Filter (math)

In mathematics , a filter is a non-empty, downward-looking upper-set within a surrounding semi-ordered set . The term filter goes back to the French mathematician Henri Cartan .

In visual terms, a filter contains elements that are too large to pass through the filter . If x is a filter element, then every element y which is larger in the given order relation is also a filter element, and two filter elements x and y each have a common core z , which itself is already too big to pass through the filter .

Filters in the reverse partial order are called ideals of order or order ideals .

Applications

Filters appear in the theory of orders and associations . An important special case are volume filters, i. H. Filter in the power set of a set that is semi-ordered by the set inclusion . Set filters are used particularly in topology and allow the concept of sequence to be generalized for topological spaces without a countable environment base . The system of the surroundings of a point in a topological space forms a special filter, the environment filter . Environment filters can be used in spaces that do not satisfy any countability axiom to define networks that partially take on the role of the sequences from elementary analysis . To do this, one understands a filter as a directed set and considers networks on this directed set. ${\ displaystyle {\ mathcal {U}} (x)}$${\ displaystyle x}$

With an ultrafilter (which is not a main filter) on the natural numbers, the hyperreal numbers of the non-standard analysis can be constructed . However, the existence of such filters themselves is only  secured by the axiom of choice - that is, not constructively.

General definitions

A non-empty subset of a quasi-order is called a filter if the following conditions are met: ${\ displaystyle F}$ ${\ displaystyle {\ varvec {P}} = (P, \ leq)}$

1. ${\ displaystyle F}$ is an above amount: ${\ displaystyle \ forall x \ in F: \ forall y \ in P \ colon x \ leq y \ Rightarrow y \ in F,}$
(That is, all (with related) elements that are larger than are part of the filter.)${\ displaystyle x}$${\ displaystyle x}$
2. ${\ displaystyle F}$is directed downwards: and${\ displaystyle \ forall x, y \ in F \ \ exists z \ in F \ colon z \ leq x}$${\ displaystyle z \ leq y.}$
(That  is, the inverse relation of the partial order considered is directed.)${\ displaystyle F}$

The filter is called the actual (or real ) filter if it is not the same , but a real subset . ${\ displaystyle F}$${\ displaystyle P}$${\ displaystyle F \ subset P}$

Every filter on a quasi or semi-ordered set is an element of the power set of . The set of filters defined on the same (weakly) semi-ordered set is in turn semi-ordered by the inclusion relation. If and filters are on the same (weakly) semi-ordered set , it is called finer than coarser than if . A real filter that is as fine as possible is called an ultrafilter . ${\ displaystyle P}$${\ displaystyle P}$${\ displaystyle \ subseteq}$${\ displaystyle F_ {1}}$${\ displaystyle F_ {2}}$${\ displaystyle P}$${\ displaystyle F_ {2}}$ ${\ displaystyle F_ {1}}$ ${\ displaystyle (F_ {1}}$ ${\ displaystyle F_ {2}),}$${\ displaystyle F_ {1} \ subseteq F_ {2}}$

Filters in associations

While this definition of filter is the most general of any quasi or semi-ordered sets, filters were originally defined for lattices . In this special case, a filter is a non-empty subset of the lattice that is an above set and closed below finite infima , i.e. H. for everyone is too . ${\ displaystyle F}$${\ displaystyle (P, \ leq)}$${\ displaystyle x, y \ in F}$${\ displaystyle x \ wedge y \ in F}$

Main filter

The smallest filter that contains a given element is . Filters of this form are called main filters , and are a main element of the filter. The main filter belonging to it is written as. ${\ displaystyle p}$${\ displaystyle \ {x \ in P \ mid p \ leq x \}}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle \ operatorname {\ uparrow} p}$

Prime filter

A real filter in an association with the additional property ${\ displaystyle F}$${\ displaystyle P}$

${\ displaystyle a \ vee b \ in F \ iff (a \ in F \; \ mathrm {or} \; b \ in F)}$

is called a prime filter .

