Filter (mathematics)

There are two competing definitions of a "filter on a set," both of which require that a filter be a dual ideal.[5] One definition defines "filter" as a synonym of "dual ideal" while the other defines "filter" to mean a dual ideal that is also proper.

Warning: It is recommended that readers always check how "filter" is defined when reading mathematical literature.

Definition: A dual ideal[5] on a set ${\displaystyle S}$ is a non-empty subset ${\displaystyle F}$ of ${\displaystyle \wp (S)}$ with the following properties:

1. ${\displaystyle F}$ is closed under finite intersections: If ${\displaystyle A,B\in F,}$ then so is their intersection.
   • This property implies that if ${\displaystyle \varnothing \not \in F}$ then ${\displaystyle F}$ has the finite intersection property.
2. ${\displaystyle F}$ is upward closed/isotone:[6] If ${\displaystyle A\in F}$ and ${\displaystyle A\subseteq B,}$ then ${\displaystyle B\in F,}$ for all subsets ${\displaystyle B\subseteq S.}$
   • This property entails that ${\displaystyle S\in F}$ (since ${\displaystyle F}$ is a non-empty subset of ${\displaystyle \wp (S)}$).

Given a set ${\displaystyle S,}$ a canonical partial ordering ${\displaystyle \,\subseteq \,}$ can be defined on the powerset ${\displaystyle \wp (S)}$ by subset inclusion, turning ${\displaystyle (\wp (S),\subseteq )}$ into a lattice. A "dual ideal" is just a filter with respect to this partial ordering. Note that if ${\displaystyle S=\varnothing }$ then there is exactly one dual ideal on ${\displaystyle S,}$ which is ${\displaystyle \wp (S)=\{\varnothing \}.}$

The article uses the following definition of "filter on a set."

Definition: A filter on a set ${\displaystyle S}$ is a dual ideal on ${\displaystyle S.}$ Equivalently, a filter on ${\displaystyle S}$ is just a filter with respect the canonical partial ordering ${\displaystyle (\wp (S),\subseteq )}$ described above.

The other definition of "filter on a set" is the original definition of a "filter" given by Henri Cartan, which required that a filter on a set be a dual ideal that does not contain the empty set:

Original/Alternative definition: A filter[5] on a set ${\displaystyle S}$ is a dual ideal on ${\displaystyle S}$ with the following additional property:

3. ${\displaystyle F}$ is proper[7]/non-degenerate:[8] The empty set is not in ${\displaystyle F}$ (i.e. ${\displaystyle \varnothing \not \in F}$).

Note: This article does not require that a filter be proper.

The only non-proper filter on ${\displaystyle S}$ is ${\displaystyle \wp (S).}$ Much mathematical literature, especially that related to Topology, defines "filter" to mean a non-degenerate dual ideal.

Filter bases and subbases

A subset ${\displaystyle B}$ of ${\displaystyle \wp (S)}$ is called a prefilter, filter base, or filter basis if ${\displaystyle B}$ is non-empty and the intersection of any two members of ${\displaystyle B}$ is a superset of some member(s) of ${\displaystyle B.}$ If the empty set is not a member of ${\displaystyle B,}$ we say ${\displaystyle B}$ is a proper filter base.

Given a filter base ${\displaystyle B,}$ the filter generated or spanned by ${\displaystyle B}$ is defined as the minimum filter containing ${\displaystyle B.}$ It is the family of all those subsets of ${\displaystyle S}$ which are supersets of some member(s) of ${\displaystyle B.}$ Every filter is also a filter base, so the process of passing from filter base to filter may be viewed as a sort of completion.

For every subset ${\displaystyle T}$ of ${\displaystyle \wp (S)}$ there is a smallest (possibly nonproper) filter ${\displaystyle F}$ containing ${\displaystyle T,}$ called the filter generated or spanned by ${\displaystyle T.}$ Similarly as for a filter spanned by a filter base, a filter spanned by a subset ${\displaystyle T}$ is the minimum filter containing ${\displaystyle T.}$ It is constructed by taking all finite intersections of ${\displaystyle T,}$ which then form a filter base for ${\displaystyle F.}$ This filter is proper if and only if every finite intersection of elements of ${\displaystyle T}$ is non-empty, and in that case we say that ${\displaystyle T}$ is a filter subbase.

