In mathematics, a filter on a set is a family of subsets such that: [1]

  1. and
  2. if and , then
  3. If , and , then

A filter on a set may be thought of as representing a "collection of large subsets",[2] one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal.

Filters were introduced by Henri Cartan in 1937[3][4] 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 filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.

Preliminaries, notation, and basic notions edit

In this article, upper case Roman letters like and denote sets (but not families unless indicated otherwise) and will denote the power set of A subset of a power set is called a family of sets (or simply, a family) where it is over if it is a subset of Families of sets will be denoted by upper case calligraphy letters such as Whenever these assumptions are needed, then it should be assumed that is non–empty and that etc. are families of sets over

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter". While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

Sets operations

The upward closure or isotonization in [5][6] of a family of sets is

and similarly the downward closure of is

Notation and Definition Name
Kernel of [6]
Dual of where is a set.[7]
Trace of [7] or the restriction of where is a set; sometimes denoted by
[8] Elementwise (set) intersection ( will denote the usual intersection)
[8] Elementwise (set) union ( will denote the usual union)
Elementwise (set) subtraction ( will denote the usual set subtraction)
Grill of [9]
Power set of a set [6]

For any two families declare that if and only if for every there exists some in which case it is said that is coarser than and that is finer than (or subordinate to) [10][11][12] The notation may also be used in place of

Two families mesh,[7] written if

Throughout, is a map and is a set.

Notation and Definition Name
[13] Image of or the preimage of under
Image of or the preimage of
[14] Image of under
Image of
Image (or range) of

Nets and their tails

A directed set is a set together with a preorder, which will be denoted by (unless explicitly indicated otherwise), that makes into an (upward) directed set;[15] this means that for all there exists some such that For any indices the notation is defined to mean while is defined to mean that holds but it is not true that (if is antisymmetric then this is equivalent to ).

A net in [15] is a map from a non–empty directed set into The notation will be used to denote a net with domain

Notation and Definition Name
Tail or section of starting at where is a directed set.
Tail or section of starting at
Set or prefilter of tails/sections of Also called the eventuality filter base generated by (the tails of) If is a sequence then is also called the sequential filter base.[16]
(Eventuality) filter of/generated by (tails of) [16]
Tail or section of a net starting at [16] where is a directed set.

Warning about using strict comparison

If is a net and then it is possible for the set which is called the tail of after , to be empty (for example, this happens if is an upper bound of the directed set ). In this case, the family would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining as rather than or even and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality may not be used interchangeably with the inequality

Filters and prefilters edit

The following is a list of properties that a family of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that

The family of sets is:
  1. Proper or nondegenerate if Otherwise, if then it is called improper[17] or degenerate.
  2. Directed downward[15] if whenever then there exists some such that
    • This property can be characterized in terms of directedness, which explains the word "directed": A binary relation on is called (upward) directed if for any two there is some satisfying Using in place of gives the definition of directed downward whereas using instead gives the definition of directed upward. Explicitly, is directed downward (resp. directed upward) if and only if for all there exists some "greater" such that (resp. such that ) − where the "greater" element is always on the right hand side,[note 1] − which can be rewritten as (resp. as ).
    • If a family has a greatest element with respect to (for example, if ) then it is necessarily directed downward.
  3. Closed under finite intersections (resp. unions) if the intersection (resp. union) of any two elements of is an element of
    • If is closed under finite intersections then is necessarily directed downward. The converse is generally false.
  4. Upward closed or Isotone in [5] if or equivalently, if whenever and some set satisfies Similarly, is downward closed if An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
    • The family which is the upward closure of is the unique smallest (with respect to ) isotone family of sets over having as a subset.

Many of the properties of defined above and below, such as "proper" and "directed downward," do not depend on so mentioning the set is optional when using such terms. Definitions involving being "upward closed in " such as that of "filter on " do depend on so the set should be mentioned if it is not clear from context.