Ideals

The term dual to the filter is that of the ideal : An ideal (also order ideal ) is a directed sub-half-set in a quasi or partial order.

If one considers the inverse relation in a semi-ordered set , then there is again a semi-ordered set. The structure resulting from dualization as noted. ${\ displaystyle {\ varvec {P}} = (P, \ leq)}$ ${\ displaystyle \ leq ^ {- 1} = {\ geq}}$${\ displaystyle (P, \ geq)}$${\ displaystyle {\ boldsymbol {P}} ^ {\ text {opp}} = (P, \ geq)}$

A filter in is an ideal in and vice versa. ${\ displaystyle {\ boldsymbol {P}} ^ {\ text {opp}}}$${\ displaystyle {\ boldsymbol {P}}}$

Likewise, a (distributive) lattice can be converted into a (distributive) lattice by swapping the two association links Supremum and Infimum . If there is a smallest element 0 and a largest element 1, they are also swapped. ${\ displaystyle (P, \ vee, \ wedge)}$ ${\ displaystyle \ vee}$ ${\ displaystyle \ wedge}$${\ displaystyle P}$

example

We consider in the so-called dotted complex plane the subsets for the (open) rays from zero (short: zero rays). On we now define a partial order by considering it as less than or equal if and are on the same ray and is less than or equal in terms of amount . I.e. ${\ displaystyle \ mathbb {C} ^ {\ times}: = \ mathbb {C} {\ setminus} \ {0 \}}$${\ displaystyle s _ {\ alpha} = \ {z \ in \ mathbb {C} ^ {\ times} \ mid \ operatorname {Arg} (z) = \ alpha \},}$${\ displaystyle 0 \ leq \ alpha <2 \ pi,}$${\ displaystyle \ mathbb {C} ^ {\ times}}$${\ displaystyle \ trianglelefteq}$${\ displaystyle z_ {1} \ in \ mathbb {C} ^ {\ times}}$${\ displaystyle z_ {2} \ in \ mathbb {C} ^ {\ times}}$${\ displaystyle z_ {1}}$${\ displaystyle z_ {2}}$${\ displaystyle z_ {1}}$${\ displaystyle z_ {2}}$

{\ displaystyle {\ begin {aligned} z_ {1} \ trianglelefteq z_ {2} &: \ Leftrightarrow & \ operatorname {Arg} (z_ {1}) = \ operatorname {Arg} (z_ {2}) & \ \ \ mathrm {and} & \ left | z_ {1} \ right | \ leq \ left | z_ {2} \ right | \ end {aligned}}}

for . ${\ displaystyle z_ {1}, z_ {2} \ in \ mathbb {C} ^ {\ times}}$

In the semi-ordered set , all filters are now given by the zero rays and their open and closed partial rays ${\ displaystyle \ left (\ mathbb {C} ^ {\ times}, \ trianglelefteq \ right)}$

${\ displaystyle s (z): = \ {z '\ in \ mathbb {C} ^ {\ times} \ mid z \ trianglelefteq z', z \ neq z '\} \ subset {\ bar {s}} ( z): = \ {z '\ in \ mathbb {C} ^ {\ times} \ mid z \ trianglelefteq z' \} \ subset s _ {\ alpha}}$

for everyone with each of these filters is real. It also follows that finer finer finer ; in particular is a maximum fine real filter and thus an ultrafilter. For every complex number , the closed ray is its main filter with its (only) main element. ${\ displaystyle z \ in \ mathbb {C} ^ {\ times}}$${\ displaystyle \ alpha = \ operatorname {Arg} (z).}$${\ displaystyle z_ {1} \ trianglelefteq z_ {2}}$${\ displaystyle {\ bar {s}} (z_ {1})}$${\ displaystyle s (z_ {1})}$${\ displaystyle {\ bar {s}} (z_ {2})}$${\ displaystyle s (z_ {2})}$${\ displaystyle s _ {\ alpha} \ (0 \ leq \ alpha <2 \ pi)}$${\ displaystyle z \ in \ mathbb {C} ^ {\ times}}$${\ displaystyle {\ bar {s}} (z)}$${\ displaystyle \ operatorname {\ uparrow} z}$${\ displaystyle z}$

