filter (mathematics)

In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal.

Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic.

Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book '' Topologie Générale'', popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ordered sets. Nevertheless, the theory of power-set filters retains interest in its own right, in part for substantial applications in topology.

Motivation

Fix a partially ordered set (poset) . Intuitively, a filter  is a subset of whose members are elements large enough to satisfy some criterion. For instance, if , then the set of elements above is a filter, called the principal filter at . (If and are incomparable elements of , then neither the principal filter at nor is contained in the other.)

Similarly, a filter on a set  contains those subsets that are sufficiently large to contain some given . For example, if is the real line and , then the family of sets including in their interior is a filter, called the neighborhood filter at . The in this case is slightly larger than , but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the definition below: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider as a "locating scheme" to find . In this interpretation, one searches in some space , and expects to describe those subsets of that contain the goal. The goal must be located somewhere; thus the empty set  can never be in . And if two subsets both contain the goal, then should "zoom in" to their common region.

An ultrafilter describes a "perfect locating scheme" where each scheme component gives new information (either "look here" or "look elsewhere"). Compactness is the property that "every search is fruitful," or, to put it another way, "every locating scheme ends in a search result."

A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space. This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.

Definition

A subset  of a partially ordered set  is a filter or dual ideal if the following are satisfied:

; Nontriviality: The set is non-empty.
; Downward directed: For every , there is some such that and .
; Upward closure: For every and , the condition implies .
If, additionally, , then is said to be a proper filter. Authors in set theory and mathematical logic often require all filters to be proper; this article will ''eschew'' that convention. An ultrafilter is a proper filter contained in no other proper filter except itself.

Filter bases

A subset  of is a base or basis for if the upper set generated by (i.e., the smallest upwards-closed set containing ) is equal to . Since every filter is upwards-closed, every filter is a base for itself.

Moreover, if is nonempty and downward directed, then generates an upper set  that is a filter (for which is a base). Such sets are called prefilters, as well as the aforementioned filter base/basis, and is said to be generated or spanned by . A prefilter is proper if and only if it generates a proper filter.

Given , the set is the smallest filter containing , and sometimes written . Such a filter is called a principal filter; is said to be the principal element of , or generate .

Refinement

Suppose and are two prefilters on , and, for each , there is a , such that . Then we say that is than (or refines) ; likewise, is coarser than (or coarsens) . Refinement is a preorder on the set of prefilters. In fact, if also refines , then and are called equivalent, for they generate the same filter. Thus passage from prefilter to filter is an instance of passing from a preordering to associated partial ordering.

Special cases

Historically, filters generalized to order-theoretic lattices before arbitrary partial orders. In the case of lattices, downward direction can be written as closure under finite meets: for all , one has .

Linear filters

A linear (ultra)filter is an (ultra)filter on the lattice of vector subspaces of a given vector space, ordered by inclusion. Explicitly, a linear filter on a vector space  is a family  of vector subspaces of such that if and is a vector subspace of that contains , then and .

A linear filter is proper if it does not contain .

Filters on a set; subbases

Given a set , the power set  is partially ordered by set inclusion; filters on this poset are often just called "filters on ," in an abuse of terminology. For such posets, downward direction and upward closure reduce to:
; Closure under finite intersections: If , then so too is .
; Isotony: If and , then .

A proper/non-degenerate filter is one that does not contain , and these three conditions (including non-degeneracy) are Henri Cartan's original definition of a filter. It is common — ''though not universal'' — to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.

Prefilters on a set are proper if and only if they do not contain either.

For every subset  of , there is a smallest filter  containing . As with prefilters, is said to generate or span ; a base for is the set  of all finite intersections of . The set is said to be a filter subbase when (and thus ) is proper.

Proper filters on sets have the finite intersection property.

If , then admits only the improper filter .

Free filters

A filter is said to be a free filter if the intersection 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. But a nonprincipal filter on an infinite set is not necessarily free: a filter is free if and only if it includes the Fréchet filter (see ).

Examples

See the image at the top of this article for a simple example of filters on the finite poset .

Partially order , the space of real-valued functions on , by pointwise comparison. Then the set of functions "large at infinity,"$\left\\text$is a filter on . One can generalize this construction quite far by compactifying the domain and completing the codomain: if is a set with distinguished subset  and is a poset with distinguished element , then is a filter in .

