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 ${\displaystyle A,B\subset X,A\in {\mathcal {B}}}$, and ${\displaystyle A\subset B}$, 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.

Preliminaries, notation, and basic notions[edit]

In this article, upper case Roman letters like ${\displaystyle S{\text{ and }}X}$ denote sets (but not families unless indicated otherwise) and ${\displaystyle \wp (X)}$ will denote the power set of ${\displaystyle X.}$ A subset of a power set is called a family of sets (or simply, a family) where it is over ${\displaystyle X}$ if it is a subset of ${\displaystyle \wp (X).}$ Families of sets will be denoted by upper case calligraphy letters such as ${\displaystyle {\mathcal {B}},{\mathcal {C}},{\text{ and }}{\mathcal {F}}.}$ Whenever these assumptions are needed, then it should be assumed that ${\displaystyle X}$ is non–empty and that ${\displaystyle {\mathcal {B}},{\mathcal {F}},}$ etc. are families of sets over ${\displaystyle X.}$

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter". While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

Sets operations

The upward closure or isotonization in ${\displaystyle X}$^[5][6] of a family of sets ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is

${\displaystyle {\mathcal {B}}^{\uparrow X}:=\{S\subseteq X~:~B\subseteq S{\text{ for some }}B\in {\mathcal {B}}\,\}=\bigcup _{B\in {\mathcal {B}}}\{S~:~B\subseteq S\subseteq X\}}$

and similarly the downward closure of ${\displaystyle {\mathcal {B}}}$ is ${\displaystyle {\mathcal {B}}^{\downarrow }:=\{S\subseteq B~:~B\in {\mathcal {B}}\,\}=\bigcup _{B\in {\mathcal {B}}}\wp (B).}$

Notation and Definition	Name
${\displaystyle \ker {\mathcal {B}}=\bigcap _{B\in {\mathcal {B}}}B}$	Kernel of ${\displaystyle {\mathcal {B}}}$[6]
${\displaystyle S\setminus {\mathcal {B}}:=\{S\setminus B~:~B\in {\mathcal {B}}\}=\{S\}\,(\setminus )\,{\mathcal {B}}}$	Dual of ${\displaystyle {\mathcal {B}}{\text{ in }}S}$ where ${\displaystyle S}$ is a set.[7]
${\displaystyle {\mathcal {B}}{\big \vert }_{S}:=\{B\cap S~:~B\in {\mathcal {B}}\}={\mathcal {B}}\,(\cap )\,\{S\}}$	Trace of ${\displaystyle {\mathcal {B}}{\text{ on }}S}$[7] or the restriction of ${\displaystyle {\mathcal {B}}{\text{ to }}S}$ where ${\displaystyle S}$ is a set; sometimes denoted by ${\displaystyle {\mathcal {B}}\cap S}$
${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {C}}=\{B\cap C~:~B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}\}}$[8]	Elementwise (set) intersection (${\displaystyle {\mathcal {B}}\cap {\mathcal {C}}}$ will denote the usual intersection)
${\displaystyle {\mathcal {B}}\,(\cup )\,{\mathcal {C}}=\{B\cup C~:~B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}\}}$[8]	Elementwise (set) union (${\displaystyle {\mathcal {B}}\cup {\mathcal {C}}}$ will denote the usual union)
${\displaystyle {\mathcal {B}}\,(\setminus )\,{\mathcal {C}}=\{B\setminus C~:~B\in {\mathcal {B}}{\text{ and }}C\in {\mathcal {C}}\}}$	Elementwise (set) subtraction (${\displaystyle {\mathcal {B}}\setminus {\mathcal {C}}}$ will denote the usual set subtraction)
${\displaystyle {\mathcal {B}}^{\#X}={\mathcal {B}}^{\#}=\{S\subseteq X~:~S\cap B\neq \varnothing {\text{ for all }}B\in {\mathcal {B}}\}}$	Grill of ${\displaystyle {\mathcal {B}}{\text{ in }}X}$[9]
${\displaystyle \wp (X)=\{S~:~S\subseteq X\}}$	Power set of a set ${\displaystyle X}$[6]

Throughout, ${\displaystyle f}$ is a map and ${\displaystyle S}$ is a set.