The order ideals in correspond to the missing beam sections between the zero and the beginning of each partial beam. If the partial beam is open, i.e. it does not contain its incident point, the incident point is also missing in the corresponding order ideal - analogously in the closed case it is contained in the partial beam and ideal. (Filter and order ideal are therefore not disjoint !) There is no corresponding order ideal from the zero ray, since the “missing” ray section would be given by the empty set (which cannot be a filter). So the ideals have the form: ${\ displaystyle \ left (\ mathbb {C} ^ {\ times}, \ trianglelefteq \ right)}$

${\ displaystyle s ^ {{-} 1} (z) = (s _ {\ alpha} {\ setminus} s (z)) \ setminus \ {z \} = \ {z '\ in \ mathbb {C} ^ {\ times} \ mid z \ trianglerighteq z ', z \ neq z' \}}$ and
${\ displaystyle {\ bar {s}} ^ {{-} 1} (z) = (s _ {\ alpha} {\ setminus} {\ bar {s}} (z)) \ cup \ {z \} = \ {z '\ in \ mathbb {C} ^ {\ times} \ mid z \ trianglerighteq z' \}}$

for everyone and . ${\ displaystyle z \ in \ mathbb {C} ^ {\ times}}$${\ displaystyle \ alpha = \ operatorname {Arg} (z)}$

Volume filter

definition

An important special case of a filter - especially in the topology - are quantity filters. In this case one starts from the semi-ordered power set of any non-empty set due to the set inclusion . A real subset is a quantity filter or filter if and only if the following properties are met: ${\ displaystyle \ left ({\ mathcal {P}} (X), \ subseteq \ right)}$${\ displaystyle X}$${\ displaystyle {\ mathcal {F}} \ subset {\ mathcal {P}} (X)}$

1. ${\ displaystyle \ emptyset \ notin {\ mathcal {F}}}$and ,${\ displaystyle X \ in {\ mathcal {F}}}$
2. ${\ displaystyle F, G \ in {\ mathcal {F}} \ \ Rightarrow \ F \ cap G \ in {\ mathcal {F}}}$,
3. ${\ displaystyle F \ in {\ mathcal {F}}, \; G \ supset F \ \ Rightarrow \ G \ in {\ mathcal {F}}}$.

A quantity filter that applies to

${\ displaystyle F \ subseteq X \ Rightarrow F \ in {\ mathcal {F}} \ lor X \! \ setminus \! {F} \ in {\ mathcal {F}}}$,

which therefore contains this or its complement for each subset, is called ultrafilter . ${\ displaystyle {\ mathcal {P}}}$

These definitions agree with those given above for real filters in lattices, since the power set of forms a lattice. ${\ displaystyle X}$

