Filter (mathematics)

The powerset lattice of the set {1,2,3,4}, with the upper set ↑{1,4} colored yellow. It is a principal filter, but not an ultrafilter, as it can be extended to the larger nontrivial filter ↑{1}, by including also the light green elements. Since ↑{1} cannot be extended any further, it is an ultrafilter.

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 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.

Motivation

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 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, and conversely.)

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 include 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 (see below). 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. The 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 (see below), 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 xy 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 ${\displaystyle \{x\in P\ |\ p\leq x\}}$ and is denoted by prefixing p with an upward arrow: ${\displaystyle \uparrow p}$.

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. 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)

If the empty set is not in F, we say F is a proper filter. [3]

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:

1. B is non-empty and the intersection of any two members of B contains a member of B (B is downward directed).
2. The empty set is not a member of 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 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 B0B, there is a C0C such that C0B0. If also B is finer than C, one says that they are equivalent filter bases.

• If B and C are filter bases, then C is finer than B if and only if the filter spanned by C contains the filter spanned by B. Therefore, B and C are equivalent filter bases if and only if they generate the same filter.
• For filter bases A, B, and C, if A is finer than B and B is finer than C then A is finer than C. Thus the refinement relation is a preorder on the set of filter bases, and the passage from filter base to filter is an instance of passing from a preordering to the associated partial ordering.

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

• Let S be a nonempty set and C be a nonempty subset of S. Then ${\displaystyle \{C\}}$ is a filter base. The filter it generates (i.e., the collection of all subsets containing C) is called the principal filter generated by C.
• A filter is said to be a free filter if the intersection of all of its members is empty. A principal filter is not free. Since the intersection of any finite number of members of a filter is also a member, no filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members. A nonprincipal filter on an infinite set is not necessarily free.
• The Fréchet filter on an infinite set S is the set of all subsets of S that have finite complement. A filter on S is free if and only if it contains the Fréchet filter.
• Every uniform structure on a set X is a filter on X×X.
• A filter in a poset can be created using the Rasiowa-Sikorski lemma, often used in forcing.
• The set ${\displaystyle \{\{N,N+1,N+2,\dots \}:N\in \{1,2,3,\dots \}\}}$ is called a filter base of tails of the sequence of natural numbers ${\displaystyle (1,2,3,\dots )}$. A filter base of tails can be made of any net ${\displaystyle (x_{\alpha })_{\alpha \in A}}$ using the construction ${\displaystyle \{\{x_{\alpha }:\alpha \in A,\alpha _{0}\leq \alpha \}:\alpha _{0}\in A\}}$ where the filter that this filter base generates is called the net's eventuality filter. Therefore, all nets generate a filter base (and therefore a filter). Since all sequences are nets, this holds for sequences as well.

Filters in model theory

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

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

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

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

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.

• Take Nx to be the neighbourhood filter at point x for X. This means that Nx is the set of all topological neighbourhoods of the point x. It can be verified that Nx is a filter. A neighbourhood system is another name for a neighbourhood filter.
• To say that N is a neighbourhood base at x for X means that each subset V0 of X is a neighbourhood of x if and only if there exists N0N such that N0V0. Every neighbourhood base at x is a filter base that generates the neighbourhood filter at x.

Convergent filter bases

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

• To say that a filter base B converges to x, denoted Bx, means that for every neighbourhood U of x, there is a B0B such that B0U. In this case, x is called a limit of B and B is called a convergent filter base.
• Every neighbourhood base N of x converges to x.
• If N is a neighbourhood base at x and C is a filter base on X, then Cx if and only if C is finer than N.
• If YX, a point p ∈ X is called a limit point of Y in X if and only if each neighborhood U of p in X intersects Y. This happens if and only if there is a filter base of subsets of Y that converges to p in X.
• For YX, the following are equivalent:
• (i) There exists a filter base F whose elements are all contained in Y such that Fx.
• (ii) There exists a filter F such that Y is an element of F and Fx.
• (iii) The point x lies in the closure of Y.

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 ${\displaystyle F=\{U\cap Y\ |\ U\in N_{x}\}}$. Then F is a filter base satisfying the properties of (i).

Clustering

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

• A filter base B on X is said to cluster at x (or have x as a cluster point) if and only if each element of B has nonempty intersection with each neighbourhood of x.
• If a filter base B clusters at x and is finer than a filter base C, then C clusters at x too.
• Every limit of a filter base is also a cluster point of the base.
• A filter base B that has x as a cluster point may not converge to x. But there is a finer filter base that does. For example the filter base of finite intersections of sets of the subbase ${\displaystyle B\cup N_{x}}$.
• For a filter base B, the set ∩{cl(B0) : B0B} is the set of all cluster points of B (note: cl(B0) is the closure of B0). Assume that X is a complete lattice.
• The limit inferior of B is the infimum of the set of all cluster points of B.
• The limit superior of B is the supremum of the set of all cluster points of B.
• B is a convergent filter base if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the filter base.

Properties of a topological space

Let X be a topological space.

• X is a Hausdorff space if and only if every filter base on X has at most one limit.
• X is compact if and only if every filter base on X clusters or has a cluster point.
• X is compact if and only if every filter base on X is a subset of a convergent filter base.
• X is compact if and only if every ultrafilter on X converges.

Functions on topological spaces

Let ${\displaystyle X}$, ${\displaystyle Y}$ be topological spaces. Let ${\displaystyle A}$ be a filter base on ${\displaystyle X}$ and ${\displaystyle f\colon X\to Y}$ be a function. The image of ${\displaystyle A}$ under ${\displaystyle f}$ is defined as the set ${\displaystyle \{f(a):a\in A\}}$. The image is denoted ${\displaystyle f[A]}$ and forms a filter base on ${\displaystyle Y}$.

• ${\displaystyle f}$ is continuous at ${\displaystyle x}$ if and only if ${\displaystyle A\to x}$ implies ${\displaystyle f[A]\to f(x)}$.

Cauchy filters

Let ${\displaystyle (X,d)}$ be a metric space.

• To say that a filter base B on X is Cauchy means that for each real number ε>0, there is a B0B such that the metric diameter of B0 is less than ε.
• Take (xn) to be a sequence in metric space X. (xn) is a Cauchy sequence if and only if the filter base {{xN, xN +1, ...} : N ∈ {1,2,3,...} } is Cauchy.

More generally, given a uniform space X, a filter F on X is called Cauchy filter if for every entourage U there is an AF with (x, y) ∈ U for all x, yA. 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 FG is Cauchy.

The Cauchy filters on a uniform space have these properties, so every uniform space (hence every metric space) defines a Cauchy space.