In mathematics, a filter or order filter is a special subset of a partially ordered set (poset). Filters appear in order and lattice theory, but can also be found in topology, from which they originate. The dual notion of a filter is an order ideal.
Filters on sets were introduced by Henri Cartan in 1937^{[1]}^{[2]} and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of this notion from sets to the more general setting of partially ordered sets. For information on order filters in the special case where the poset consists of the power set ordered by set inclusion, see the article Filter (set theory).
1. Intuitively, a filter in a partially ordered set (poset), is a subset of that includes as members those elements that are large enough to satisfy some given criterion. For example, if is an element of the poset, then the set of elements that are above is a filter, called the principal filter at (If and are incomparable elements of the poset, then neither of the principal filters at and 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 is one of its points, then the family of sets that include in their interior is a filter, called the filter of neighbourhoods of The thing in this case is slightly larger than but it still does not 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?
2. Alternatively, a filter can be viewed as a "locating scheme": When trying to locate something (a point or a subset) in the space call a filter the collection of subsets of 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 of the space can be used in deciding whether "what is looked for" might lie in
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.
3. A common use for a filter is to define properties that are satisfied by "almost all" elements of some topological space ^{[3]} The entire space definitely contains almost-all elements in it; If some contains almost all elements of then any superset of it definitely does; and if two subsets, and contain almost-all elements of then so does their intersection. In a measure-theoretic terms, the meaning of " contains almost-all elements of " is that the measure of is 0.
A subset of a partially ordered set is an order filter or dual ideal if the following conditions hold:
is said to be a proper filter if in addition is not equal to the whole set Depending on the author, the term filter is either a synonym of order filter or else it refers to a proper order filter. This article defines filter to mean order 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 of a lattice is a filter, if and only if it is a non-empty upper set that is closed under finite infima (or meets), that is, for all it is also the case that ^{[4]} A subset of is a filter basis if the upper set generated by is all of Note that every filter is its own basis.
The smallest filter that contains a given element is a principal filter and is a principal element in this situation. The principal filter for is just given by the set and is denoted by prefixing with an upward arrow:
The dual notion of a filter, that is, 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.
Applying these definitions to the case where is a vector space and is the set of all vector subspaces of ordered by inclusion gives rise to the notion of linear filters and linear ultrafilters. Explicitly, a linear filter on a vector space is a family of vector subspaces of such that if and if is a vector subspace of that contains then ^{[5]} A linear filter is called proper if it does not contain a linear ultrafilter on is a maximal proper linear filter on ^{[5]}
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
π-system | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
𝜆-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
δ-Ring | Never | |||||||||
𝜎-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
𝜎-Algebra (𝜎-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
Topology | (even arbitrary unions) |
Never | ||||||||
Is necessarily true of or, is closed under: |
directed downward |
finite intersections |
finite unions |
relative complements |
complements in |
countable intersections |
countable unions |
contains | contains | Finite Intersection Property |
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in |
There are two competing definitions of a "filter on a set," both of which require that a filter be a dual ideal.^{[6]} One definition defines "filter" as a synonym of "dual ideal" while the other defines "filter" to mean a dual ideal that is also proper.
A dual ideal^{[6]} on a set is a non-empty subset of with the following properties:
Given a set a canonical partial ordering can be defined on the powerset by subset inclusion, turning into a lattice. A "dual ideal" is just a filter with respect to this partial ordering. Note that if then there is exactly one dual ideal on which is
A filter on a set may be thought of as representing a "collection of large subsets".^{[8]}
The article uses the following definition of "filter on a set."
Definition as a dual ideal: A filter on a set is a dual ideal on Equivalently, a filter on is just a filter with respect the canonical partial ordering described above.
The other definition of "filter on a set" is the original definition of a "filter" given by Henri Cartan, which required that a filter on a set be a dual ideal that does not contain the empty set:
Original/Alternative definition as a proper dual ideal: A filter^{[6]} on a set is a dual ideal on with the following additional property:
The only non-proper filter on is Much mathematical literature, especially that related to Topology, defines "filter" to mean a non-degenerate dual ideal.
Filter bases and subbases
A subset of is called a prefilter, filter base, or filter basis if is non-empty and the intersection of any two members of is a superset of some member(s) of If the empty set is not a member of we say is a proper filter base.
Given a filter base the filter generated or spanned by is defined as the minimum filter containing It is the family of all those subsets of which are supersets of some member(s) of 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.
For every subset of there is a smallest (possibly non-proper) filter containing called the filter generated or spanned by Similarly as for a filter spanned by a filter base, a filter spanned by a subset is the minimum filter containing It is constructed by taking all finite intersections of which then form a filter base for This filter is proper if and only if every finite intersection of elements of is non-empty, and in that case we say that is a filter subbase.
Finer/equivalent filter bases
If and are two filter bases on one says is finer than (or that is a refinement of ) if for each there is a such that For filter bases and if is finer than and is finer than then is finer than 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.
If also is finer than one says that they are equivalent filter bases. If and are filter bases, then is finer than if and only if the filter spanned by contains the filter spanned by Therefore, and are equivalent filter bases if and only if they generate the same filter.
A filter in a poset can be created using the Rasiowa–Sikorski lemma, which is often used in forcing. The set is called a filter base of tails of the sequence of natural numbers A filter base of tails can be made of any net using the construction 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.
Let be a set and be a non-empty subset of Then is a filter base. The filter it generates (that is, the collection of all subsets containing ) is called the principal filter generated by A filter is said to be a free filter if the intersection of all of its members is empty. A proper principal filter is not free. Since the intersection of any finite number of members of a filter is also a member, no proper 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 is the set of all subsets of that have finite complement. A filter on is free if and only if it includes the Fréchet filter. More generally, if is a measure space for which the collection of all such that forms a filter. The Fréchet filter is the case where and is the counting measure.
Every uniform structure on a set is a filter on
For every filter on a set the set function defined by
In topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. 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. Nets generalize the notion of a sequence by requiring the index set simply be a directed set. If working only with certain categories of topological spaces, such as first-countable spaces for instance, sequences suffice to characterize most topological properties, but this is not true in general. However, filters (as well as nets) do always suffice to characterize most topological properties. An advantage to using filters is that they do not involve any set other than (and its subsets) whereas sequences and nets rely on directed sets that may be unrelated to Moreover, the set of all filters on is a set whereas the class of all nets valued in is not (it is a proper class).
Neighbourhood bases
Let be the neighbourhood filter at point in a topological space This means that is the set of all topological neighbourhoods of the point It can be verified that is a filter. A neighbourhood system is another name for a neighbourhood filter. A family of neighbourhoods of is a neighbourhood base at if generates the filter This means that each subset of is a neighbourhood of if and only if there exists such that
Convergent filters and cluster points
We say that a filter base converges to a point written if the neigbourhood filter is contained in the filter generated by that is, if is finer than In particular, a filter (which is a filter base that generates itself) converges to if Explicitly, to say that a filter base converges to means that for every neighbourhood of there is a such that If a filter base converges to a point then is called a limit (point) of and is called a convergent filter base.
A filter base on is said to cluster at (or have as a cluster point) if and only if each element of has non-empty intersection with each neighbourhood of Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an ultrafilter is a limit point.
By definition, every neighbourhood base at a given point generates so converges to If is a filter base on then if is finer than any neighbourhood base at For the neighborhood filter at that point, the converse holds as well: any basis of a convergent filter refines the neighborhood filter.