Examples of volume filters

• ${\ displaystyle {\ mathcal {F}} _ {C}: = \ {M \ subseteq X \ mid C \ subseteq M \}}$is the name of the main filter created by.${\ displaystyle C \ subseteq X}$
• If a topological space with topology is used , then the environment filter is called by .${\ displaystyle (X, \ tau)}$${\ displaystyle \ tau}$${\ displaystyle {\ mathcal {U}} (x): = \ left \ {U \ subseteq X \ mid \ exists O \ in \ tau \ colon O \ subseteq U \ land x \ in O \ right \}}$${\ displaystyle x}$
• Is an infinite set, then called Fréchet filter the amount .${\ displaystyle S}$${\ displaystyle \ {M \ subseteq S \ mid S \ setminus M {\ text {finite}} \}}$ ${\ displaystyle S}$
• Is a non-empty system of sets of having the following properties ${\ displaystyle {\ mathcal {B}}}$${\ displaystyle {\ mathcal {P}} (X)}$
1. ${\ displaystyle \ emptyset \ notin {\ mathcal {B}}}$ and
2. ${\ displaystyle \ forall B_ {1}, B_ {2} \ in {\ mathcal {B}} \ \ exists B_ {3} \ in {\ mathcal {B}} \ colon B_ {3} \ subseteq B_ {1 } \ cap B_ {2}}$,
that's the name of the filter base in . Such a system of quantities naturally creates a filter ${\ displaystyle {\ mathcal {B}}}$ ${\ displaystyle X}$
${\ displaystyle {\ mathcal {F}} _ {\ mathcal {B}}: = \ langle {\ mathcal {B}} \ rangle: = \ left \ {M \ subseteq X \ mid \ exists B \ in {\ mathcal {B}} \ colon B \ subseteq M \ right \}}$
This is the name of the filter produced.${\ displaystyle {\ mathcal {B}}}$
• If there is a mapping between two non-empty sets and a filter , then this is the name of the filter created by the filter base . This is called the image filter of .${\ displaystyle f \ colon X \ rightarrow Y}$${\ displaystyle {\ mathcal {F}}}$${\ displaystyle X}$${\ displaystyle f ({\ mathcal {F}})}$${\ displaystyle \ {B \ subseteq Y \ mid \ exists F \ in {\ mathcal {F}} \ colon f (F) = B \}}$${\ displaystyle f}$

Applications in topology

In the topology, filters and networks replace the consequences that are insufficient for a satisfactory convergence theory . In particular, the filters as constricting quantity systems have proven to be well suited for convergence measurement. In this way one often receives analogous sentences to sentences about sequences in metric spaces .

If a topological space, then a filter is called convergent to a if and only if , i. i.e. , if is finer than the environmental filter of , i. H. all (open) environments of contains. Notation: One speaks of the refinement of decompositions especially in connection with integration theories . ${\ displaystyle (X, \ tau)}$${\ displaystyle {\ mathcal {F}}}$${\ displaystyle x \ in X}$${\ displaystyle {\ mathcal {U}} (x) \ subseteq {\ mathcal {F}}}$${\ displaystyle {\ mathcal {F}}}$ ${\ displaystyle {\ mathcal {U}} (x)}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle {\ mathcal {F}} \ rightarrow x.}$

For instance, a mapping between two topological spaces is continuous if for each filter to apply that . ${\ displaystyle f \ colon X \ rightarrow Y}$${\ displaystyle {\ mathcal {F}}}$${\ displaystyle {\ mathcal {F}} \ rightarrow x}$${\ displaystyle f ({\ mathcal {F}}) \ rightarrow f (x)}$

In a non- Hausdorff space , a filter can converge to several points . Hausdorff spaces can even be characterized precisely by the fact that there is no filter in them which converges to two different points.

literature

On the general, order and association theoretical concept formations and their applications: On the applications in the set theoretical topology:

• Boto von Querenburg : Set theoretical topology. 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .
• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (= Berlin study series on mathematics. Vol. 15). Heldermann, Lemgo 2006, ISBN 3-88538-115-X .
• Lutz Führer: General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Horst Schubert : Topology . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .

Original work

• Henri Cartan: Théorie des filtres. In: Comptes rendus hebdomadaires des séances de l'Académie des Sciences. Volume 205, 1937, , pp. 595-598, digitized .
• Henri Cartan: Filtres et ultrafiltres. In: Comptes rendus hebdomadaires des séances de l'Académie des Sciences. Volume 205, 1937, pp. 777-779, digitized .

2. d. H. a set with a reflexive and transitive relation , also called preorder , weak partial order or weak partial order . In particular, every semi-organized set falls under this condition.${\ displaystyle P}$${\ displaystyle \ leq}$${\ displaystyle \ leq}$