Notation and Definition	Name
${\displaystyle f^{-1}({\mathcal {B}})=\left\{f^{-1}(B)~:~B\in {\mathcal {B}}\right\}}$[13]	Image of ${\displaystyle {\mathcal {B}}{\text{ under }}f^{-1},}$ or the preimage of ${\displaystyle {\mathcal {B}}}$ under ${\displaystyle f}$
${\displaystyle f^{-1}(S)=\{x\in \operatorname {domain} f~:~f(x)\in S\}}$	Image of ${\displaystyle S{\text{ under }}f^{-1},}$ or the preimage of ${\displaystyle S{\text{ under }}f}$
${\displaystyle f({\mathcal {B}})=\{f(B)~:~B\in {\mathcal {B}}\}}$[14]	Image of ${\displaystyle {\mathcal {B}}}$ under ${\displaystyle f}$
${\displaystyle f(S)=\{f(s)~:~s\in S\cap \operatorname {domain} f\}}$	Image of ${\displaystyle S{\text{ under }}f}$
${\displaystyle \operatorname {image} f=f(\operatorname {domain} f)}$	Image (or range) of ${\displaystyle f}$

Nets and their tails

A directed set is a set ${\displaystyle I}$ together with a preorder, which will be denoted by ${\displaystyle \,\leq \,}$ (unless explicitly indicated otherwise), that makes ${\displaystyle (I,\leq )}$ into an (upward) directed set;^[15] this means that for all ${\displaystyle i,j\in I,}$ there exists some ${\displaystyle k\in I}$ such that ${\displaystyle i\leq k{\text{ and }}j\leq k.}$ For any indices ${\displaystyle i{\text{ and }}j,}$ the notation ${\displaystyle j\geq i}$ is defined to mean ${\displaystyle i\leq j}$ while ${\displaystyle i<j}$ is defined to mean that ${\displaystyle i\leq j}$ holds but it is not true that ${\displaystyle j\leq i}$ (if ${\displaystyle \,\leq \,}$ is antisymmetric then this is equivalent to ${\displaystyle i\leq j{\text{ and }}i\neq j}$).

A net in ${\displaystyle X}$^[15] is a map from a non–empty directed set into ${\displaystyle X.}$ The notation ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ will be used to denote a net with domain ${\displaystyle I.}$

Notation and Definition	Name
${\displaystyle I_{\geq i}=\{j\in I~:~j\geq i\}}$	Tail or section of ${\displaystyle I}$ starting at ${\displaystyle i\in I}$ where ${\displaystyle (I,\leq )}$ is a directed set.
${\displaystyle x_{\geq i}=\left\{x_{j}~:~j\geq i{\text{ and }}j\in I\right\}}$	Tail or section of ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ starting at ${\displaystyle i\in I}$
${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)=\left\{x_{\geq i}~:~i\in I\right\}}$	Set or prefilter of tails/sections of ${\displaystyle x_{\bullet }.}$ Also called the eventuality filter base generated by (the tails of) ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}.}$ If ${\displaystyle x_{\bullet }}$ is a sequence then ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ is also called the sequential filter base.[16]
${\displaystyle \operatorname {TailsFilter} \left(x_{\bullet }\right)=\operatorname {Tails} \left(x_{\bullet }\right)^{\uparrow X}}$	(Eventuality) filter of/generated by (tails of) ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$[16]
${\displaystyle f\left(I_{\geq i}\right)=\{f(j)~:~j\geq i{\text{ and }}j\in I\}}$	Tail or section of a net ${\displaystyle f:I\to X}$ starting at ${\displaystyle i\in I}$[16] where ${\displaystyle (I,\leq )}$ is a directed set.

Warning about using strict comparison

If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net and ${\displaystyle i\in I}$ then it is possible for the set ${\displaystyle x_{>i}=\left\{x_{j}~:~j>i{\text{ and }}j\in I\right\},}$ which is called the tail of ${\displaystyle x_{\bullet }}$ after ${\displaystyle i}$, to be empty (for example, this happens if ${\displaystyle i}$ is an upper bound of the directed set ${\displaystyle I}$). In this case, the family ${\displaystyle \left\{x_{>i}~:~i\in I\right\}}$ would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining ${\displaystyle \operatorname {Tails} \left(x_{\bullet }\right)}$ as ${\displaystyle \left\{x_{\geq i}~:~i\in I\right\}}$ rather than ${\displaystyle \left\{x_{>i}~:~i\in I\right\}}$ or even ${\displaystyle \left\{x_{>i}~:~i\in I\right\}\cup \left\{x_{\geq i}~:~i\in I\right\}}$ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality ${\displaystyle \,<\,}$ may not be used interchangeably with the inequality ${\displaystyle \,\leq .}$

