In mathematics, a 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 where they originate. The dual notion of a filter is an 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.

Intuitively, a filter in a partially ordered set (poset), X, is a subset of X that includes as members those elements that are large enough to satisfy some given criterion. For example, if x is an element of the poset, then the set of elements that are above x is a filter, called the principal filter at x. (If x and y are incomparable elements of the poset, then neither of the principal filters at x and y is contained in the other one, and conversely.)

Similarly, a filter on a set contains those subsets that are sufficiently large to contain some given thing. For example, if the set is the real line and x is one of its points, then the family of sets that include x in their interior is a filter, called the filter of neighbourhoods of x. The thing in this case is slightly larger than x, but it still doesn't contain any other specific point of the line.

The above interpretations explain conditions 1 and 3: 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 the condition 2 of the general definition of filter (see below).

For, why should two "large enough" things contain a common "large enough" thing?

Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something (a point or a subset) in the space X, call a filter the collection of subsets of X that might contain "what is looked for". Then this "filter" should possess the following natural structure:

An ultrafilter can be viewed as a "perfect locating scheme" where each subset E of the space X can be used in deciding whether "what is looked for" might lie in E.

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.

A subset F of a partially ordered set (P,≤) is a filter if the following conditions hold:

A filter is proper if it is not equal to the whole set P. This condition is sometimes added to the definition of a filter.

While the above definition is the most general way to define a filter for arbitrary posets, it was originally defined for lattices only. In this case, the above definition can be characterized by the following equivalent statement: A subset F of a lattice (P,≤) is a filter, if and only if it is a nonempty upper set that is closed under finite intersection (infima or meet), i.e., for all x, y in F, then x ∧ y is also in F.^{[3] [666666]}

The dual notion of a filter, i.e. the concept obtained by reversing all ≤ and exchanging ∧ with ∨, is ideal. Because of this duality, the discussion of filters usually boils down to the discussion of ideals. Hence, most additional information on this topic (including the definition of maximal filters and prime filters) is to be found in the article on ideals. There is a separate article on ultrafilters.

A special case of a filter is a filter defined on a set.

Given a set S, a partial ordering ⊆ can be defined on the powerset P (S) by subset inclusion, turning (P (S),⊆) into a lattice. Define a filter F on S as a nonempty subset of P (S) with the following properties:

The second property entails that S is in F (since F is a non-empty subset of P (S)).

If the empty set is not in F, we say F is a proper filter. ^[4]

The first two properties imply that a filter on a set has the finite intersection property. With this definition, a filter on a set is indeed a filter. The only nonproper filter on S is P (S).

A filter base (or filter basis) is a subset B of P (S) with the following properties:

Given a filter base B, the filter generated or spanned by B is defined as the minimum filter containing B. It is the family of all the subsets of S which contain a member of B. Every filter is also a filter base, so the process of passing from filter base to filter may be viewed as a sort of completion.

If B and C are two filter bases on S, one says C is finer than B (or that C is a refinement of B) if for each B0 ∈ B, there is a C0 ∈ C such that C0 ⊆ B0. If also B is finer than C, one says that they are equivalent filter bases.

For every subset T of P (S) there is a smallest (possibly nonproper) filter F containing T, called the filter generated or spanned by T. It is constructed by taking all finite intersections of T, which then form a filter base for F. This filter is proper if and only if every finite intersection of elements of T is non-empty, and in that case we say that T is a filter subbase.

For every filter F on a set S, the set function defined by

is finitely additive — a "measure" if that term is construed rather loosely. Therefore the statement

can be considered somewhat analogous to the statement that φ holds "almost everywhere".

That interpretation of membership in a filter is used (for motivation, although it is not needed for actual proofs) in the theory of ultraproducts in model theory, a branch of mathematical logic.

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 sequence is usually indexed by the natural numbers, which are a totally ordered set. Thus, limits in first-countable spaces can be described by sequences. However, if the space is not first-countable, nets or filters must be used. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. Filters can be thought of as sets built from multiple nets. Therefore, both the limit of a filter and the limit of a net are conceptually the same as the limit of a sequence.

Let X be a topological space and x a point of X.

Let X be a topological space and x a point of X.

Indeed:

(i) implies (ii): if F is a filter base satisfying the properties of (i), then the filter associated to F satisfies the properties of (ii).

(ii) implies (iii): if U is any open neighborhood of x then by the definition of convergence U contains an element of F; since also Y is an element of F, U and Y have nonempty intersection.

Let X be a topological space and x a point of X.

Let X be a topological space.

More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an A ∈ F with (x, y) ∈ U for all x, y ∈ A. In a metric space this agrees with the previous definition. X is said to be complete if every Cauchy filter converges. Conversely, on a uniform space every convergent filter is a Cauchy filter. Moreover, every cluster point of a Cauchy filter is a limit point.

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.

^[13] is apparently the world-best reference on the topic of filters.