A family is/is a(n):
  1. Ideal[17][18] if is downward closed and closed under finite unions.
  2. Dual ideal on [19] if is upward closed in and also closed under finite intersections. Equivalently, is a dual ideal if for all [9]
    • Explanation of the word "dual": A family is a dual ideal (resp. an ideal) on if and only if the dual of which is the family
      is an ideal (resp. a dual ideal) on In other words, dual ideal means "dual of an ideal". The family should not be confused with because these two sets are not equal in general; for instance, The dual of the dual is the original family, meaning The set belongs to the dual of if and only if [17]
  3. Filter on [19][7] if is a proper dual ideal on That is, a filter on is a non−empty subset of that is closed under finite intersections and upward closed in Equivalently, it is a prefilter that is upward closed in In words, a filter on is a family of sets over that (1) is not empty (or equivalently, it contains ), (2) is closed under finite intersections, (3) is upward closed in and (4) does not have the empty set as an element.
    • Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a proper/non–degenerate dual ideal.[20] It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always require non-degeneracy. This article uses Henri Cartan's original definition of "filter",[3][4] which required non–degeneracy.
    • A dual filter on is a family whose dual is a filter on Equivalently, it is an ideal on that does not contain as an element.
    • The power set is the one and only dual ideal on that is not also a filter. Excluding from the definition of "filter" in topology has the same benefit as excluding from the definition of "prime number": it obviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-") in many important results, thereby making their statements less awkward.
  4. Prefilter or filter base[7][21] if is proper and directed downward. Equivalently, is called a prefilter if its upward closure is a filter. It can also be defined as any family that is equivalent (with respect to ) to some filter.[8] A proper family is a prefilter if and only if [8] A family is a prefilter if and only if the same is true of its upward closure.
    • If is a prefilter then its upward closure is the unique smallest (relative to ) filter on containing and it is called the filter generated by A filter is said to be generated by a prefilter if in which is called a filter base for
    • Unlike a filter, a prefilter is not necessarily closed under finite intersections.
  5. π–system if is closed under finite intersections. Every non–empty family is contained in a unique smallest π–system called the π–system generated by which is sometimes denoted by It is equal to the intersection of all π–systems containing and also to the set of all possible finite intersections of sets from :
    • A π–system is a prefilter if and only if it is proper. Every filter is a proper π–system and every proper π–system is a prefilter but the converses do not hold in general.
    • A prefilter is equivalent (with respect to ) to the π–system generated by it and both of these families generate the same filter on
  6. Filter subbase[7][22] and centered[8] if and satisfies any of the following equivalent conditions:
    1. has the finite intersection property, which means that the intersection of any finite family of (one or more) sets in is not empty; explicitly, this means that whenever then
    2. The π–system generated by is proper; that is,
    3. The π–system generated by is a prefilter.
    4. is a subset of some prefilter.
    5. is a subset of some filter.
    • Assume that is a filter subbase. Then there is a unique smallest (relative to ) filter containing called the filter generated by , and is said to be a filter subbase for this filter. This filter is equal to the intersection of all filters on that are supersets of The π–system generated by denoted by will be a prefilter and a subset of Moreover, the filter generated by is equal to the upward closure of meaning [8] However, if and only if is a prefilter (although is always an upward closed filter subbase for ).
    • A  –smallest (meaning smallest relative to  ) prefilter containing a filter subbase will exist only under certain circumstances. It exists, for example, if the filter subbase happens to also be a prefilter. It also exists if the filter (or equivalently, the π–system) generated by is principal, in which case is the unique smallest prefilter containing Otherwise, in general, a  –smallest prefilter containing might not exist. For this reason, some authors may refer to the π–system generated by as the prefilter generated by However, if a  –smallest prefilter does exist (say it is denoted by ) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by " (that is, is possible). And if the filter subbase happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning ) will not be (that is, is possible even when is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ".
  7. Subfilter of a filter and that is a superfilter of [17][23] if is a filter and where for filters,
    • Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of." However, can also be written which is described by saying " is subordinate to " With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of,"[24] which makes this one situation where using the term "subordinate" and symbol may be helpful.

There are no prefilters on (nor are there any nets valued in ), which is why this article, like most authors, will automatically assume without comment that whenever this assumption is needed.

Basic examples edit

Named examples

  • The singleton set is called the indiscrete or trivial filter on [25][10] It is the unique minimal filter on because it is a subset of every filter on ; however, it need not be a subset of every prefilter on
  • The dual ideal is also called the degenerate filter on [9] (despite not actually being a filter). It is the only dual ideal on that is not a filter on
  • If is a topological space and then the neighborhood filter at is a filter on By definition, a family is called a neighborhood basis (resp. a neighborhood subbase) at if and only if is a prefilter (resp. is a filter subbase) and the filter on that generates is equal to the neighborhood filter The subfamily of open neighborhoods is a filter base for Both prefilters also form a bases for topologies on with the topology generated being coarser than This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets
  • is an elementary prefilter[26] if for some sequence
  • is an elementary filter or a sequential filter on [27] if is a filter on generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily not an ultrafilter.[28] Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set.[9] The intersection of finitely many sequential filters is again sequential.[9]
  • The set of all cofinite subsets of (meaning those sets whose complement in is finite) is proper if and only if is infinite (or equivalently, is infinite), in which case is a filter on known as the Fréchet filter or the cofinite filter on [10][25] If is finite then is equal to the dual ideal which is not a filter. If is infinite then the family of complements of singleton sets is a filter subbase that generates the Fréchet filter on As with any family of sets over that contains the kernel of the Fréchet filter on is the empty set:
  • The intersection of all elements in any non–empty family is itself a filter on called the infimum or greatest lower bound of which is why it may be denoted by Said differently, Because every filter on has as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to ) filter contained as a subset of each member of [10]
    • If are filters then their infimum in is the filter [8] If are prefilters then is a prefilter that is coarser (with respect to ) than both (that is, ); indeed, it is one of the finest such prefilters, meaning that if is a prefilter such that then necessarily [8] More generally, if are non−empty families and if then and is a greatest element (with respect to ) of [8]
  • Let and let The supremum or least upper bound of denoted by is the smallest (relative to ) dual ideal on containing every element of as a subset; that is, it is the smallest (relative to ) dual ideal on containing as a subset. This dual ideal is where is the π–system generated by As with any non–empty family of sets, is contained in some filter on if and only if it is a filter subbase, or equivalently, if and only if is a filter on in which case this family is the smallest (relative to ) filter on containing every element of as a subset and necessarily
  • Let and let The supremum or least upper bound of denoted by if it exists, is by definition the smallest (relative to ) filter on containing every element of as a subset. If it exists then necessarily [10] (as defined above) and will also be equal to the intersection of all filters on containing This supremum of exists if and only if the dual ideal is a filter on The least upper bound of a family of filters may fail to be a filter.[10] Indeed, if contains at least 2 distinct elements then there exist filters for which there does not exist a filter that contains both If is not a filter subbase then the supremum of does not exist and the same is true of its supremum in but their supremum in the set of all dual ideals on will exist (it being the degenerate filter ).[9]
    • If are prefilters (resp. filters on ) then is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filter) on (with respect to ) that is finer (with respect to ) than both this means that if is any prefilter (resp. any filter) such that then necessarily [8] in which case it is denoted by [9]
  • Let be non−empty sets and for every let be a dual ideal on If is any dual ideal on then is a dual ideal on called Kowalsky's dual ideal or Kowalsky's filter.[17]
  • The club filter of a regular uncountable cardinal is the filter of all sets containing a club subset of It is a -complete filter closed under diagonal intersection.