Filters and prefilters[edit]

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$

The following is a list of properties that a family ${\displaystyle {\mathcal {B}}}$ of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that ${\displaystyle {\mathcal {B}}\subseteq \wp (X).}$

The family of sets ${\displaystyle {\mathcal {B}}}$ is:

1. Proper or nondegenerate if ${\displaystyle \varnothing \not \in {\mathcal {B}}.}$ Otherwise, if ${\displaystyle \varnothing \in {\mathcal {B}},}$ then it is called improper^[17] or degenerate.
2. Directed downward^[15] if whenever ${\displaystyle A,B\in {\mathcal {B}}}$ then there exists some ${\displaystyle C\in {\mathcal {B}}}$ such that ${\displaystyle C\subseteq A\cap B.}$
   • This property can be characterized in terms of directedness, which explains the word "directed": A binary relation ${\displaystyle \,\preceq \,}$ on ${\displaystyle {\mathcal {B}}}$ is called (upward) directed if for any two ${\displaystyle A{\text{ and }}B,}$ there is some ${\displaystyle C}$ satisfying ${\displaystyle A\preceq C{\text{ and }}B\preceq C.}$ Using ${\displaystyle \,\supseteq \,}$ in place of ${\displaystyle \,\preceq \,}$ gives the definition of directed downward whereas using ${\displaystyle \,\subseteq \,}$ instead gives the definition of directed upward. Explicitly, ${\displaystyle {\mathcal {B}}}$ is directed downward (resp. directed upward) if and only if for all ${\displaystyle A,B\in {\mathcal {B}},}$ there exists some "greater" ${\displaystyle C\in {\mathcal {B}}}$ such that ${\displaystyle A\supseteq C{\text{ and }}B\supseteq C}$ (resp. such that ${\displaystyle A\subseteq C{\text{ and }}B\subseteq C}$) − where the "greater" element is always on the right hand side,^{[note 1]} − which can be rewritten as ${\displaystyle A\cap B\supseteq C}$ (resp. as ${\displaystyle A\cup B\subseteq C}$).
   • If a family ${\displaystyle {\mathcal {B}}}$ has a greatest element with respect to ${\displaystyle \,\supseteq \,}$ (for example, if ${\displaystyle \varnothing \in {\mathcal {B}}}$) then it is necessarily directed downward.
3. Closed under finite intersections (resp. unions) if the intersection (resp. union) of any two elements of ${\displaystyle {\mathcal {B}}}$ is an element of ${\displaystyle {\mathcal {B}}.}$
   • If ${\displaystyle {\mathcal {B}}}$ is closed under finite intersections then ${\displaystyle {\mathcal {B}}}$ is necessarily directed downward. The converse is generally false.
4. Upward closed or Isotone in ${\displaystyle X}$^[5] if ${\displaystyle {\mathcal {B}}\subseteq \wp (X){\text{ and }}{\mathcal {B}}={\mathcal {B}}^{\uparrow X},}$ or equivalently, if whenever ${\displaystyle B\in {\mathcal {B}}}$ and some set ${\displaystyle C}$ satisfies ${\displaystyle B\subseteq C\subseteq X,{\text{ then }}C\in {\mathcal {B}}.}$ Similarly, ${\displaystyle {\mathcal {B}}}$ is downward closed if ${\displaystyle {\mathcal {B}}={\mathcal {B}}^{\downarrow }.}$ An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
   • The family ${\displaystyle {\mathcal {B}}^{\uparrow X},}$ which is the upward closure of ${\displaystyle {\mathcal {B}}{\text{ in }}X,}$ is the unique smallest (with respect to ${\displaystyle \,\subseteq }$) isotone family of sets over ${\displaystyle X}$ having ${\displaystyle {\mathcal {B}}}$ as a subset.

