Filter on a set

{{short description|Family of subsets representing "large" sets}} {{for|filters on a poset|Filter (mathematics)}} {{other uses|Filter (disambiguation)}} In mathematics, a '''filter''' on a set is a family of subsets which is closed under supersets and finite intersections. The concept originates in topology, where the neighborhoods of a point form a filter on the space. Filters were introduced by Henri Cartan in 1937{{sfn|Cartan|1937a}}{{sfn|Cartan|1937b}} 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. They have also found applications in model theory and set theory.

Filters on a set were later generalized to order filters. Specifically, a filter on a set <math>X</math> is an order filter on the power set of <math>X</math> ordered by inclusion.

The notion dual to a filter is an ideal. Ultrafilters are a particularly important subclass of filters.

==Definition==

Given a set <math>X</math>, a '''filter''' <math>\mathcal{F}</math> on <math>X</math> is a set of subsets of <math>X</math> such that:{{sfn|Császár|1978|p=56}}{{sfn|Schechter|1996|p=100}}{{sfn|Willard|2004|p=78}}

* <math>\mathcal{F}</math> is upwards-closed: If <math>A, B \subseteq X</math> are such that <math>A \in \mathcal{F}</math> and <math>A \subseteq B</math> then <math>B \in \mathcal{F}</math>, * <math>\mathcal{F}</math> is closed under finite intersections: <math>X \in \mathcal{F}</math>,{{efn|The intersection of zero subsets of <math>X</math> is <math>X</math> itself.}}, and if <math>A \in \mathcal{F}</math> and <math>B \in \mathcal{F}</math> then <math>A \cap B \in \mathcal{F}</math>.

A ''{{visible anchor|proper}}'' (or ''non-degenerate'') filter is a filter which is proper as a subset of the powerset <math>\mathcal{P}(X)</math> (i.e., the only improper filter is <math>\mathcal{P}(X)</math>, consisting of all possible subsets). By upwards-closure, a filter is proper if and only if it does not contain the empty set.{{sfn|Schechter|1996|p=100}} Many authors adopt the convention that a filter must be proper by definition.{{sfn|Dolecki|Mynard|2016|p=29}}{{sfn|Joshi|1983|p=241}}{{sfn|Köthe|1983|p=11}}{{sfn|Schubert|1968|p=48}}

When <math>\mathcal{F}</math> and <math>\mathcal{G}</math> are two filters on the same set such that <math>\mathcal{F} \subseteq \mathcal{G}</math> holds, <math>\mathcal{F}</math> is said to be ''coarser''{{sfn|Schubert|1968|p=49}} than <math>\mathcal{G}</math> (or a ''subfilter'' of <math>\mathcal{G}</math>) while <math>\mathcal{G}</math> is said to be ''finer''{{sfn|Schubert|1968|p=49}} than <math>\mathcal{F}</math> (or ''{{visible anchor|subordinate}}'' to <math>\mathcal{F}</math> or a ''superfilter''{{sfn|Schechter|1996|p=102}} of <math>\mathcal{F}</math>).

