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.

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 common "large enough" thing?

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.

There are two competing definitions of a "filter on a set," both of which require that a filter be a dual ideal.[5] One definition defines "filter" as a synonym of "dual ideal" while the other defines "filter" to mean a dual ideal that is also proper.

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

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 not contain the empty set:

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.

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:

The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.