Finer/equivalent filter bases

If ${\displaystyle B}$ and ${\displaystyle C}$ are two filter bases on ${\displaystyle S,}$ one says ${\displaystyle C}$ is finer than ${\displaystyle B}$ (or that ${\displaystyle C}$ is a refinement of ${\displaystyle B}$) if for each ${\displaystyle B_{0}\in B,}$ there is a ${\displaystyle C_{0}\in C}$ such that ${\displaystyle C_{0}\subseteq B_{0}.}$ If also ${\displaystyle B}$ is finer than ${\displaystyle C,}$ one says that they are equivalent filter bases.

• If ${\displaystyle B}$ and ${\displaystyle C}$ are filter bases, then ${\displaystyle C}$ is finer than ${\displaystyle B}$ if and only if the filter spanned by ${\displaystyle C}$ contains the filter spanned by ${\displaystyle B.}$ Therefore, ${\displaystyle B}$ and ${\displaystyle C}$ are equivalent filter bases if and only if they generate the same filter.
• For filter bases ${\displaystyle A,}$ ${\displaystyle B,}$ and ${\displaystyle C,}$ if ${\displaystyle A}$ is finer than ${\displaystyle B}$ and ${\displaystyle B}$ is finer than ${\displaystyle C}$ then ${\displaystyle A}$ is finer than ${\displaystyle C.}$ Thus the refinement relation is a preorder on the set of filter bases, and the passage from filter base to filter is an instance of passing from a preordering to the associated partial ordering.

• Let ${\displaystyle S}$ be a set and ${\displaystyle C}$ be a non-empty subset of ${\displaystyle S.}$ Then ${\displaystyle \{C\}}$is a filter base. The filter it generates (that is,, the collection of all subsets containing ${\displaystyle C}$) is called the principal filter generated by ${\displaystyle C.}$
• A filter is said to be a free filter if the intersection of all 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. A nonprincipal filter on an infinite set is not necessarily free.
• The Fréchet filter on an infinite set ${\displaystyle S}$ is the set of all subsets of ${\displaystyle S}$ that have finite complement. A filter on ${\displaystyle S}$ is free if and only if it includes the Fréchet filter.
  • More generally, if ${\displaystyle (X,\mu )}$ is a measure space for which ${\displaystyle \mu (X)=\infty ,}$ the collection of all ${\displaystyle A\subseteq X}$ such that ${\displaystyle \mu (X\setminus A)<\infty }$ forms a filter. The Fréchet filter is the case where ${\displaystyle X=S}$ and ${\displaystyle \mu }$ is the counting measure.
• Every uniform structure on a set ${\displaystyle X}$ is a filter on ${\displaystyle X\times X.}$
• A filter in a poset can be created using the Rasiowa–Sikorski lemma, often used in forcing.
• The set ${\displaystyle \{\{N,N+1,N+2,\dots \}:N\in \mathbb {N} \}}$ is called a filter base of tails of the sequence of natural numbers ${\displaystyle (1,2,3,\dots ).}$ A filter base of tails can be made of any net ${\displaystyle (x_{\alpha })_{\alpha \in A}}$ using the construction ${\displaystyle \{\{x_{\alpha }:\alpha \in A,\alpha _{0}\leq \alpha \}:\alpha _{0}\in A\},}$ where the filter that this filter base generates is called the net's eventuality filter. Therefore, all nets generate a filter base (and therefore a filter). Since all sequences are nets, this holds for sequences as well.