Many of the properties of ${\displaystyle {\mathcal {B}}}$ defined above and below, such as "proper" and "directed downward," do not depend on ${\displaystyle X,}$ so mentioning the set ${\displaystyle X}$ is optional when using such terms. Definitions involving being "upward closed in ${\displaystyle X,}$" such as that of "filter on ${\displaystyle X,}$" do depend on ${\displaystyle X}$ so the set ${\displaystyle X}$ should be mentioned if it is not clear from context.

$${\displaystyle {\textrm {Filters}}(X)\quad =\quad {\textrm {DualIdeals}}(X)\,\setminus \,\{\wp (X)\}\quad \subseteq \quad {\textrm {Prefilters}}(X)\quad \subseteq \quad {\textrm {FilterSubbases}}(X).}$$

A family ${\displaystyle {\mathcal {B}}}$ is/is a(n):

1. Ideal^[17][18] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is downward closed and closed under finite unions.
2. Dual ideal on ${\displaystyle X}$^[19] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is upward closed in ${\displaystyle X}$ and also closed under finite intersections. Equivalently, ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a dual ideal if for all ${\displaystyle R,S\subseteq X,}$ ${\displaystyle R\cap S\in {\mathcal {B}}\;{\text{ if and only if }}\;R,S\in {\mathcal {B}}.}$^[9]
   • Explanation of the word "dual": A family ${\displaystyle {\mathcal {B}}}$ is a dual ideal (resp. an ideal) on ${\displaystyle X}$ if and only if the dual of ${\displaystyle {\mathcal {B}}{\text{ in }}X,}$ which is the family
     $${\displaystyle X\setminus {\mathcal {B}}:=\{X\setminus B~:~B\in {\mathcal {B}}\},}$$

     
     is an ideal (resp. a dual ideal) on ${\displaystyle X.}$ In other words, dual ideal means "dual of an ideal". The family ${\displaystyle X\setminus {\mathcal {B}}}$ should not be confused with ${\displaystyle \wp (X)\setminus {\mathcal {B}}=\{S\subseteq X~:~S\notin {\mathcal {B}}\}}$ because these two sets are not equal in general; for instance, ${\displaystyle X\setminus {\mathcal {B}}=\wp (X){\text{ if and only if }}{\mathcal {B}}=\wp (X).}$ The dual of the dual is the original family, meaning ${\displaystyle X\setminus (X\setminus {\mathcal {B}})={\mathcal {B}}.}$ The set ${\displaystyle X}$ belongs to the dual of ${\displaystyle {\mathcal {B}}}$ if and only if ${\displaystyle \varnothing \in {\mathcal {B}}.}$^[17]
3. Filter on ${\displaystyle X}$^[19][7] if ${\displaystyle {\mathcal {B}}}$ is a proper dual ideal on ${\displaystyle X.}$ That is, a filter on ${\displaystyle X}$ is a non−empty subset of ${\displaystyle \wp (X)\setminus \{\varnothing \}}$ that is closed under finite intersections and upward closed in ${\displaystyle X.}$ Equivalently, it is a prefilter that is upward closed in ${\displaystyle X.}$ In words, a filter on ${\displaystyle X}$ is a family of sets over ${\displaystyle X}$ that (1) is not empty (or equivalently, it contains ${\displaystyle X}$), (2) is closed under finite intersections, (3) is upward closed in ${\displaystyle X,}$ and (4) does not have the empty set as an element.
   • Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a proper/non–degenerate dual ideal.^[20] It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always require non-degeneracy. This article uses Henri Cartan's original definition of "filter",^[3][4] which required non–degeneracy.
   • A dual filter on ${\displaystyle X}$ is a family ${\displaystyle {\mathcal {B}}}$ whose dual ${\displaystyle X\setminus {\mathcal {B}}}$ is a filter on ${\displaystyle X.}$ Equivalently, it is an ideal on ${\displaystyle X}$ that does not contain ${\displaystyle X}$ as an element.
   • The power set ${\displaystyle \wp (X)}$ is the one and only dual ideal on ${\displaystyle X}$ that is not also a filter. Excluding ${\displaystyle \wp (X)}$ from the definition of "filter" in topology has the same benefit as excluding ${\displaystyle 1}$ from the definition of "prime number": it obviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-${\displaystyle 1}$") in many important results, thereby making their statements less awkward.
4. Prefilter or filter base^[7][21] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is proper and directed downward. Equivalently, ${\displaystyle {\mathcal {B}}}$ is called a prefilter if its upward closure ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is a filter. It can also be defined as any family that is equivalent (with respect to ${\displaystyle \leq }$) to some filter.^[8] A proper family ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is a prefilter if and only if ${\displaystyle {\mathcal {B}}\,(\cap )\,{\mathcal {B}}\leq {\mathcal {B}}.}$^[8] A family is a prefilter if and only if the same is true of its upward closure.
   • If ${\displaystyle {\mathcal {B}}}$ is a prefilter then its upward closure ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is the unique smallest (relative to ${\displaystyle \subseteq }$) filter on ${\displaystyle X}$ containing ${\displaystyle {\mathcal {B}}}$ and it is called the filter generated by ${\displaystyle {\mathcal {B}}.}$ A filter ${\displaystyle {\mathcal {F}}}$ is said to be generated by a prefilter ${\displaystyle {\mathcal {B}}}$ if ${\displaystyle {\mathcal {F}}={\mathcal {B}}^{\uparrow X},}$ in which ${\displaystyle {\mathcal {B}}}$ is called a filter base for ${\displaystyle {\mathcal {F}}.}$
   • Unlike a filter, a prefilter is not necessarily closed under finite intersections.
5. π–system if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ is closed under finite intersections. Every non–empty family ${\displaystyle {\mathcal {B}}}$ is contained in a unique smallest π–system called the π–system generated by ${\displaystyle {\mathcal {B}},}$ which is sometimes denoted by ${\displaystyle \pi ({\mathcal {B}}).}$ It is equal to the intersection of all π–systems containing ${\displaystyle {\mathcal {B}}}$ and also to the set of all possible finite intersections of sets from ${\displaystyle {\mathcal {B}}}$:
   $${\displaystyle \pi ({\mathcal {B}})=\left\{B_{1}\cap \cdots \cap B_{n}~:~n\geq 1{\text{ and }}B_{1},\ldots ,B_{n}\in {\mathcal {B}}\right\}.}$$

