Filter (mathematics)


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 order 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.

Motivation

Intuitively, a filter in a partially ordered set, P, is a subset of P 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.
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 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?
Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something 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:
  1. A locating scheme must be non-empty in order to be of any use at all.
  2. If two subsets, E and F, both might contain "what is looked for", then so might their intersection. Thus the filter should be closed with respect to finite intersection.
  3. If a set E might contain "what is looked for", so does every superset of it. Thus the filter is upward-closed.
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 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.

General definition

A subset F of a partially ordered set is a filter if the following conditions hold:
  1. F is non-empty.
  2. For every x, y in F, there is some element z in F such that z ≤ x and z ≤ y.
  3. For every x in F and y in P, x ≤ y implies that y is in F.
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 is a filter, if and only if it is a non-empty upper set that is closed under finite infima, i.e., for all x, y in F, it is also the case that x ∧ y is in F. A subset S of F is a filter basis if the upper set generated by S is all of F.
The smallest filter that contains a given element p is a principal filter and p is a principal element in this situation. The principal filter for p is just given by the set and is denoted by prefixing p with an upward arrow:
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 is to be found in the article on ideals. There is a separate article on ultrafilters.

Filter on a set

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 by subset inclusion, turning into a lattice. Define a filter F on S as a non-empty subset of P with the following properties:
  1. if A and B are in F, then so is their intersection.
  2. If A is in F and A is a subset of B, then B is in F, for all subsets B of S.
With this definition, a filter on a set is indeed a filter.
The second property entails that S is in F.
If the empty set is not in F, we say F is a proper filter. Property 1 implies that a proper filter on a set has the finite intersection property. The only nonproper filter on S is P.
A filter base is a subset B of P with the properties that B is non-empty and the intersection of any two members of B includes a member of B. If the empty set is not a member of B, we say B is a proper filter base.
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 that include 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 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 there is a smallest 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.

Examples

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 in the theory of ultraproducts in model theory, a branch of mathematical logic.

Filters in topology

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.

Neighbourhood bases

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:
implies : if F is a filter base satisfying the properties of, then the filter associated to F satisfies the properties of.
implies : 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 non-empty intersection.
implies : Define. Then F is a filter base satisfying the properties of.

Clustering

Let X be a topological space and x a point of X.
Let X be a topological space.
Let, be topological spaces. Let be a filter base on and be a function. The image of under is defined as the set. The image is denoted and forms a filter base on.
Let be a metric space.
More generally, given a uniform space X, a filter F on X is called a Cauchy filter if for every entourage U there is an A ∈ F with ∈ 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:
  1. for each x in X, the ultrafilter at x, U, is Cauchy.
  2. if F is a Cauchy filter, and F is a subset of a filter G, then G is Cauchy.
  3. if F and G are Cauchy filters and each member of F intersects each member of G, then F ∩ G is Cauchy.
The Cauchy filters on a uniform space have these properties, so every uniform space defines a Cauchy space.