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 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[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.

Contents

General definition

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

  1. For every x, y in F, there is some element z in F, such that z ≤ x and z ≤ y. (F is a filter base)
  2. For every x in F and y in P, x ≤ y implies that y is in F. (F is an upper set)
  3. A filter is proper if it is not equal to the whole set P. This is sometimes omitted from 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 non-empty subset F of a lattice (P,≤) is a filter, if and only if it is an upper set that is closed under finite meets (infima), i.e., for all x, y in F, we find that x ∧ y is also in 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 {x in P | p ≤ x} and is denoted by prefixing p with an upward arrow: \uparrow p.

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.

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(S) by subset inclusion, turning (P(S),⊆) into a lattice. Define a filter F on S as a subset of P(S) with the following properties:

  1. S is in F. (F is non-empty)
  2. The empty set is not in F. (F is proper)
  3. If A and B are in F, then so is their intersection. (F is closed under finite meets)
  4. If A is in F and A is a subset of B, then B is in F, for all subsets B of S. (F is an upper set)

The first three properties imply that a filter on a set has the finite intersection property. Note that with this definition, a filter on a set is indeed a filter; in fact, it is a proper filter. Because of this, sometimes this is called a proper filter on a set; however, as long as the set context is clear, the shorter name is sufficient.

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

  1. The intersection of any two sets of B contains a set of B
  2. B is non-empty and the empty set is not in B

Given a filter base B, one may obtain a (proper) filter by including all sets of P(S) which contain a set of B. The resulting filter is said to be generated by or spanned by filter base B. Every filter is a fortiori 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.

Given a subset T of P(S) we can ask whether there exists a smallest filter F containing T. Such a filter exists if and only if the finite intersection of subsets of T is non-empty. We call T a subbase of F and say F is generated by T. F can be constructed by taking all finite intersections of T which is then filter base for F.

Examples

Filters in model theory

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

m(A)= \begin{cases} 1 & \text{if }A\in F \\ 0 & \text{if }S\setminus A\in F \\ \text{undefined} & \text{otherwise} \end{cases}

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

\left\{\,x\in S: \varphi(x)\,\right\}\in F

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.

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 is conceptually the same as the limit of a sequence.

Neighbourhood bases

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

Convergent filter bases

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 is an element of F; since also Y is an element of F, U and Y have nonempty intersection.

(iii) implies (i): Define F = \{ U \cap Y \ | \ U \in N_x \}. Then F is a filter base satisfying the properties of (i).

Clustering

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

Properties of a topological space

Let X be a topological space.

Functions on topological spaces

Let E, X, Y be topological spaces with E \subset X. Let B be a filter base on E and f: E \to Y be a function. The image of B under f is f[B] is the set \{ f(x)�: x \in B \}. The image f[B] forms a filter base on Y.

Cauchy filters

Let (X,d) be a metric 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:

  1. for each x in X, the ultrafilter at x, U(x), 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 (hence every metric space) defines a Cauchy space.

See also

Notes

  1. ^ H. Cartan, "Théorie des filtres". CR Acad. Paris, 205, (1937) 595–598.
  2. ^ H. Cartan, "Filtres et ultrafiltres" CR Acad. Paris, 205, (1937) 777–779.

References