Filter (mathematics)

The powerset algebra of the set \{1,2,3,4\} with the upset \uparrow\!\{1\} colored green. The green elements make a principal ultrafilter on the lattice.

In mathematics, a filter is a special subset of a partially ordered set. For example, the power set of some set, partially ordered by set inclusion, is a filter. 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 Garrett Birkhoff in 1935[1] and Henri Cartan in 1937[2][3] 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 on a partially ordered set (poset) contains those elements that are large enough to satisfy some 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. (Notice that 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.)

Similarly, a filter on a set contains those subsets that are sufficiently large to contain something. For example, if the set is the real line and x is one of its points, then the family of sets that contain x in their interior is a filter, called the filter of neighbourhoods of x. (Notice that 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 do not really, without elaboration, explain the condition 2. of the general definition of filter. For, why should two "large enough" things contain a common "large enough" thing? (Note, however, that they do 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".)

Alternatively, a filter can be viewed as a "locating scheme": Suppose we try 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 we are looking for". Then this "filter" should possess the following natural structure: 1. Empty set cannot contain anything so it will not belong to our filter. 2. If two subsets, E and F, both might contain "what we are looking for", then so might their intersection. Thus our filter should be closed with respect to finite intersection. 3. If a set E might contain "what we are looking for", so might any superset of it. Thus our 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 we are looking 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, we will always find something.

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 (P,≤) is a filter if the following conditions hold:

  1. F is nonempty.
  2. 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, or downward directed)
  3. For every x in F and y in P, x ≤ y implies that y is in F. (F is an upper set, or upward closed)

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 an upper set that is closed under finite intersection (infima or meet), 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 \leq 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 nonempty subset of P(S) with the following properties:

  1. S is in F, and if A and B are in F, then so is their intersection. (F is closed under finite intersection)
  2. The empty set is not in F. (F is a proper filter)
  3. 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 upward closed)

The first two 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, the adjective "proper" is generally omitted and considered implicit. 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:

  1. B is non-empty and the intersection of any two sets of B contains a set of B. (B is downward directed)
  2. The empty set is not in B. (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 which contain some set 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 any 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 any finite intersection of elements of T is non-empty, and in that case we say that T is a filter subbase.

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 are 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 contains 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 X, Y be topological spaces. Let A be a filter base on X and f\colon X \to Y be a function. The image of A under f is defined as the set \{ f[a] : a \in A \}. The image is denoted f[A] and 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. ↑ G. Birkhoff, "A new definition of limit", Bull. Amer. Math. Soc., 41, (1935) 636.
  2. ↑ H. Cartan, "Théorie des filtres", CR Acad. Paris, 205, (1937) 595–598.
  3. ↑ H. Cartan, "Filtres et ultrafiltres", CR Acad. Paris, 205, (1937) 777–779.

References

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