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- Filter (mathematics)

In mathematics, a **filter** or **order filter** is a special subset of a partially ordered set. Filters appear in order and lattice theory, but can also be found in topology, from which they originate. The dual notion of a filter is an order ideal.

Filters were introduced by Henri Cartan in 1937^{[1]} ^{[2]} and subsequently used by Bourbaki in their book *Topologie Générale* as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith.

**1.** Intuitively, a filter in a partially ordered set,

*P,*

*P*

*x*

*x*

*x.*

*x*

*y*

*x*

*y*

Similarly, a filter on a set contains those subsets that are sufficiently large to contain some given . For example, if the set is the real line and

*x*

*x*

*x.*

*x,*

The above interpretations explain conditions 1 and 3 in the section General definition: Clearly the empty set is not "large enough", and clearly the collection of "large enough" things should be "upward-closed". However, they do not really, without elaboration, explain condition 2 of the general definition. For, why should two "large enough" things contain a "large enough" thing?

**2.** Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something (a point or a subset) in the space

*X,*

*X*

- A locating scheme must be non-empty in order to be of any use at all.
- If two subsets,

*E*

*F,*

- If a set

*E*

An **ultrafilter** can be viewed as a "perfect locating scheme" where subset

*E*

*X*

*E.*

From this interpretation, **compactness** (see the mathematical characterization below) can be viewed as the property that "no location scheme can end up with nothing", or, to put it another way, "always something will be found".

The mathematical notion of **filter** provides a precise language to treat these situations in a rigorous and general way, which is useful in analysis, general topology and logic.

**3.** A common use for a filter is to define properties that are satisfied by "almost all" elements of some topological space

*X*

*X*

*E\subseteq**X*

*X*

*E*

*F,*

*X*

*E*

*X*

*X\setminus**E*

A subset

*F*

*(P,**\leq)*

*F*

*F*

*x,**y**\in**F,*

*z**\in**F*

*z**\leq**x*

*z**\leq**y.*

*F*

*x**\in**F*

*p**\in**P,*

*x**\leq**p*

*p**\in**F.*

*F*

*F*

*P.*

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In this case, the above definition can be characterized by the following equivalent statement:A subset

*F*

*(P,**\leq)*

*x,**y**\in**F,*

*x**\wedge**y**\in**F.*

*S*

*F*

*S*

*F.*

The smallest filter that contains a given element

*p**\in**P*

*p*

*p*

*\{x**\in**P**:**p**\leq**x\}*

*p*

The dual notion of a filter, that is, the concept obtained by reversing all

*\leq*

*\wedge*

*\vee,*

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

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

Given a set

*S,*

*\subseteq*

*\wp(S)*

*(\wp(S),**\subseteq)*

*S*=*\varnothing*

*S,*

*\wp(S)*=*\{**\varnothing**\}.*

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

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 contain the empty set:

**Note**: This article does require that a filter be proper.

The only non-proper filter on

*S*

*\wp(S).*

- Filter bases and subbases

A subset

*B*

*\wp(S)*

*B*

*B*

*B.*

*B,*

*B*

Given a filter base

*B,*

*B*

*B.*

*S*

*B.*

For every subset

*T*

*\wp(S)*

*F*

*T,*

*T.*

*T*

*T.*

*T,*

*F.*

*T*

*T*

- Finer/equivalent filter bases

If

*B*

*C*

*S,*

*C*

*B*

*C*

*B*

*B*_{0}*\in**B,*

*C*_{0}*\in**C*

*C*_{0}*\subseteq**B*_{0.}

*B*

*C,*

- If

*B*

*C*

*C*

*B*

*C*

*B.*

*B*

*C*

- For filter bases

*A,*

*B,*

*C,*

*A*

*B*

*B*

*C*

*A*

*C.*

- Let

*S*

*C*

*S.*

*\{**C**\}*

*C*

*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

*S*

*S*

*S*

- More generally, if

*(X,\mu)*

*\mu(X)*=inf*ty,*

*A**\subseteq**X*

*\mu(X**\setminus**A)**<*inf*ty*

*X*=*\N*

*\mu*

- Every uniform structure on a set

*X*

*X* x *X.*

- A filter in a poset can be created using the Rasiowa–Sikorski lemma, often used in forcing.
- The set

*\{**\{**N,**N*+1*,**N*+2*,*...*\}**:**N**\in**\N**\}*

*(*1*,*2*,*3*,*...*).*

*(x*_{\alpha)}_{\alpha}

*\{**\{**x*_{\alpha}*:**\alpha**\in**A,**\alpha*_{0}*\leq**\alpha**\}**:**\alpha*_{0}*\in**A**\},*

See also: Filter quantifier.

