Filter (math)
In mathematics , a filter is a nonempty, downwardlooking upperset within a surrounding semiordered 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 semiordered 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.
With an ultrafilter (which is not a main filter) on the natural numbers, the hyperreal numbers of the nonstandard 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 nonempty subset of a quasiorder is called a filter if the following conditions are met:

is an above amount:
 (That is, all (with related) elements that are larger than are part of the filter.)

is directed downwards: and
 (That is, the inverse relation of the partial order considered is directed.)
The filter is called the actual (or real ) filter if it is not the same , but a real subset .
Every filter on a quasi or semiordered set is an element of the power set of . The set of filters defined on the same (weakly) semiordered set is in turn semiordered by the inclusion relation. If and filters are on the same (weakly) semiordered set , it is called finer than coarser than if . A real filter that is as fine as possible is called an ultrafilter .
Filters in associations
While this definition of filter is the most general of any quasi or semiordered sets, filters were originally defined for lattices . In this special case, a filter is a nonempty subset of the lattice that is an above set and closed below finite infima , i.e. H. for everyone is too .
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.
Prime filter
A real filter in an association with the additional property
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 subhalfset in a quasi or partial order.
If one considers the inverse relation in a semiordered set , then there is again a semiordered set. The structure resulting from dualization as noted.
A filter in is an ideal in and vice versa.
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.
example
We consider in the socalled 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.
for .
In the semiordered set , all filters are now given by the zero rays and their open and closed partial rays
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.
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:
 and
for everyone and .
Volume filter
definition
An important special case of a filter  especially in the topology  are quantity filters. In this case one starts from the semiordered power set of any nonempty set due to the set inclusion . A real subset is a quantity filter or filter if and only if the following properties are met:
 and ,
 ,
 .
A quantity filter that applies to
 ,
which therefore contains this or its complement for each subset, is called ultrafilter .
These definitions agree with those given above for real filters in lattices, since the power set of forms a lattice.
Examples of volume filters
 is the name of the main filter created by.
 If a topological space with topology is used , then the environment filter is called by .
 Is an infinite set, then called Fréchet filter the amount .
 Is a nonempty system of sets of having the following properties
 and
 ,
 that's the name of the filter base in . Such a system of quantities naturally creates a filter
 This is the name of the filter produced.
 If there is a mapping between two nonempty sets and a filter , then this is the name of the filter created by the filter base . This is called the image filter of .
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 .
For instance, a mapping between two topological spaces is continuous if for each filter to apply that .
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.
See also
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 3540677909 .
 Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to settheoretical and algebraic topology (= Berlin study series on mathematics. Vol. 15). Heldermann, Lemgo 2006, ISBN 388538115X .
 Lutz Führer: General topology with applications . Vieweg, Braunschweig 1977, ISBN 3528030593 .
 Horst Schubert : Topology . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3519122006 .
Original work
 Henri Cartan: Théorie des filtres. In: Comptes rendus hebdomadaires des séances de l'Académie des Sciences. Volume 205, 1937, ISSN 00014036 , pp. 595598, digitized .
 Henri Cartan: Filtres et ultrafiltres. In: Comptes rendus hebdomadaires des séances de l'Académie des Sciences. Volume 205, 1937, pp. 777779, digitized .
References and comments
 ↑ Cartan: Comptes rendus . tape 205 , p. 595598, 777779 .
 ↑ d. H. a set with a reflexive and transitive relation , also called preorder , weak partial order or weak partial order . In particular, every semiorganized set falls under this condition.
 ↑ ^{a } ^{b } ^{c} Stefan Bold: AD and super compactness , Mathematical Institute of the Rheinische FriedrichWilhelmUniversität Bonn, April 2002, pages 2–3
 ↑ weak semiordered syn. quasiordered
 ↑ Analogous for ideals.
 ↑ Guide: General Topology with Applications. 1977, p. 9.
 ^ Schubert: Topology. 1975, p. 44.