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| [[Image:Upset.svg|thumb|The powerset algebra of the set <math>\{1,2,3,4\}</math> with the upset <math>\uparrow\!\{1\}</math> colored green. The green elements make a ''principal ultrafilter'' on the lattice.]]
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| In [[mathematics]], a '''filter''' is a special [[subset]] of a [[partially ordered set]]. For example, the [[power set]] of some set, partially ordered by set inclusion, is a filter. Filters appear in [[order theory|order]] and [[lattice theory]], but can also be found in [[topology]] whence they originate. The [[duality (order theory)|dual]] notion of a filter is an [[ideal (order theory)|ideal]].
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| Filters were introduced by [[Henri Cartan]] in 1937<ref>H. Cartan, [http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f594.image "Théorie des filtres"], ''CR Acad. Paris'', '''205''', (1937) 595–598.</ref><ref>H. Cartan, [http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image "Filtres et ultrafiltres"], ''CR Acad. Paris'', '''205''', (1937) 777–779.</ref> and subsequently used by [[Bourbaki]] in their book ''[[Topologie Générale]]'' as an alternative to the similar notion of a [[net (topology)|net]] developed in 1922 by [[E. H. Moore]] and [[H. L. Smith]].
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| == Motivation ==
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| Intuitively, a filter on a partially ordered set (''poset'') contains those elements that satisfy some criterion. For example, if ''x'' is an element of the poset, then the set of elements that are above ''x'' is a filter, called the '''principal filter''' at ''x''. (Notice that if ''x'' and ''y'' are incomparable elements of the poset, then neither of the principal filters at ''x'' and ''y'' is contained in the other one.)
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| Similarly, a filter on a set contains those subsets that are sufficiently large to contain ''something''. For example, if the set is the [[real line]] and ''x'' is one of its points, then the family of sets that contain ''x'' in their interior is a filter, called the '''filter of neighbourhoods''' of ''x''. (Notice that the ''thing'' in this case is slightly larger than ''x'', but it still doesn't contain any other specific point of the line.)
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| 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.
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| == General definition ==
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| A subset ''F'' of a partially ordered set (''P'',≤) is a '''filter''' if the following conditions hold:
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| # For every ''x'', ''y'' in ''F'', there is some element ''z'' in ''F'' such that ''z'' ≤ ''x'' and ''z'' ≤ ''y''. (''F'' is a '''filter base''', or downward [[directed set|directed]])
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| # For every ''x'' in ''F'' and ''y'' in ''P'', ''x'' ≤ ''y'' implies that ''y'' is in ''F''. (''F'' is an ''[[upper set]]'', or upward closed)
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| A filter is '''proper''' if it is not equal to the whole set ''P''. This condition is sometimes added to the definition of a filter.
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| While the above definition is the most general way to define a filter for arbitrary [[Partially ordered set|posets]], it was originally defined for [[lattice (order)|lattice]]s only. In this case, the above definition can be characterized by the following equivalent statement:
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| A subset ''F'' of a lattice (''P'',≤) is a filter, [[if and only if]] it is an upper set that is closed under finite meets ([[infimum|infima]]), i.e., for all ''x'', ''y'' in ''F'', we find that ''x'' ∧ ''y'' is also in ''F''.
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| The smallest filter that contains a given element ''p'' is a '''principal filter''' and ''p'' is a '''principal element''' in this situation. The principal filter for ''p'' is just given by the set {''x'' in ''P'' | ''p'' ≤ ''x''} and is denoted by prefixing ''p'' with an upward arrow: <math>\uparrow p</math>.
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| The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ∧ with ∨, is '''ideal'''. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of '''maximal filters''' and '''prime filters''') is to be found in the article on [[ideal (order theory)|ideals]]. There is a separate article on [[ultrafilter]]s.
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| == Filter on a set ==
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| A special case of a filter is a filter defined on a set. Given a set ''S'', a partial ordering ⊆ can be defined on the [[powerset]] '''P'''(''S'') by subset inclusion, turning ('''P'''(''S''),⊆) into a lattice. Define a '''filter''' ''F'' on ''S'' as a subset of '''P'''(''S'') with the following properties:
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| # ''S'' is in ''F'', and if ''A'' and ''B'' are in ''F'', then so is their intersection. (''F is closed under finite meets'')
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| # The empty set is not in ''F''. (''F is a proper filter'')
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| # If ''A'' is in ''F'' and ''A'' is a subset of ''B'', then ''B'' is in ''F'', for all subsets ''B'' of ''S''. (''F is upward closed'')
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| The first two properties imply that a '''filter on a set''' has the [[finite intersection property]]. Note that with this definition, a filter on a set is indeed a filter; in fact, it is a proper filter. Because of this, sometimes this is called a '''proper filter on a set'''; however, the adjective "proper" is generally omitted and considered implicit. The only nonproper filter on ''S'' is '''P'''(''S'').