==Examples==

* The singleton set <math>\mathcal{F} = \{X\}</math> is called the ''trivial'' or ''indiscrete'' filter on <math>X</math>.{{sfn|Bourbaki|1987|pp=57–68}} * If <math>Y</math> is a subset of <math>X</math>, the subsets of <math>X</math> which are supersets of <math>Y</math> form a ''principal filter''.{{sfn|Császár|1978|p=56}} * If <math>X</math> is a topological space and <math>x \in X</math>, then the set of neighborhoods of <math>x</math> is a filter on <math>X</math>, the ''neighborhood filter''{{sfn|Joshi|1983|p=242}} or ''vicinity filter''{{sfn|Dolecki|Mynard|2016|p=30}} of <math>x</math>. * Many examples arise from various "largeness" conditions: ** If <math>X</math> is a set, the set of all cofinite subsets of <math>X</math> (i.e., those sets whose complement in <math>X</math> is finite) is a filter on <math>X</math>, the Fréchet filter{{sfn|Bourbaki|1987|pp=57–68}}{{sfn|Schechter|1996|p=103}}{{sfn|Willard|2004|p=78}} (or ''cofinite filter''{{sfn|Joshi|1983|p=242}}). ** Similarly, if <math>X</math> is a set, the cocountable subsets of <math>X</math> (those whose complement is countable) form a filter, the ''cocountable filter''{{sfn|Dolecki|Mynard|2016|p=30}} which is finer than the Fréchet filter. More generally, for any cardinal <math>\kappa</math>, the subsets whose complement has cardinal at most <math>\kappa</math> form a filter. ** If <math>X</math> is a metric space, e.g., <math>\R^n</math>, the co-bounded subsets of <math>X</math> (those whose complement is bounded set) form a filter on <math>X</math>.{{sfn|Schechter|1996|p=104}} ** If <math>X</math> is a complete measure space (e.g., <math>\R^n</math> with the Lebesgue measure), the conull subsets of <math>X</math>, i.e., the subsets whose complement has measure zero, form a filter on <math>X</math>. (For a non-complete measure space, one can take the subsets which, while not necessarily measurable, are contained in a measurable subset of measure zero.) ** Similarly, if <math>X</math> is a measure space, the subsets whose complement is contained in a measurable subset of finite measure form a filter on <math>X</math>. ** If <math>X</math> is a topological space, the comeager subsets of <math>X</math>, i.e., those whose complement is meager, form a filter on <math>X</math>. ** The subsets of <math>\mathbb{N}</math> which have a natural density of 1 form a filter on <math>\mathbb{N}</math>.<ref>{{cite book|last=Jech|first=Thomas|author-link=Thomas Jech|title=Set Theory: The Third Millennium Edition, Revised and Expanded|publisher=Springer Science & Business Media|publication-place=Berlin New York|year=2006|isbn=978-3-540-44085-7|oclc=50422939|page=74}}</ref> * The club filter of a regular uncountable cardinal <math>\kappa</math> is the filter of all sets containing a club subset of <math>\kappa</math>. * If <math>(\mathcal{F}_i)_{i \in I}</math> is a family of filters on <math>X</math> and <math>\mathcal{J}</math> is a filter on <math>I</math> then <math>\bigcup_{A \in \mathcal{J}} \bigcap_{i \in A} \mathcal{F}_i</math> is a filter on <math>X</math> called ''Kowalsky's filter''.{{sfn|Schechter|1996|pp=100–130}}

==Principal and free filters==

The ''kernel'' of a filter <math>\mathcal{F}</math> on <math>X</math> is the intersection of all the subsets of <math>X</math> in <math>\mathcal{F}</math>.

A filter <math>\mathcal{F}</math> on <math>X</math> is ''principal''{{sfn|Császár|1978|p=56}} (or ''atomic''{{sfn|Joshi|1983|p=242}}) when it has a particularly simple form: it contains exactly the supersets of <math>Y</math>, for some fixed subset <math>Y \subseteq X</math>. When <math>Y = \varnothing</math>, this yields the improper filter. When <math>Y = \{y\}</math> is a singleton, this filter (which consists of all subsets that contain <math>y</math>) is called the ''fundamental filter''{{sfn|Császár|1978|p=56}} (or ''discrete filter''{{sfn|Wilansky|2013|p=44}}) associated with <math>y</math>.

A filter <math>\mathcal{F}</math> is principal if and only if the kernel of <math>\mathcal{F}</math> is an element of <math>\mathcal{F}</math>, and when this is the case, <math>\mathcal{F}</math> consists of the supersets of its kernel.{{sfn|Dolecki|Mynard|2016|p=33}} On a finite set, every filter is principal (since the intersection defining the kernel is finite).