The set is a filter in . More generally, if is any directed set, then$\$is a filter in , called the tail filter. Likewise any net  generates the eventuality filter . A tail filter is the eventuality filter for .

The Fréchet filter on an infinite set  is$\\text$If is a measure space, then the collection is a filter. If , then is also a filter; the Fréchet filter is the case where is counting measure.

Given an ordinal , a subset of is called a club if it is closed in the order topology of but has net-theoretic limit . The clubs of form a filter: the club filter, .

The previous construction generalizes as follows: any club  is also a collection of dense subsets (in the ordinal topology) of , and meets each element of . Replacing with an arbitrary collection  of dense sets, there "typically" exists a filter meeting each element of , called a generic filter. For countable , the Rasiowa–Sikorski lemma implies that such a filter must exist; for "small" uncountable , the existence of such a filter can be forced through Martin's axiom.

Let denote the set of partial orders of limited cardinality, modulo isomorphism. Partially order by:
: if there exists a strictly increasing .
Then the subset of non-atomic partial orders forms a filter. Likewise, if is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism, then a partial order on is:
: if there exists an injective linear map .
Given any infinite cardinal , the modules in that cannot be generated by fewer than elements form a filter.

Every uniform structure on a set  is a filter on .

Relationship to ideals

The dual notion to a filter — that is, the concept obtained by reversing all and exchanging with  — is an order ideal. Because of this duality, any question of filters can be mechanically translated to a question about ideals and vice versa; in particular, a prime or maximal filter is a filter whose corresponding ideal is (respectively) prime or maximal.

A filter is an ultrafilter if and only if the corresponding ideal is minimal.

In model theory

For every filter  on a set , the set function defined by$m(A) = \begin 1 & \textA \in F \\ 0 & \textS \smallsetminus A \in F \\ \text & \text \end$is finitely additive — a " measure," if that term is construed rather loosely. Moreover, the measures so constructed are defined everywhere if is an ultrafilter. Therefore, the statement$\left\ \in F$can be considered somewhat analogous to the statement that holds "almost everywhere." That interpretation of membership in a filter is used (for motivation, not actual ) in the theory of ultraproducts in model theory, a branch of mathematical logic.

In topology

In general topology and analysis, filters are used to define convergence in a manner similar to the role of sequences in a metric space. They unify the concept of a limit across the wide variety of arbitrary topological spaces.

To understand the need for filters, begin with the equivalent concept of a net. A sequence is usually indexed by the natural numbers , which are a totally ordered set. Nets generalize the notion of a sequence by replacing with an arbitrary directed set. In certain categories of topological spaces, such as first-countable spaces, sequences characterize most topological properties, but this is not true in general. However, nets — as well as filters — always do characterize those topological properties.

Filters do not involve any set external to the topological space , whereas sequences and nets rely on other directed sets. For this reason, the collection of all filters on is always a set, whereas the collection of all -valued nets is a proper class.

Neighborhood bases

Any point  in the topological space  defines a neighborhood filter or system : namely, the family of all sets containing in their interior. A set  of neighborhoods of is a neighborhood base at if generates . Equivalently, is a neighborhood of if and only if there exists such that .

Convergent filters and cluster points

A prefilter  converges to a point , written , if and only if generates a filter  that contains the neighborhood filter  — explicitly, for every neighborhood  of , there is some such that . Less explicitly, if and only if refines , and any neighborhood base at can replace in this condition. Clearly, every neighborhood base at converges to .

A filter  (which generates itself) converges to if . The above can also be reversed to characterize the neighborhood filter : is the finest filter coarser than each filter converging to .

If , then is called a limit (point) of . The prefilter is said to cluster at (or have as a cluster point) if and only if each element of has non-empty intersection with each neighborhood of . Every limit point is a cluster point but the converse is not true in general. However, every cluster point of an filter is a limit point.

See also

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Notes

References

* Nicolas Bourbaki, General Topology (Topologie Générale), (Ch. 1–4): Provides a good reference for filters in general topology (Chapter I) and for Cauchy filters in uniform spaces (Chapter II)
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* (Provides an introductory review of filters in topology and in metric spaces.)
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Further reading

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{{Order theory

General topology
Order theory