View source for Filter (mathematics)

{{short description|In mathematics, a special subset of a partially ordered set}}
{{for|filters on sets|Filter (set theory)}}
{{other uses|Filter (disambiguation)}}
{{more footnotes|date=June 2017}}
[[File:Filter vs ultrafilter.svg|thumb|The power set lattice of the set {{Math|{{brace|1, 2, 3, 4}}}}, with [[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]], a '''filter''' or '''order filter''' is a special [[subset]] of a [[partially ordered set]] (poset), describing "large" or "eventual" elements. Filters appear in [[Order theory|order]] and [[lattice theory]], but also [[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|ultrafilters]], which are filters that cannot be enlarged, and describe nonconstructive techniques in [[mathematical logic]].  

[[Filter (set theory)|Filters on sets]] were introduced by [[Henri Cartan]] in 1937.  [[Nicolas Bourbaki]], in their book ''[[Topologie Générale]]'', popularized filters as an alternative to [[E. H. Moore]] and [[Herman L. Smith]]'s 1922 notion of a [[Net (topology)|net]]; order filters generalize this notion from the specific case of a [[power set]] under [[Inclusion (set theory)|inclusion]] to arbitrary [[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]] 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]]&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#Ordered Spaces|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:

; 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 {{Math|''F'' &NotEqual; ''P''}} as well, then {{Mvar|F}} is said to be a '''proper filter'''. Authors in [[set theory]] and [[mathematical logic]] often require all filters to be proper; this article will ''eschew'' that convention.{{sfn|Dugundji|1966|pp=211-213}}  An [[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]] generated by {{Mvar|S}} (i.e., the smallest upwards-closed 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]] 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 [[Meet (mathematics)|meet]]s: 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 subspace|vector subspaces]] of a given [[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)}}
{{Families of sets}}
Given a set&nbsp;{{Mvar|S}}, the [[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 [[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 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]].  

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 [[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;(''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|&clubs;(''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|&clubs;(''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|injective modules]] 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''}}.

== 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==
{{See also|Filter quantifier}}
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]].  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]]s in [[model theory]], a branch of [[mathematical logic]].

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

To understand the need for filters, begin with the equivalent concept of a [[Net (mathematics)|net]].   A [[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]]. In certain categories of topological spaces, such as [[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 [[neighbourhood base|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}} [[Convergent prefilter|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 [[neighbourhood base|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 of a filter|limit]] (point) of {{Mvar|B}}.  The prefilter {{Mvar|B}} is said to cluster at {{Mvar|x}} (or have {{Mvar|x}} as a [[Cluster point of a filter|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|Filtration (abstract algebra)}}
* {{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|author-link=Henri Cartan|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|author-link=Henri Cartan|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.|author-link1=George Mark Bergman|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=}} -->

{{Order theory}}

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