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 neighborhoodss 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 is an order filter on the power set ordered by inclusion.

The notion dual to a filter is an ideal. Ultrafilters are a particularly important subclass of filters.

Definition

, a filter such that:

• is upwards-closed,

• ,.

A (or non-degenerate, consisting of all possible subsets). By upwards-closure, a filter is proper if and only if it does not contain the empty set. Many authors adopt the convention that a filter must be proper by definition.

is said to be coarser (or a subfilter is said to be finer (or or a superfilter).

Examples

• is called the trivial or indiscrete.

• form a principal filter.

• is a topological space, the neighborhood filter or vicinity filter.

Many examples arise from various "largeness" conditions:

• is a set, the set of all cofinite subsets, the Fréchet filter (or cofinite filter).

• is a set, the cocountable subsets (those whose complement is countable) form a filter, the cocountable filter which is finer than the Fréchet filter. More generally, for any infinite cardinal form a filter.

• (those whose complement is bounded set.

• is a complete measure space with the Lebesgue measure), the conull. (For a non-complete measure space, one can take the subsets which, while not necessarily measurable, are contained in a measurable subset of measure zero.)

• is a measure space.

• is a topological space, the comeager, i.e., those whose complement is meager.

• which have a natural density.

• The club filter of a regular uncountable cardinal is the filter of all sets containing a club subset.

• called Kowalsky's filter.

Principal and free filters

The kernel.

is principal (or atomic) is called the fundamental filter (or discrete filter.

consists of the supersets of its kernel. On a finite set, every filter is principal (since the intersection defining the kernel is finite).

A filter is said to be free when it has empty kernel, otherwise it is fixed is an element of the kernel, it is ). is free if and only if it contains the Fréchet filter.

mesh. (the free part (the principal part with any number of elements from the kernel possibly removed.

is countably deep.

Correspondence with order filters

The concept of a filter on a set is a special case of the more general concept of a filter on a partially ordered set. By definition, a filter on a partially ordered set which is upwards-closed) and downwards-directed ordered by inclusion.

Constructions of filters

Intersection of filters

. The intersection is a greatest lower bound with a complete lattice structure.

.

Filter generated by a family of subsets

is said to be a filter subbase.

.

has the finite intersection property is non-empty.

ordered by inclusion, the least upper bound.

is proper.

Filter bases

. A filter base is the upwards closure.

is downwards-directed. is also called a prefilter..

Examples

• , a filter base of the neighborhood filter is known as a neighborhood base is known as a neighborhood subbase. The open, the closed balls.

• is a filter subbase of the Fréchet filter, which consists of the cofinite subsets which contain all points but a finite odd number. In fact, for every base of the Fréchet filter, removing any subset yields another base of the Fréchet filter.

• is a topological space, the dense open subsets, because they are closed under finite intersection. The filter they generate consists of the complements of nowhere dense, restricting to the null dense open subsets yields another filter base for the same filter.

• is a topological space, the countable intersections of dense open subsets form a filter base which generates the filter of comeager subsets.

• be a net is a directed set. The filter base of tails. The generated filter is called the eventuality filter or filter of tails. A sequential filter or is a filter which is the eventuality filter of some net. This example is fundamental in the application of filters in topology.

• Every π-system is a filter base.

Trace of a filter on a subset

, the trace, which is a filter.

Image of a filter by a function

be a function.

is defined as

The image filter. is its upwards closure..

.

Product of filters

-th component. This construction is similar to the product topology.

.

See also

• Axiomatic foundations of topological spaces, for a definition of topological spaces in terms of filters

• Convergence space, a generalization of topological spaces using filters

• Generic filter, a kind of filter used in set-theoretic forcing

Notes

Citations

References

• (Provides an introductory review of filters in topology and in metric spaces.)