For every filter

*F*

*S*

*\varphi*

See main article: Filters in topology.

In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space.

In topology and related areas of mathematics, a filter is a generalization of a net. Both nets and filters provide very general contexts to unify the various notions of limit to arbitrary topological spaces.

*\N,*

Throughout,

*X*

*x**\in**X.*

Take

l{N}_{x}

*x*

*X.*

l{N}_{x}

*x.*

l{N}_{x}

l{N}

*x*

*X*

*S*

*X*

*x*

*N**\in*l{N}suchthat*N**\subseteq**S.*

*x*

*x.*

To say that a filter base

*B*

*x,*

*B**\to**x,*

*U*

*x,*

*B*_{0}*\in**B*

*B*_{0}*\subseteq**U*_{0.}

*x*

*B*

*B*

Every neighbourhood base

*N*

*x*

*x.*

- If

l{N}

*x*

*C*

*X,*

*C**\to**x*

*C*

l{N}.

l{N}

- If

*Y**\subseteq**X,*

*p**\in**X*

*Y*

*X*

*U*

*p*

*X*

*Y.*

*Y*

*p*

*X.*

For

*Y**\subseteq**X,*

- (i) There exists a filter base

*F*

*Y*

*F**\to**x.*

- (ii) There exists a filter

*F*

*Y*

*F*

*F**\to**x.*

- (iii) The point

*x*

*Y.*

Indeed:

(i) implies (ii): if

*F*

*F*

(ii) implies (iii): if

*U*

*x*

*U*

*F*

*Y**\in**F,*

*U*

*Y*

(iii) implies (i): Define

*F*=*\left\{**U**\cap**Y**:**U**\in*l{N}_{x}*\right\}.*

*F*

A filter base

*B*

*X*

*x*

*x*

*B*

*x.*

- If a filter base

*B*

*x*

*C,*

*C*

*x.*

- Every limit of a filter base is also a cluster point of the base.
- A filter base

*B*

*x*

*x.*

*B**\cap*l{N}_{x.}

For a filter base

*B,*

*\cap**\{**\operatorname{cl}\left(B*_{0\right)}*:**B*_{0}*\in**B**\}*

*B*

*B*_{0}

*\operatorname{cl}\left(B*_{0\right).}

*X*

- The limit inferior of

*B*

*B.*

- The limit superior of

*B*

*B.*

*B*

If

*X*

*X*

*X*

*X*

*X*

*X*

*X*

*X*

*X*

Let

*X*

*Y*

*A*

*X,*

*f**:**X**\to**Y*

*A*

*f,*

*f(A),*

*f(A)*=*\{**f(a)**:**a**\in**A**\},*

*Y.*

*f*

*x**\in**X*

*A*

*X,*

*A**\to**x*implies*f(A)**\to**f(x).*

Let

*(X,**d)*

- To say that a filter base

*B*

*X*

*r**>*0*,*

*B*_{0}*\in**B*

*B*_{0}

*r.*

- Take

*\left(x*_{n\right)}_{n}

*X.*

*\left(x*_{n\right)}_{n}

*\left\{**\{**x*_{n,}*X*_{N}*,**\ldots**\}* *:* *N**\in**\N**\right\}*

*X,*

*F*

*X*

*U*

*A**\in**F*

*(x,**y)**\in**U*forall*x,**y**\in**A.*

*X*

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:

- for each

*x**\in**X,*

*x,*

*U(x),*

- if

*F*

*F*

*G,*

*G*

- if

*F*

*G*

*F*

*G,*

*F**\cap**G*

- Nicolas Bourbaki, General Topology (Topologie Générale), (Ch. 1-4): Provides a good reference for filters in general topology (Chapter I) and for Cauchy filters in uniform spaces (Chapter II)
- Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981.
*A Course in Universal Algebra.*Springer-Verlag. .- MacIver, David,
*Filters in Analysis and Topology*(2004)*(Provides an introductory review of filters in topology and in metric spaces.)*

- MacIver, David,
- Stephen Willard,
*General Topology*, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.*(Provides an introductory review of filters in topology.)*

- George M. Bergman; Ehud Hrushovski: Linear ultrafilters, Comm. Alg., 26 (1998) 4079–4113.

- H. Cartan, "Théorie des filtres",
*CR Acad. Paris*,**205**, (1937) 595–598. - H. Cartan, "Filtres et ultrafiltres",
*CR Acad. Paris*,**205**, (1937) 777–779. - Igarashi. Ayumi. Zwicker. William S.. 2021-02-16. Fair division of graphs and of tangled cakes. math.CO. 2102.08560.
- Book: B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Introduction to Lattices and Order. Cambridge University Press. Cambridge Mathematical Textbooks. 1990 . 184.