   
   • A π–system is a prefilter if and only if it is proper. Every filter is a proper π–system and every proper π–system is a prefilter but the converses do not hold in general.
   • A prefilter is equivalent (with respect to ${\displaystyle \leq }$) to the π–system generated by it and both of these families generate the same filter on ${\displaystyle X.}$
6. Filter subbase^[7][22] and centered^[8] if ${\displaystyle {\mathcal {B}}\neq \varnothing }$ and ${\displaystyle {\mathcal {B}}}$ satisfies any of the following equivalent conditions:
   1. ${\displaystyle {\mathcal {B}}}$ has the finite intersection property, which means that the intersection of any finite family of (one or more) sets in ${\displaystyle {\mathcal {B}}}$ is not empty; explicitly, this means that whenever ${\displaystyle n\geq 1{\text{ and }}B_{1},\ldots ,B_{n}\in {\mathcal {B}}}$ then ${\displaystyle \varnothing \neq B_{1}\cap \cdots \cap B_{n}.}$
   2. The π–system generated by ${\displaystyle {\mathcal {B}}}$ is proper; that is, ${\displaystyle \varnothing \not \in \pi ({\mathcal {B}}).}$
   3. The π–system generated by ${\displaystyle {\mathcal {B}}}$ is a prefilter.
   4. ${\displaystyle {\mathcal {B}}}$ is a subset of some prefilter.
   5. ${\displaystyle {\mathcal {B}}}$ is a subset of some filter.
   • Assume that ${\displaystyle {\mathcal {B}}}$ is a filter subbase. Then there is a unique smallest (relative to ${\displaystyle \subseteq }$) filter ${\displaystyle {\mathcal {F}}_{\mathcal {B}}{\text{ on }}X}$ containing ${\displaystyle {\mathcal {B}}}$ called the filter generated by ${\displaystyle {\mathcal {B}}}$, and ${\displaystyle {\mathcal {B}}}$ is said to be a filter subbase for this filter. This filter is equal to the intersection of all filters on ${\displaystyle X}$ that are supersets of ${\displaystyle {\mathcal {B}}.}$ The π–system generated by ${\displaystyle {\mathcal {B}},}$ denoted by ${\displaystyle \pi ({\mathcal {B}}),}$ will be a prefilter and a subset of ${\displaystyle {\mathcal {F}}_{\mathcal {B}}.}$ Moreover, the filter generated by ${\displaystyle {\mathcal {B}}}$ is equal to the upward closure of ${\displaystyle \pi ({\mathcal {B}}),}$ meaning ${\displaystyle \pi ({\mathcal {B}})^{\uparrow X}={\mathcal {F}}_{\mathcal {B}}.}$^[8] However, ${\displaystyle {\mathcal {B}}^{\uparrow X}={\mathcal {F}}_{\mathcal {B}}}$ if and only if ${\displaystyle {\mathcal {B}}}$ is a prefilter (although ${\displaystyle {\mathcal {B}}^{\uparrow X}}$ is always an upward closed filter subbase for ${\displaystyle {\mathcal {F}}_{\mathcal {B}}}$).
   • A ${\displaystyle \subseteq }$ –smallest (meaning smallest relative to ${\displaystyle \subseteq }$ ) prefilter containing a filter subbase ${\displaystyle {\mathcal {B}}}$ will exist only under certain circumstances. It exists, for example, if the filter subbase ${\displaystyle {\mathcal {B}}}$ happens to also be a prefilter. It also exists if the filter (or equivalently, the π–system) generated by ${\displaystyle {\mathcal {B}}}$ is principal, in which case ${\displaystyle {\mathcal {B}}\cup \{\ker {\mathcal {B}}\}}$ is the unique smallest prefilter containing ${\displaystyle {\mathcal {B}}.}$ Otherwise, in general, a ${\displaystyle \subseteq }$ –smallest prefilter containing ${\displaystyle {\mathcal {B}}}$ might not exist. For this reason, some authors may refer to the π–system generated by ${\displaystyle {\mathcal {B}}}$ as the prefilter generated by ${\displaystyle {\mathcal {B}}.}$ However, if a ${\displaystyle \subseteq }$ –smallest prefilter does exist (say it is denoted by ${\displaystyle \operatorname {minPre} {\mathcal {B}}}$) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by ${\displaystyle {\mathcal {B}}}$" (that is, ${\displaystyle \operatorname {minPre} {\mathcal {B}}\neq \pi ({\mathcal {B}})}$ is possible). And if the filter subbase ${\displaystyle {\mathcal {B}}}$ happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning ${\displaystyle \pi ({\mathcal {B}})}$) will not be ${\displaystyle {\mathcal {B}}=\operatorname {minPre} {\mathcal {B}}}$ (that is, ${\displaystyle \pi ({\mathcal {B}})\neq {\mathcal {B}}}$ is possible even when ${\displaystyle {\mathcal {B}}}$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ${\displaystyle {\mathcal {B}}}$".
7. Subfilter of a filter ${\displaystyle {\mathcal {F}}}$ and that ${\displaystyle {\mathcal {F}}}$ is a superfilter of ${\displaystyle {\mathcal {B}}}$^[17][23] if ${\displaystyle {\mathcal {B}}}$ is a filter and ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {F}}}$ where for filters, ${\displaystyle {\mathcal {B}}\subseteq {\mathcal {F}}{\text{ if and only if }}{\mathcal {B}}\leq {\mathcal {F}}.}$
   • Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of." However, ${\displaystyle {\mathcal {B}}\leq {\mathcal {F}}}$ can also be written ${\displaystyle {\mathcal {F}}\vdash {\mathcal {B}}}$ which is described by saying "${\displaystyle {\mathcal {F}}}$ is subordinate to ${\displaystyle {\mathcal {B}}.}$" With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of,"^[24] which makes this one situation where using the term "subordinate" and symbol ${\displaystyle \,\vdash \,}$ may be helpful.