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| A '''filter base''' (or '''filter basis''') is a subset ''B'' of '''P'''(''S'') with the following properties:
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| # ''B'' is non-empty and the intersection of any two sets of ''B'' contains a set of ''B''. (''B is downward directed'')
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| # The empty set is not in ''B''. (''B is a proper filter base'')
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| Given a filter base ''B'', the filter generated or spanned by ''B'' is defined as the minimum filter containing ''B''. It is the family of all the subsets of ''S'' which contain some set of ''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.
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| If ''B'' and ''C'' are two filter bases on ''S'', one says ''C'' is '''finer''' than ''B'' (or that ''C'' is a '''refinement''' of ''B'') if for each ''B''<sub>0</sub> ∈ ''B'', there is a ''C''<sub>0</sub> ∈ ''C'' such that ''C''<sub>0</sub> ⊆ ''B''<sub>0</sub>. If also ''B'' is finer than ''C'', one says that they are '''equivalent filter bases'''.
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| * If ''B'' and ''C'' are filter bases, then ''C'' is finer than ''B'' if and only if the filter spanned by ''C'' contains the filter spanned by ''B''. Therefore, ''B'' and ''C'' are equivalent filter bases if and only if they generate the same filter.
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| * For filter bases ''A'', ''B'', and ''C'', if ''A'' is finer than ''B'' and ''B'' is finer than ''C'' then ''A'' is finer than ''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.
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| For any subset ''T'' of '''P'''(''S'') there is a smallest (possibly nonproper) filter ''F'' containing ''T'', called the filter generated or spanned by ''T''. It is constructed by taking all finite intersections of ''T'', which then form a filter base for ''F''. This filter is proper if and only if any finite intersection of elements of ''T'' is non-empty, and in that case we say that ''T'' is a '''filter subbase'''.
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| === Examples ===
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| * Let ''S'' be a nonempty set and ''C'' be a nonempty subset. Then <math>\{ C \}</math> is a filter base. The filter it generates (i.e., the collection of all subsets containing ''C'') is called the '''principal filter''' generated by ''C''.
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| * A filter is said to be a '''free filter''' if the intersection of all of its members is empty. A principal filter is not free. Since the intersection of any finite number of members of a filter is also a member, no 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.
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| * The [[Fréchet filter]] on an infinite set ''S'' is the set of all subsets of ''S'' that have finite complement. A filter on ''S'' is free if and only if it contains the Fréchet filter.
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| * Every [[uniform structure]] on a set ''X'' is a filter on ''X''×''X''.
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| * A filter in a [[poset]] can be created using the [[Rasiowa-Sikorski lemma]], often used in [[forcing (mathematics)|forcing]].
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| * The set <math>\{ \{ N, N+1, N+2, \dots \} : N \in \{1,2,3,\dots\} \}</math> is called a ''filter base of tails'' of the sequence of natural numbers <math>(1,2,3,\dots)</math>. A filter base of tails can be made of any [[net (mathematics)|net]] <math>(x_\alpha)_{\alpha \in A}</math> using the construction <math>\{ \{ x_\alpha : \alpha \in A, \alpha_0 \leq a \} : \alpha_0 \in A \}\,</math>. Therefore, all nets generate a filter base (and therefore a filter). Since all sequences are nets, this holds for sequences as well.
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| === Filters in model theory ===
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| For any filter ''F'' on a set ''S'', the set function defined by
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| :<math>
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| m(A)=
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| \begin{cases}
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| 1 & \text{if }A\in F \\
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| 0 & \text{if }S\setminus A\in F \\
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| \text{undefined} & \text{otherwise}
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| \end{cases}
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| </math>
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| is finitely additive — a "[[measure (mathematics)|measure]]" if that term is construed rather loosely. Therefore the statement
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| :<math>\left\{\,x\in S: \varphi(x)\,\right\}\in F</math>
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| can be considered somewhat analogous to the statement that φ holds "almost everywhere". That interpretation of membership in a filter is used (for motivation, although it is not needed for actual ''proofs'') in the theory of [[ultraproduct]]s in [[model theory]], a branch of [[mathematical logic]].