A filter is said to be ''free'' when it has empty kernel, otherwise it is ''fixed'' (and if <math>x</math> is an element of the kernel, it is ''fixed by <math>x</math>'').{{sfn|Schechter|1996|p=16}} A filter on a set <math>X</math> is free if and only if it contains the Fréchet filter on <math>X</math>.{{sfn|Dolecki|Mynard|2016|p=34}}

Two filters <math>\mathcal{F}_1</math> and <math>\mathcal{F}_2</math> on <math>X</math> ''mesh'' when every member of <math>\mathcal{F}_1</math> intersects every member of <math>\mathcal{F}_2</math>.{{sfn|Dolecki|Mynard|2016|p=31}} For every filter <math>\mathcal{F}</math> on <math>X</math>, there exists a unique pair of filters <math>\mathcal{F}_f</math> (the ''free part'' of <math>\mathcal{F}</math>) and <math>\mathcal{F}_p</math> (the ''principal part'' of <math>\mathcal{F}</math>) on <math>X</math> such that <math>\mathcal{F}_f</math> is free, <math>\mathcal{F}_p</math> is principal, <math>\mathcal{F}_f \cap \mathcal{F}_p = \mathcal{F}</math>, and <math>\mathcal{F}_p</math> does not mesh with <math>\mathcal{F}_f</math>. The principal part <math>\mathcal{F}_p</math> is the principal filter generated by the kernel of <math>\mathcal{F}</math>, and the free part <math>\mathcal{F}_f</math> consists of elements of <math>\mathcal{F}</math> with any number of elements from the kernel possibly removed.{{sfn|Dolecki|Mynard|2016|p=34}}

A filter <math>\mathcal{F}</math> is ''countably deep'' if the kernel of any countable subset of <math>\mathcal{F}</math> belongs to <math>\mathcal{F}</math>.{{sfn|Dolecki|Mynard|2016|p=30}}

==Correspondence with order filters==

The concept of a filter on a set is a special case of the more general concept of a filter on a partially ordered set. By definition, a filter on a partially ordered set <math>P</math> is a subset <math>\mathcal{F}</math> of <math>P</math> which is upwards-closed (if <math>x \in \mathcal{F}</math> and <math>x \leq y</math> then <math>y \in \mathcal{F}</math>) and downwards-directed (every finite subset of <math>\mathcal{F}</math> has a lower bound in <math>\mathcal{F}</math>). A filter on a set <math>X</math> is the same as a filter on the powerset <math>\mathcal{P}(X)</math> ordered by inclusion.{{efn|It is immediate that a filter on <math>X</math> is an order filter on <math>\mathcal{P}(X)</math>. For the converse, let <math>\mathcal{F}</math> be an order filter on <math>\mathcal{P}(X)</math>. It is upwards-closed by definition. We check closure under finite intersections. If <math>A_0, \dots, A_{n-1}</math> is a finite family of subsets from <math>\mathcal{F}</math>, it has a lower bound in <math>\mathcal{F}</math> by downwards-closure, which is some <math>B \in \mathcal{F}</math> such that <math>B \subseteq A_0, \dots, B \subseteq A_{n-1}</math>. Then <math>B \subseteq A_0 \cap \dots \cap A_{n-1}</math>, hence <math>A_0 \cap \dots \cap A_{n-1} \in \mathcal{F}</math> by upwards-closure.}}

==Constructions of filters==

===Intersection of filters===

If <math>(\mathcal{F}_i)_{i \in I}</math> is a family of filters on <math>X</math>, its intersection <math>\bigcap_{i \in I} \mathcal{F}_i</math> is a filter on <math>X</math>. The intersection is a greatest lower bound operation in the set of filters on <math>X</math> partially ordered by inclusion, which endows the filters on <math>X</math> with a complete lattice structure.{{sfn|Dolecki|Mynard|2016|p=30}}{{sfn|Schubert|1968|p=50}}

The intersection <math>\bigcap_{i \in I} \mathcal{F}_i</math> consists of the subsets which can be written as <math>\bigcup_{i \in I} A_i</math> where <math>A_i \in \mathcal{F}_i</math> for each <math>i \in I</math>.