There are no prefilters on ${\displaystyle X=\varnothing }$ (nor are there any nets valued in ${\displaystyle \varnothing }$), which is why this article, like most authors, will automatically assume without comment that ${\displaystyle X\neq \varnothing }$ whenever this assumption is needed.

Basic examples[edit]

Named examples

• The singleton set ${\displaystyle {\mathcal {B}}=\{X\}}$ is called the indiscrete or trivial filter on ${\displaystyle X.}$^[25][10] It is the unique minimal filter on ${\displaystyle X}$ because it is a subset of every filter on ${\displaystyle X}$; however, it need not be a subset of every prefilter on ${\displaystyle X.}$
• The dual ideal ${\displaystyle \wp (X)}$ is also called the degenerate filter on ${\displaystyle X}$^[9] (despite not actually being a filter). It is the only dual ideal on ${\displaystyle X}$ that is not a filter on ${\displaystyle X.}$
• If ${\displaystyle (X,\tau )}$ is a topological space and ${\displaystyle x\in X,}$ then the neighborhood filter ${\displaystyle {\mathcal {N}}(x)}$ at ${\displaystyle x}$ is a filter on ${\displaystyle X.}$ By definition, a family ${\displaystyle {\mathcal {B}}\subseteq \wp (X)}$ is called a neighborhood basis (resp. a neighborhood subbase) at ${\displaystyle x{\text{ for }}(X,\tau )}$ if and only if ${\displaystyle {\mathcal {B}}}$ is a prefilter (resp. ${\displaystyle {\mathcal {B}}}$ is a filter subbase) and the filter on ${\displaystyle X}$ that ${\displaystyle {\mathcal {B}}}$ generates is equal to the neighborhood filter ${\displaystyle {\mathcal {N}}(x).}$ The subfamily ${\displaystyle \tau (x)\subseteq {\mathcal {N}}(x)}$ of open neighborhoods is a filter base for ${\displaystyle {\mathcal {N}}(x).}$ Both prefilters ${\displaystyle {\mathcal {N}}(x){\text{ and }}\tau (x)}$ also form a bases for topologies on ${\displaystyle X,}$ with the topology generated ${\displaystyle \tau (x)}$ being coarser than ${\displaystyle \tau .}$ This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets ${\displaystyle S\subseteq X.}$
• ${\displaystyle {\mathcal {B}}}$ is an elementary prefilter^[26] if ${\displaystyle {\mathcal {B}}=\operatorname {Tails} \left(x_{\bullet }\right)}$ for some sequence ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }{\text{ in }}X.}$
• ${\displaystyle {\mathcal {B}}}$ is an elementary filter or a sequential filter on ${\displaystyle X}$^[27] if ${\displaystyle {\mathcal {B}}}$ is a filter on ${\displaystyle X}$ generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily not an ultrafilter.^[28] Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.^[9] The intersection of finitely many sequential filters is again sequential.^[9]
• The set ${\displaystyle {\mathcal {F}}}$ of all cofinite subsets of ${\displaystyle X}$ (meaning those sets whose complement in ${\displaystyle X}$ is finite) is proper if and only if ${\displaystyle {\mathcal {F}}}$ is infinite (or equivalently, ${\displaystyle X}$ is infinite), in which case ${\displaystyle {\mathcal {F}}}$ is a filter on ${\displaystyle X}$ known as the Fréchet filter or the cofinite filter on ${\displaystyle X.}$^[10][25] If ${\displaystyle X}$ is finite then ${\displaystyle {\mathcal {F}}}$ is equal to the dual ideal ${\displaystyle \wp (X),}$ which is not a filter. If ${\displaystyle X}$ is infinite then the family ${\displaystyle \{X\setminus \{x\}~:~x\in X\}}$ of complements of singleton sets is a filter subbase that generates the Fréchet filter on ${\displaystyle X.}$ As with any family of sets over ${\displaystyle X}$ that contains ${\displaystyle \{X\setminus \{x\}~:~x\in X\},}$ the kernel of the Fréchet filter on ${\displaystyle X}$ is the empty set: ${\displaystyle \ker {\mathcal {F}}=\varnothing .}$
• The intersection of all elements in any non–empty family ${\displaystyle \mathbb {F} \subseteq \operatorname {Filters} (X)}$ is itself a filter on ${\displaystyle X}$ called the infimum or greatest lower bound of ${\displaystyle \mathbb {F} {\text{ in }}\operatorname {Filters} (X),}$ which is why it may be denoted by ${\displaystyle \bigwedge _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}.}$ Said differently, ${\displaystyle \ker \mathbb {F} =\bigcap _{{\mathcal {F}}\in \mathbb {F} }{\mathcal {F}}\in \operatorname {Filters} (X).}$ Because every filter on ${\displaystyle X}$ has