Filter (mathematics)

Filter (mathematics)

In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the partially ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in topology whence they originate. The dual notion of a filter is an ideal.

Filters were introduced by Henri Cartan in 1937 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.

Read more about Filter (mathematics):  General Definition, Filter On A Set

Other articles related to "filter":

Filter (mathematics) - Filter On A Set - Filters in Topology - Cauchy Filters
... To say that a filterbase B on X is Cauchy means that for each real number ε>0, there is a B0 ∈ B such that the metric diameter of B0 is less than ε ... (xn) is a Cauchy sequence if and only if the filterbase {{xN,xN+1...} N ∈ {1,2,3...} } is Cauchy ... More generally, given a uniform space X, a filterF on X is called Cauchy filterif for every entourage U there is an A ∈ F with (x,y) ∈ U for all x,y ∈ A ...