===Filter generated by a family of subsets===

Given a family of subsets <math>\mathcal{S} \subseteq \mathcal{P}(X)</math>, there exists a minimum filter on <math>X</math> (in the sense of inclusion) which contains <math>\mathcal{S}</math>. It can be constructed as the intersection (greatest lower bound) of all filters on <math>X</math> containing <math>\mathcal{S}</math>. This filter <math>\langle \mathcal{S} \rangle</math> is called the filter generated by <math>\mathcal{S}</math>, and <math>\mathcal{S}</math> is said to be a ''filter subbase'' of <math>\langle \mathcal{S} \rangle</math>. {{sfn|Császár|1978|p=57}}

The generated filter can also be described more explicitly: <math>\langle \mathcal{S} \rangle</math> is obtained by closing <math>\mathcal{S}</math> under finite intersections, then upwards, i.e., <math>\langle \mathcal{S} \rangle</math> consists of the subsets <math>Y \subseteq X</math> such that <math>A_0 \cap \dots \cap A_{n-1} \subseteq Y</math> for some <math>A_0, \dots, A_{n-1} \in \mathcal{B}</math>.{{sfn|Schechter|1996|p=102}}

Since these operations preserve the kernel, it follows that <math>\langle \mathcal{S} \rangle</math> is a proper filter if and only if <math>\mathcal{S}</math> has the finite intersection property: the intersection of a finite subfamily of <math>\mathcal{S}</math> is non-empty.{{sfn|Schechter|1996|p=104}}

In the complete lattice of filters on <math>X</math> ordered by inclusion, the least upper bound of a family of filters <math>(\mathcal{F}_i)_{i \in I}</math> is the filter generated by <math>\bigcup_{i \in I} \mathcal{F}_i</math>.{{sfn|Dolecki|Mynard|2016|p=33}}

Two filters <math>\mathcal{F}_1</math> and <math>\mathcal{F}_2</math> on <math>X</math> mesh if and only if <math>\langle \mathcal{F}_1 \cup \mathcal{F}_2 \rangle</math> is proper.{{sfn|Dolecki|Mynard|2016|p=31}}

===Filter bases===

Let <math>\mathcal{F}</math> be a filter on <math>X</math>. A ''filter base'' of <math>\mathcal{F}</math> is a family of subsets <math>\mathcal{B} \subseteq \mathcal{P}(X)</math> such that <math>\mathcal{F}</math> is the upwards closure of <math>\mathcal{B}</math>, i.e., <math>\mathcal{F}</math> consists of those subsets <math>Y \subseteq X</math> for which <math>A \subseteq Y</math> for some <math>A \in \mathcal{B}</math>.{{sfn|Dolecki|Mynard|2016|p=29}}

This upwards closure is a filter if and only if <math>\mathcal{B}</math> is downwards-directed, i.e., <math>\mathcal{B}</math> is non-empty and for all <math>A, B \in \mathcal{B}</math> there exists <math>C \in \mathcal{B}</math> such that <math>C \subseteq A \cap B</math>.{{sfn|Dolecki|Mynard|2016|p=29}}{{sfn|Joshi|1983|p=242}} When this is the case, <math>\mathcal{B}</math> is also called a ''prefilter'', and the upwards closure is also equal to the generated filter <math>\langle \mathcal{B} \rangle</math>.{{sfn|Schechter|1996|p=104}} Hence, being a filter base of <math>\mathcal{F}</math> is a stronger property than being a filter subbase of <math>\mathcal{F}</math>.