Take ${\displaystyle {\mathcal {N}}_{x}}$ to be the neighbourhood filter at point ${\displaystyle x}$ for ${\displaystyle X.}$ This means that ${\displaystyle {\mathcal {N}}_{x}}$ is the set of all topological neighbourhoods of the point ${\displaystyle x.}$ It can be verified that ${\displaystyle {\mathcal {N}}_{x}}$ is a filter. A neighbourhood system is another name for a neighbourhood filter. To say that ${\displaystyle {\mathcal {N}}}$ is a neighbourhood base at ${\displaystyle x}$ for ${\displaystyle X}$ means that each subset ${\displaystyle S}$ of ${\displaystyle X}$ is a neighbourhood of ${\displaystyle x}$ if and only if there exists ${\displaystyle N\in {\mathcal {N}}{\text{ such that }}N\subseteq S.}$ Every neighbourhood base at ${\displaystyle x}$ is a filter base that generates the neighbourhood filter at ${\displaystyle x.}$

To say that a filter base ${\displaystyle B}$ converges to ${\displaystyle x,}$ denoted ${\displaystyle B\to x,}$ means that for every neighbourhood ${\displaystyle U}$ of ${\displaystyle x,}$ there is a ${\displaystyle B_{0}\in B}$ such that ${\displaystyle B_{0}\subseteq U_{0}.}$ In this case, ${\displaystyle x}$ is called a limit of ${\displaystyle B}$ and ${\displaystyle B}$ is called a convergent filter base.

Every neighbourhood base ${\displaystyle N}$ of ${\displaystyle x}$ converges to ${\displaystyle x.}$

• If ${\displaystyle {\mathcal {N}}}$ is a neighbourhood base at ${\displaystyle x}$ and ${\displaystyle C}$ is a filter base on ${\displaystyle X,}$ then ${\displaystyle C\to x}$ if ${\displaystyle C}$ is finer than ${\displaystyle {\mathcal {N}}.}$ If ${\displaystyle {\mathcal {N}}}$ is the upward closed neighborhood filter, then the converse holds as well: any basis of a convergent filter refines the neighborhood filter.
• If ${\displaystyle Y\subseteq X,}$ a point ${\displaystyle p\in X}$ is called a limit point of ${\displaystyle Y}$ in ${\displaystyle X}$ if and only if each neighborhood ${\displaystyle U}$ of ${\displaystyle p}$ in ${\displaystyle X}$ intersects ${\displaystyle Y.}$ This happens if and only if there is a filter base of subsets of ${\displaystyle Y}$ that converges to ${\displaystyle p}$ in ${\displaystyle X.}$

For ${\displaystyle Y\subseteq X,}$ the following are equivalent:

• (i) There exists a filter base ${\displaystyle F}$ whose elements are all contained in ${\displaystyle Y}$ such that ${\displaystyle F\to x.}$
• (ii) There exists a filter ${\displaystyle F}$ such that ${\displaystyle Y}$ is an element of ${\displaystyle F}$ and ${\displaystyle F\to x.}$
• (iii) The point ${\displaystyle x}$ lies in the closure of ${\displaystyle Y.}$

Indeed:

(i) implies (ii): if ${\displaystyle F}$ is a filter base satisfying the properties of (i), then the filter associated to ${\displaystyle F}$ satisfies the properties of (ii).

(ii) implies (iii): if ${\displaystyle U}$ is any open neighborhood of ${\displaystyle x}$ then by the definition of convergence, ${\displaystyle U}$ contains an element of ${\displaystyle F}$; since also ${\displaystyle Y\in F,}$ ${\displaystyle U}$ and ${\displaystyle Y}$ have non-empty intersection.

(iii) implies (i): Define ${\displaystyle F=\left\{U\cap Y:U\in {\mathcal {N}}_{x}\right\}.}$ Then ${\displaystyle F}$ is a filter base satisfying the properties of (i).

A filter base ${\displaystyle B}$ on ${\displaystyle X}$ is said to cluster at ${\displaystyle x}$ (or have ${\displaystyle x}$ as a cluster point) if and only if each element of ${\displaystyle B}$ has non-empty intersection with each neighbourhood of ${\displaystyle x.}$

• If a filter base ${\displaystyle B}$ clusters at ${\displaystyle x}$ and is finer than a filter base ${\displaystyle C,}$ then ${\displaystyle C}$ also clusters at ${\displaystyle x.}$
• Every limit of a filter base is also a cluster point of the base.
• A filter base ${\displaystyle B}$ that has ${\displaystyle x}$ as a cluster point may not converge to ${\displaystyle x.}$ But there is a finer filter base that does. For example, the filter base of finite intersections of sets of the subbase ${\displaystyle B\cap {\mathcal {N}}_{x}.}$

For a filter base ${\displaystyle B,}$ the set ${\displaystyle \cap \{\operatorname {cl} \left(B_{0}\right):B_{0}\in B\}}$ is the set of all cluster points of ${\displaystyle B}$ (the closure of ${\displaystyle B_{0}}$ is ${\displaystyle \operatorname {cl} \left(B_{0}\right).}$ Assume that ${\displaystyle X}$ is a complete lattice.