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| === Filters in topology ===
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| In [[topology]] and analysis, filters are used to define convergence in a manner similar to the role of [[sequence]]s in a [[metric space]].
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| In topology and related areas of mathematics, a filter is a generalization of a [[net (mathematics)|net]]. Both nets and filters provide very general contexts to unify the various notions of [[Limit (mathematics)|limit]] to arbitrary [[topological space]]s.
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| A [[sequence]] is usually indexed by the [[natural numbers]], which are a [[totally ordered set]]. Thus, limits in [[first-countable space]]s can be described by sequences. However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a [[directed set]]. Filters can be thought of as sets built from multiple nets. Therefore, both the limit of a filter and the limit of a net are conceptually the same as the limit of a sequence.<!--An advantage to using filters is that many results can be shown without using the [[axiom of choice]].
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| This sentence is very misleading. Whenever invoking the ultrafilter lemma, you are essentially using the axiom of choice (if my understanding is correct.) Besides, regardless of the use of filter, you can't avoid ac to prove, say, Tychonoff's theorem. TakuyaMurata -->
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| ====Neighbourhood bases====
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| Let ''X'' be a topological space and ''x'' a point of ''X''.
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| * Take ''N''<sub>''x''</sub> to be the '''[[Neighbourhood system|neighbourhood filter]]''' at point ''x'' for ''X''. This means that ''N''<sub>''x''</sub> is the set of all topological [[neighbourhood (mathematics)|neighbourhood]]s of the point ''x''. It can be verified that ''N''<sub>''x''</sub> is a filter. A '''neighbourhood system''' is another name for a '''neighbourhood filter'''.
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| * To say that ''N'' is a '''neighbourhood base''' at ''x'' for ''X'' means that each subset ''V''<sub>0</sub> of X is a neighbourhood of ''x'' if and only if there exists ''N''<sub>0</sub> ∈ ''N'' such that ''N''<sub>0</sub> ⊆ ''V''<sub>0</sub>. Note that every neighbourhood base at ''x'' is a filter base that generates the neighbourhood filter at ''x''.
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| ====Convergent filter bases====
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| Let ''X'' be a topological space and ''x'' a point of ''X''.
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| * To say that a filter base ''B'' '''converges''' to ''x'', denoted ''B'' → ''x'', means that for every neighbourhood ''U'' of ''x'', there is a ''B''<sub>0</sub> ∈ ''B'' such that ''B''<sub>0</sub> ⊆ ''U''. In this case, ''x'' is called a [[Limit (mathematics)|limit]] of ''B'' and ''B'' is called a '''convergent filter base'''.
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| * Every neighbourhood base ''N'' of ''x'' converges to ''x''.
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| ** If ''N'' is a neighbourhood base at ''x'' and ''C'' is a filter base on ''X'', then ''C'' → ''x'' [[if and only if]] ''C'' is finer than ''N''.
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| ** If ''Y'' ⊆ ''X'', a point ''p ∈ X'' is called a '''limit point''' of ''Y'' in ''X'' if and only if each neighborhood ''U'' of ''p'' in ''X'' intersects ''Y''. This happens if and only if there is a filter base of subsets of ''Y'' that converges to ''p'' in ''X''.
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| * For ''Y'' ⊆ ''X'', the following are equivalent:
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| ** (i) There exists a filter base ''F'' whose elements are all contained in ''Y'' such that ''F'' → ''x''.
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| ** (ii) There exists a filter ''F'' such that ''Y'' is an element of ''F'' and ''F'' → ''x''.
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| ** (iii) The point ''x'' lies in the closure of ''Y''.
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| Indeed:
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| (i) implies (ii): if ''F'' is a filter base satisfying the properties of (i), then the filter associated to ''F'' satisfies the properties of (ii).
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| (ii) implies (iii): if ''U'' is any open neighborhood of ''x'' then by the definition of convergence ''U'' contains an element of ''F''; since also ''Y'' is an element of ''F'',
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| ''U'' and ''Y'' have nonempty intersection.
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| (iii) implies (i): Define <math> F = \{ U \cap Y \ | \ U \in N_x \}</math>. Then ''F'' is a filter base satisfying the properties of (i).
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| ====Clustering====
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| Let ''X'' be a topological space and ''x'' a point of ''X''.
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| * A filter base ''B'' on ''X'' is said to '''cluster''' at ''x'' (or have ''x'' as a [[cluster point]]) if and only if each element of ''B'' has nonempty intersection with each neighbourhood of ''x''.
