</math>can be considered somewhat analogous to the statement that {{math|&phi;}} holds "almost everywhere."  That interpretation of membership in a filter is used (for motivation, not actual {{em|proofs}}) in the theory of [[Ultraproduct|ultraproduct]]s in [[Model theory|model theory]], a branch of [[Mathematical logic|mathematical logic]].

==In topology==
{{Main|Filters in topology}}
In [[General topology|general topology]] and analysis, filters are used to define convergence in a manner similar to the role of [[Sequence|sequence]]s in a [[Metric space|metric space]].  They unify the concept of a [[Limit (mathematics)|limit]] across the wide variety of arbitrary [[Topological space|topological space]]s.

To understand the need for filters, begin with the equivalent concept of a [[Net (mathematics)|net]].   A [[Sequence|sequence]] is usually indexed by the natural numbers&nbsp;{{Math|{{mathbb|N}}}}, which are a totally ordered set. Nets generalize the notion of a sequence by replacing {{Math|{{mathbb|N}}}} with an arbitrary [[Directed set|directed set]]. In certain categories of topological spaces, such as [[First-countable space|first-countable space]]s, sequences characterize most topological properties, but this is not true in general.  However, nets&nbsp;— as well as filters&nbsp;— always do characterize those topological properties.

Filters do not involve any set external to the topological space&nbsp;{{Mvar|X}}, whereas sequences and nets rely on other directed sets.  For this reason, the collection of all filters on {{Mvar|X}} is always a [[Set (mathematics)|set]], whereas the collection of all {{Mvar|X}}-valued nets is a proper class.

=== Neighborhood bases ===
Any point&nbsp;{{Mvar|x}} in the topological space&nbsp;{{Mvar|X}} defines a [[Neighbourhood system|neighborhood filter or system]]&nbsp;{{Math|{{mathcal|N}}<sub>''x''</sub>}}: namely, the family of all sets containing {{Mvar|x}} in their [[Interior (topology)|interior]].  A set&nbsp;{{Math|{{mathcal|N}}}} of neighborhoods of {{Mvar|x}} is a neighborhood base at {{Mvar|x}} if {{Math|{{mathcal|N}}}} generates {{Math|{{mathcal|N}}<sub>''x''</sub>}}.  Equivalently, {{Math|''S'' &subseteq; ''X''}} is a neighborhood of {{Mvar|x}} if and only if there exists {{Math|''N'' &isin; {{mathcal|N}}}} such that {{Math|''N'' &subseteq; ''S''}}.

==== Convergent filters and cluster points ====
A prefilter&nbsp;{{Mvar|B}} converges to a point&nbsp;{{Mvar|x}}, written {{Math|''B'' &rarr; ''x''}}, if and only if {{Mvar|B}} generates a filter&nbsp;{{Mvar|F}} that contains the neighborhood filter {{Math|{{mathcal|N}}<sub>''x''</sub>}}&nbsp;— explicitly, for every neighborhood&nbsp;{{Mvar|U}} of {{Mvar|x}}, there is some {{Math|''V'' &isin; ''B''}} such that {{Math|''V'' &subseteq; ''U''}}.  Less explicitly, {{Math|''B'' &rarr; ''x''}} if and only if {{Mvar|B}} refines {{Math|{{mathcal|N}}<sub>''x''</sub>}}, and any neighborhood base at {{Mvar|x}} can replace {{Math|{{mathcal|N}}<sub>''x''</sub>}} in this condition.  Clearly, every neighborhood base at {{Mvar|x}} converges to {{Mvar|x}}.

A filter&nbsp;{{Mvar|F}} (which generates itself) converges to {{Mvar|x}} if {{Math|{{mathcal|N}}<sub>''x''</sub> &subseteq; ''F''}}.  The above can also be reversed to characterize the neighborhood filter {{Math|{{mathcal|N}}<sub>''x''</sub>}}: {{Math|{{mathcal|N}}<sub>''x''</sub>}} is the finest filter coarser than each filter converging to {{Mvar|x}}.

If {{Math|''B'' &rarr; ''x''}}, then {{Mvar|x}} is called a limit (point) of {{Mvar|B}}.  The prefilter {{Mvar|B}} is said to cluster at {{Mvar|x}} (or have {{Mvar|x}} as a cluster point) if and only if each element of {{Mvar|B}} has non-empty intersection with each neighborhood of {{Mvar|x}}.  Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an {{em|ultra}}filter is a limit point.

== See also ==
* {{annotated link|Filtration (mathematics)}}
* {{annotated link|Filtration (probability theory)}}
* {{annotated link|Generic filter}}
* {{annotated link|Ideal (set theory)}}

==Notes==

{{reflist|group=note}}
{{reflist}}

==References==

* Nicolas Bourbaki, <cite>General Topology</cite> (<cite>Topologie Générale</cite>), {{ISBN|0-387-19374-X}} (Ch. 1–4): Provides a good reference for filters in general topology (Chapter I) and for Cauchy filters in uniform spaces (Chapter II)
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!--{{sfn|Bourbaki|1987|p=}}-->
* {{cite book|last1=Burris|first1=Stanley|last2=Sankappanavar|first2=Hanamantagouda P.|year=2012|title=A Course in Universal Algebra|publisher=Springer-Verlag|isbn=978-0-9880552-0-9|url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html|archive-url=https://web.archive.org/web/20220401154440/https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf|archive-date=1 April 2022}}
* {{cite journal|last=Cartan|first=Henri|title=Théorie des filtres|title-link=|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=205|year=1937a|pages=595–598|url=http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f594.image}} <!--{{sfn|Cartan|1937a|p=}}-->
* {{cite journal|last=Cartan|first=Henri|title=Filtres et ultrafiltres|title-link=|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=205|year=1937b|pages=777–779|url=http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image}} <!--{{sfn|Cartan|1937b|p=}}-->
* {{Dolecki Mynard Convergence Foundations Of Topology}}
* {{Dugundji Topology}} <!-- {{sfn|Dugundji|1966|p=}} -->
* {{cite journal|last1=Koutras|first1=Costas D.|last2=Moyzes|first2=Christos|last3=Nomikos|first3=Christos|last4=Tsaprounis|first4=Konstantinos|last5=Zikos|first5=Yorgos|title=On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation|journal=Logic Journal of the IGPL|date=20 October 2021|doi=10.1093/jigpal/jzab030}} <!-- {{sfn|Koutras|Moyzes|Nomikos|2021|p=}} -->
* {{cite web|last=MacIver R.|first=David|title=Filters in Analysis and Topology|date=1 July 2004|url=http://www.efnet-math.org/~david/mathematics/filters.pdf |archive-url=https://web.archive.org/web/20071009170540/http://www.efnet-math.org/~david/mathematics/filters.pdf |archive-date=2007-10-09 }} (Provides an introductory review of filters in topology and in metric spaces.)
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Koutras|Moyzes|Nomikos|Tsaprounis|2021|p=}} -->
* {{Willard General Topology}} <!-- {{sfn|Willard|2004|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->

==Further reading==

* {{cite journal|last1=Bergman|first1=George M.|last2=Hrushovski|first2=Ehud|author-link2=Ehud Hrushovski|title=Linear ultrafilters|journal=Communications in Algebra|volume=26|issue=12|year=1998|pages=4079–4113|doi=10.1080/00927879808826396 |citeseerx=10.1.1.54.9927 }} <!-- {{sfn|Bergman|Hrushovski|1998|p=}} -->


[[Category:General topology]]
[[Category:Order theory]]

{{Sourceattribution|Filter (mathematics)}}