• The limit inferior of ${\displaystyle B}$ is the infimum of the set of all cluster points of ${\displaystyle B.}$
• The limit superior of ${\displaystyle B}$ is the supremum of the set of all cluster points of ${\displaystyle B.}$
• ${\displaystyle B}$ is a convergent filter base if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the filter base.

If ${\displaystyle X}$ is a topological space then:

• ${\displaystyle X}$ is a Hausdorff space if and only if every filter base on ${\displaystyle X}$ has at most one limit.
• ${\displaystyle X}$ is compact if and only if every filter base on ${\displaystyle X}$ clusters or has a cluster point.
• ${\displaystyle X}$ is compact if and only if every filter base on ${\displaystyle X}$ is a subset of a convergent filter base.
• ${\displaystyle X}$ is compact if and only if every ultrafilter on ${\displaystyle X}$ converges.

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological spaces, let ${\displaystyle A}$ be a filter base on ${\displaystyle X,}$ and let ${\displaystyle f:X\to Y}$ be a function. The image of ${\displaystyle A}$ under ${\displaystyle f,}$ denoted by ${\displaystyle f(A),}$ is defined as the set ${\displaystyle f(A)=\{f(a):a\in A\},}$ which necessarily forms a filter base on ${\displaystyle Y.}$

${\displaystyle f}$ is continuous at ${\displaystyle x\in X}$ if and only if for every filter base ${\displaystyle A}$ on ${\displaystyle X,}$ ${\displaystyle A\to x{\text{ implies }}f(A)\to f(x).}$

Let ${\displaystyle (X,d)}$ be a metric space.

• To say that a filter base ${\displaystyle B}$ on ${\displaystyle X}$ is Cauchy means that for each real number ${\displaystyle r>0,}$ there is a ${\displaystyle B_{0}\in B}$ such that the metric diameter of ${\displaystyle B_{0}}$ is less than ${\displaystyle r.}$
• Take ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ to be a sequence in metric space ${\displaystyle X.}$ Then ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ is a Cauchy sequence if and only if the filter base ${\displaystyle \left\{\,\{x_{N},x_{N+1},\ldots \}\;:\;N\in \mathbb {N} \right\}}$ is Cauchy.

More generally, given a uniform space ${\displaystyle X,}$ a filter ${\displaystyle F}$ on ${\displaystyle X}$ is called a Cauchy filter if for every entourage ${\displaystyle U}$ there is an ${\displaystyle A\in F}$ with ${\displaystyle (x,y)\in U{\text{ for all }}x,y\in A.}$ In a metric space this agrees with the previous definition. ${\displaystyle X}$ is said to be complete if every Cauchy filter converges. Conversely, on a uniform space every convergent filter is a Cauchy filter. Moreover, every cluster point of a Cauchy filter is a limit point.

A compact uniform space is complete: on a compact space each filter has a cluster point, and if the filter is Cauchy, such a cluster point is a limit point. Further, a uniformity is compact if and only if it is complete and totally bounded.

Most generally, a Cauchy space is a set equipped with a class of filters declared to be Cauchy. These are required to have the following properties:

1. for each ${\displaystyle x\in X,}$ the ultrafilter at ${\displaystyle x,}$ ${\displaystyle U(x),}$ is Cauchy.
2. if ${\displaystyle F}$ is a Cauchy filter, and ${\displaystyle F}$ is a subset of a filter ${\displaystyle G,}$ then ${\displaystyle G}$ is Cauchy.
3. if ${\displaystyle F}$ and ${\displaystyle G}$ are Cauchy filters and each member of ${\displaystyle F}$ intersects each member of ${\displaystyle G,}$ then ${\displaystyle F\cap G}$ is Cauchy.

The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.