Topic summary
Filter on a set
In mathematics, a filter on a set is a collection of nonempty subsets which is closed under taking supersets and finite intersections. An example of filter is the collection of neighborhoods of a point in a topological space.
Filters were introduced by Henri Cartan in 1937 in the context of general topological spaces and were subsequently developed by Nicolas Bourbaki in their book Topologie Générale (first edition in 1940) as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. They later found applications in many fields outside of topology, including set theory, mathematical logic, model theory (ultraproducts for example), abstract algebra, and others.
Filters on a set were later generalized to order filters. Specifically, a filter on a set is an order filter on the power set of ordered by inclusion.
The notion dual to a filter is an ideal. Ultrafilters are a particularly important subclass of filters.