====Examples====

* When <math>X</math> is a topological space and <math>x \in X</math>, a filter base of the neighborhood filter of <math>x</math> is known as a neighborhood base for <math>x</math>, and similarly, a filter subbase of the neighborhood filter of <math>x</math> is known as a neighborhood subbase for <math>x</math>. The open neighborhoods of <math>x</math> always form a neighborhood base for <math>x</math>, by definition of the neighborhood filter. In <math>X = \R^n</math>, the closed balls of positive radius around <math>x</math> also form a neighborhood base for <math>x</math>. * Let <math>X</math> be an infinite set and let <math>\mathcal{F}</math> consist of the subsets of <math>X</math> which contain all points but one. Then <math>\mathcal{F}</math> is a filter subbase of the Fréchet filter on <math>X</math>, which consists of the cofinite subsets. Its closure under finite intersections is the entire Fréchet filter, but there are smaller bases of the Fréchet filter which contain the subbase <math>\mathcal{F}</math>, such as the one formed by the subsets of <math>X</math> which contain all points but a finite odd number. In fact, for every base of the Fréchet filter, removing any subset yields another base of the Fréchet filter. * If <math>X</math> is a topological space, the dense open subsets of <math>X</math> form a filter base on <math>X</math>, because they are closed under finite intersection. The filter they generate consists of the complements of nowhere dense subsets. On <math>X = \R^n</math>, restricting to the null dense open subsets yields another filter base for the same filter.{{cn|date=September 2025}} * Similarly, if <math>X</math> is a topological space, the countable intersections of dense open subsets form a filter base which generates the filter of comeager subsets. * Let <math>X</math> be a set and let <math>(x_i)_{i \in I}</math> be a net with values in <math>X</math>, i.e., a family whose domain <math>I</math> is a directed set. The filter base of ''tails'' of <math>(x_i)</math> consists of the sets <math>\{x_j, j \geq i\}</math> for <math>i \in I</math>; it is downwards-closed by directedness of <math>I</math>. The generated filter is called the ''eventuality filter'' or ''filter of tails'' of <math>(x_n)</math>. A ''sequential filter''{{sfn|Dolecki|Mynard|2016|p=35}} or ''{{visible anchor|elementary filter}}''{{fact|date=May 2026}} is a filter which is the eventuality filter of some net. This example is fundamental in the application of filters in topology.{{sfn|Joshi|1983|p=242}}{{sfn|Narici|Beckenstein|2011|p=5}} * Every π-system is a filter base.

===Trace of a filter on a subset===

If <math>\mathcal{F}</math> is a filter on <math>X</math> and <math>Y \subseteq X</math>, the ''trace'' of <math>\mathcal{F}</math> on <math>Y</math> is <math>\{A \cap Y, A \in \mathcal{F}\}</math>, which is a filter.{{sfn|Schechter|1996|p=103}}

===Image of a filter by a function===

Let <math>f : X \to Y</math> be a function.

When <math>\mathcal{F}</math> is a family of subsets of <math>X</math>, its image by <math>f</math> is defined as

<math display=block>f(\mathcal{F}) = \{\{f(x), x \in A\}, A \in \mathcal{F}\}</math>

The ''image filter'' by <math>f</math> of a filter <math>\mathcal{F}</math> on <math>X</math> is defined as the generated filter <math>\langle f(\mathcal{F}) \rangle</math>.{{sfn|Joshi|1983|p=246}} If <math>f</math> is surjective, then <math>f(\mathcal{F})</math> is already a filter. In the general case, <math>f(\mathcal{F})</math> is a filter base and hence <math>\langle f(\mathcal{F}) \rangle</math> is its upwards closure.{{sfn|Dolecki|Mynard|2016|p=37}} Furthermore, if <math>\mathcal{B}</math> is a filter base of <math>\mathcal{F}</math> then <math>f(\mathcal{B})</math> is a filter base of <math>\langle f(\mathcal{F}) \rangle</math>.

The kernels of <math>\mathcal{F}</math> and <math>\langle f(\mathcal{F}) \rangle</math> are linked by <math>f\left(\bigcap \mathcal{F}\right) \subseteq \bigcap \langle f(\mathcal{F}) \rangle</math>.

===Product of filters===

Given a family of sets <math>(X_i)_{i \in I}</math> and a filter <math>\mathcal{F}_i</math> on each <math>X_i</math>, the product filter <math>\prod_{i \in I} \mathcal{F}_i</math> on the product set <math>\prod_{i \in I} X_i</math> is defined as the filter generated by the sets <math>\pi_i^{-1}(A)</math> for <math>i \in I</math> and <math>A \in \mathcal{F}_i</math>, where <math>\pi_i : \left(\prod_{j \in I} X_j\right) \to X_i</math> is the projection from the product set onto the <math>i</math>-th component.{{sfn|Bourbaki|1987|pp=57–68}}{{sfn|Dolecki|Mynard|2016|p=39}} This construction is similar to the product topology.