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| ** If a filter base ''B'' clusters at ''x'' and is finer than a filter base ''C'', then ''C'' clusters at ''x'' too.
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| ** Every limit of a filter base is also a cluster point of the base.
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| ** A filter base ''B'' that has ''x'' as a cluster point may not converge to ''x''. But there is a finer filter base that does. For example the filter base of finite intersections of sets of the subbase <math>B\cup N_x</math>.
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| ** For a filter base ''B'', the set ∩{cl(''B''<sub>0</sub>) : ''B''<sub>0</sub>∈''B''} is the set of all cluster points of ''B'' (note: cl(''B''<sub>0</sub>) is the [[closure (topology)|closure]] of ''B''<sub>0</sub>). Assume that ''X'' is a [[complete lattice]].
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| *** The [[limit inferior]] of ''B'' is the [[infimum]] of the set of all cluster points of ''B''.
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| *** The [[limit superior]] of ''B'' is the [[supremum]] of the set of all cluster points of ''B''.
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| *** ''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.
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| ====Properties of a topological space====
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| Let ''X'' be a topological space.
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| * ''X'' is a [[Hausdorff space]] [[if and only if]] every filter base on ''X'' has at most one limit.
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| * ''X'' is [[Compact space|compact]] if and only if every filter base on ''X'' clusters.
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| * ''X'' is compact if and only if every filter base on ''X'' is a subset of a convergent filter base.
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| * ''X'' is compact if and only if every [[ultrafilter]] on ''X'' converges.
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| ====Functions on topological spaces====
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| Let <math>X</math>, <math>Y</math> be topological spaces. Let <math>B</math> be a filter base on <math>X</math> and <math>f\colon X \to Y</math> be a function. The [[Image (mathematics)|image]] of <math>B</math> under <math>f</math> is <math>f[B]</math> is the set <math>\{ f[x] : x \in B \}</math>. The image <math>f[B]</math> forms a filter base on <math>Y</math>.
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| * <math>f</math> is [[Continuous function (topology)|continuous]] at <math>x</math> if and only if <math>B \to x</math> implies <math>f[B] \to f(x)</math>.
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| ==== Cauchy filters ====
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| Let <math>(X,d)</math> be a [[metric space]].
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| * To say that a filter base ''B'' on ''X'' is '''Cauchy''' means that for each [[real number]] ε>0, there is a ''B''<sub>0</sub> ∈ ''B'' such that the metric [[diameter]] of ''B''<sub>0</sub> is less than ε.
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| * Take (''x<sub>n</sub>'') to be a [[sequence]] in metric space ''X''. (''x<sub>n</sub>'') is a [[Cauchy sequence]] if and only if the filter base <nowiki>{{</nowiki>''x<sub>N</sub>,x<sub>N''+1</sub>,...} : ''N'' ∈ {1,2,3,...} } is Cauchy.
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| More generally, given a [[uniform space]] ''X'', a filter ''F'' on ''X'' is called '''Cauchy filter''' if for every [[entourage (topology)|entourage]] ''U'' there is an ''A'' ∈ ''F'' with (''x,y'') ∈ ''U'' for all ''x,y'' ∈ ''A''. In a metric space this agrees with the previous definition. ''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.
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| 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]].
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| 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:
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| # for each ''x'' in ''X'', the [[ultrafilter]] at ''x'', ''U''(''x''), is Cauchy.
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| # if ''F'' is a Cauchy filter, and ''F'' is a subset of a filter ''G'', then ''G'' is Cauchy.
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| # if ''F'' and ''G'' are Cauchy filters and each member of ''F'' intersects each member of ''G'', then ''F'' ∩ ''G'' is Cauchy.
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| The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.
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| == See also ==
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| * [[Ultrafilter]]
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| * [[Filtration (mathematics)]]
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| * [[Net (mathematics)]]
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| * [[Generic filter]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| *[[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)
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| * Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. ''(Provides an introductory review of filters in topology.)''
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| *David MacIver, ''[http://www.efnet-math.org/~david/mathematics/filters.pdf Filters in Analysis and Topology]'' (2004) ''(Provides an introductory review of filters in topology and in metric spaces.)''
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| * Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. ISBN 3-540-90578-2.
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| * [[Victor Porton]]. ''[http://ijpam.eu/contents/2012-74-1/6/6.pdf Filters on Posets and Generalizations]'' (2012). [http://ijpam.eu IJPAM]
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| [[Category:Order theory]]
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| [[Category:General topology]]
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