Filter (mathematics)

In mathematics, a filter is a non-empty downward Above amount within a surrounding partially ordered set. The term of the filter goes back to the French mathematician Henri Cartan.

Clearly considered includes a filter items that are too large that they could pass through the filter. If x is a filter element, including each major in the given order relation element y is a filter element, and two filter elements x and y have a common core z, which is itself too large, as that he could pass through the filter.

Filter in the reverse partial order called ideals of order or order ideals.

• 4.1 Definition
• 4.2 Examples of quantity filters
• 4.3 Applications in the topology

• 6.1 Original Articles

Applications

Filter occur in the theory of systems and organizations. An important special case are lot of filters, ie filter in the partially ordered by set inclusion power set of a set. Amount filters are used particularly in the topology and there allow the generalization of the concept of sequence for topological spaces without countable neighborhood base. Thus the system of neighborhoods of a point in a topological space forms a special filter, the filter area. Environment filters can be used in rooms that meet not countable, the definition of networks, which take over the role of the consequences of the elemental analysis partially. It sums up to a filter as directed set up and considered networks on this directed set.

With an ultra-filter ( the filter is not a major ) on the natural numbers can be the hyper- real numbers of nonstandardanalysis construct. However, the existence of such filters is itself only by the axiom of choice - secured - not constructive.

General Definitions

A non-empty subset of a partially ordered set is called filter if the following conditions are met:

• Above is a - lot:

• Is directed downward: and

A filter is called real filter when it is not fully ( not equal) to.

Each filter on a partially ordered set is an element of the power set of. The set of defined on the same partially ordered set filter is partially ordered by the inclusion relation in turn. Are the filter and on the same partially ordered set, then is called finer than coarser than if. A maximum fine of real filter is called an ultrafilter.

Filter in associations

While this definition of filter is the most general for arbitrary partially ordered sets, filters were originally defined for associations. In this special case, a filter is a non-empty subset of the Association, which is an above - volume and closed under finite infima, that is, for all is well.

Main filter

The smallest filter has a predetermined element. Filters of this form are called main filter, and a major element of the filter. The belonging to the main filter is written as.

Primfilter

A real filter in an association with the additional property

Ie Primfilter.

Ideal

Looking in a partially ordered set, the inverse relation, so is again a partially ordered set, as obtained from a ( distributive ) Association by interchanging the two supremum and infimum Verbandsverküpfungen again a ( distributive ) Association. Are in a least element 0 and a greatest element 1 is present, it will also be reversed. In all these cases the structure is thus formed by dualization as quoted.

A filter is called an order ideal or ideal in short.

Example

We consider the so-called punctured complex plane, the subsets of the (open) rays from the zero (short: null rays). Now on, we define a partial order, by considering as less than or equal if and lie on the same ray and on amount is less than or equal. That is,

For.

In the partially ordered set, all the filters are now given by the null rays and their open and closed partial beams

For all with Each of these filters is real. Furthermore follows that fine fine fine; In particular, a maximum real -fine filter, and thus an ultrafilter. For each complex number of the completed beam is their main filter ( the only ) as the main element.

The order ideals corresponding to the missing section of the beams between the zero and the start of each sub- beam. If the partial beam open, he therefore does not contain its start point, so lacking in the corresponding order ideal of the model point - it is analogous to the closed case each contained in sub-beam and ideal. (Filter and order ideal are therefore not disjoint! ) From the zero beam results in no corresponding order ideal, as would be given the "missing" radiation section through the empty set (which may be no filter ). Thus, the ideals have the form:

For all and.

Amount of filter

Definition

An important special case of a filter - especially in the topology - are lot of filters. It is made ​​in this case of the partially ordered by set inclusion power set of an arbitrary non -empty sets. A proper subset is then just a lot of filters or filter if the following properties are satisfied

This definition is consistent with the specification given above for real filter in associations since the power set form of a bandage.

Examples of quantity filters

• Means of the generated main filter.
• Is a topological space with topology, then is called a neighborhood of filters.
• Is an infinite set, then is called the Fréchet filter the crowd.
• Is a non-empty class of sets of the following properties   and
• ,

• Is a mapping between two non- empty sets and a filter, so called the filter generated by the filter base. This is called image filters.

Applications in the topology

In the topology filter and replace the networks there is insufficient for a satisfactory convergence theory consequences. In particular, the filter as a narrowing of sets here have proven to be well suited for convergence measurement. Obtained in this way often analogous sets of sentences about consequences in metric spaces.

If a topological space is called a filter if and only convergent to a if, that is, if finer than the neighborhood filter of, that is, all ( it suffice open ) containing environments. Notation: From the refinement of decompositions is called particularly in the context of theories of integration.

For example, a mapping between two topological spaces is continuous if and only if for each filter with that.

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