If each <math>\mathcal{B}_i</math> is a filter base on <math>\mathcal{F}_i</math>, a filter base of <math>\prod_{i \in I} \mathcal{F}_i</math> is given by the sets <math>\prod_{i \in I} A_i</math> where <math>(A_i)</math> is a family such that <math>A_i \in \mathcal{F}_i</math> for all <math>i \in I</math> and <math>A_i = X_i</math> for all but finitely many <math>i \in I</math>.{{sfn|Bourbaki|1987|pp=57–68}}{{sfn|Köthe|1983|p=14}}

==See also==

* Axiomatic foundations of topological spaces, for a definition of topological spaces in terms of filters * {{annotated link|Filters in topology}} * Convergence space, a generalization of topological spaces using filters * {{annotated link|Filter quantifier}} * {{annotated link|Ultrafilter}} * Generic filter, a kind of filter used in set-theoretic forcing

==Notes==

==Citations==

==References==

{{refbegin|30em}} * {{Bourbaki General Topology Part I Chapters 1-4}}  * {{Bourbaki General Topology Part II Chapters 5-10}}  * {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}}  * {{cite book|last1=Burris|first1=Stanley|author-link1=Stanley Burris|last2=Sankappanavar|first2=Hanamantagouda P.|year=2012|title=A Course in Universal Algebra|publisher=Springer-Verlag|isbn=978-0-9880552-0-9|url=https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf|archive-url=https://web.archive.org/web/20220401154440/https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf|archive-date=1 April 2022|url-status=live|pages=127–135}} * {{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}}  * {{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}}  * {{cite book | last1=Császár | first1=Ákos | author-link=Ákos Császár | translator-last=Császár | translator-first=Klára | title=General topology | publisher=Adam Hilger Ltd | location=Bristol England | year=1978 | isbn=0-85274-275-4 | oclc=4146011 | pages=55–59}} * {{cite book | last1=Dolecki | first1=Szymon | last2=Mynard | first2=Frédéric | author-link1=Szymon Dolecki | title=Convergence Foundations Of Topology | publisher=World Scientific Publishing Company | location=New Jersey | year=2016 | isbn=978-981-4571-52-4 | oclc=945169917 | pages=29–39}} * {{cite book | last=Joshi | first=K. D. | author-link=K. D. Joshi | title=Introduction to General Topology | publisher=John Wiley and Sons Ltd | location=New York | year=1983 | isbn=978-0-85226-444-7 | oclc=9218750 | pages=241–248}} * {{cite book |last=Köthe |first=Gottfried |author-link=Gottfried Köthe |translator-last1=Garling |translator-first1=D. J. H. |title=Topological Vector Spaces I |publisher=Springer Science & Business Media |location=New York |year=1983 |orig-year=1969 |volume=159 |series=Grundlehren der mathematischen Wissenschaften |isbn=978-3-642-64988-2 |oclc=840293704 |mr=0248498 |pages=11–15}} * {{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|volume=31 |pages=68–95 |doi=10.1093/jigpal/jzab030}}  * {{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.) * {{cite book | last1=Narici | first1=Lawrence | last2=Beckenstein | first2=Edward | title=Topological Vector Spaces | edition=Second | publisher=CRC Press | location=Boca Raton, FL | year=2011 | series=Pure and applied mathematics | isbn=978-1584888666 | oclc=144216834 | pages=2–5}} * {{cite book | last1=Schechter | first1=Eric | author-link=Eric Schechter | title=Handbook of Analysis and Its Foundations | publisher=Academic Press | location=San Diego, CA | year=1996 | isbn=978-0-12-622760-4 | oclc=175294365 | pages=100–105 }} * {{cite book | last=Schubert | first=Horst | author-link=Horst Schubert | title=Topology | publisher=Macdonald & Co | location=London | year=1968 | isbn=978-0-356-02077-8 | oclc=463753 | pages=48–51 }} * {{cite book | last=Wilansky | first=Albert | author-link=Albert Wilansky | title=Modern Methods in Topological Vector Spaces | publisher=Dover Publications, Inc | location=Mineola, New York | year=2013 | isbn=978-0-486-49353-4 | oclc=849801114 | pages= }} * {{cite book |last=Willard |first=Stephen |title=General Topology |edition= |publisher=Dover Publications |location=Mineola, N.Y. |year=2004 |orig-year=1970 |isbn=978-0-486-43479-7 |oclc=115240 |url=https://books.google.com/books?id=-o8xJQ7Ag2cC |pages=77–84 }} {{refend}}

Category:General topology Category:Order theory Category:Set theory