Other examples

  • Let and let which makes a prefilter and a filter subbase that is not closed under finite intersections. Because is a prefilter, the smallest prefilter containing is The π–system generated by is In particular, the smallest prefilter containing the filter subbase is not equal to the set of all finite intersections of sets in The filter on generated by is All three of the π–system generates, and are examples of fixed, principal, ultra prefilters that are principal at the point is also an ultrafilter on
  • Let be a topological space, and define where is necessarily finer than [29] If is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of If is a filter on then is a prefilter but not necessarily a filter on although is a filter on equivalent to
  • The set of all dense open subsets of a (non–empty) topological space is a proper π–system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than If (with ) then the set of all such that has finite Lebesgue measure is a proper π–system and free prefilter that is also a proper subset of The prefilters and are equivalent and so generate the same filter on The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense subsets of Since is a Baire space, every countable intersection of sets in is dense in (and also comeagre and non–meager) so the set of all countable intersections of elements of is a prefilter and π–system; it is also finer than, and not equivalent to,
  • A filter subbase with no smallest prefilter containing it: In general, if a filter subbase is not a π–system then an intersection of sets from will usually require a description involving variables that cannot be reduced down to only two (consider, for instance when ). This example illustrates an atypical class of a filter subbases where all sets in both and its generated π–system can be described as sets of the form so that in particular, no more than two variables (specifically, ) are needed to describe the generated π–system. For all let
    where always holds so no generality is lost by adding the assumption For all real if is non-negative then [note 2] For every set of positive reals, let[note 3]
    Let and suppose is not a singleton set. Then is a filter subbase but not a prefilter and is the π–system it generates, so that is the unique smallest filter in containing However, is not a filter on (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and is a proper subset of the filter If are non−empty intervals then the filter subbases generate the same filter on if and only if If is a prefilter satisfying [note 4] then for any the family is also a prefilter satisfying This shows that there cannot exist a minimal/least (with respect to ) prefilter that both contains and is a subset of the π–system generated by This remains true even if the requirement that the prefilter be a subset of is removed; that is, (in sharp contrast to filters) there does not exist a minimal/least (with respect to ) prefilter containing the filter subbase

Ultrafilters edit

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.

A non–empty family of sets is/is an:
  1. Ultra[7][30] if and any of the following equivalent conditions are satisfied:
    1. For every set there exists some set such that (or equivalently, such that ).
    2. For every set there exists some set such that
      • This characterization of " is ultra" does not depend on the set so mentioning the set is optional when using the term "ultra."
    3. For every set (not necessarily even a subset of ) there exists some set such that
      • If satisfies this condition then so does every superset For example, if is any singleton set then is ultra and consequently, any non–degenerate superset of (such as its upward closure) is also ultra.
  2. Ultra prefilter[7][30] if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter is ultra if and only if it satisfies any of the following equivalent conditions:
    1. is maximal in with respect to which means that
      • Although this statement is identical to that given below for ultrafilters, here is merely assumed to be a prefilter; it need not be a filter.
    2. is ultra (and thus an ultrafilter).
    3. is equivalent (with respect to ) to some ultrafilter.
    • A filter subbase that is ultra is necessarily a prefilter. A filter subbase is ultra if and only if it is a maximal filter subbase with respect to (as above).[17]
  3. Ultrafilter on [7][30] if it is a filter on that is ultra. Equivalently, an ultrafilter on is a filter that satisfies any of the following equivalent conditions:
    1. is generated by an ultra prefilter.
    2. For any [17]
    3. This condition can be restated as: is partitioned by and its dual
      • The sets are disjoint whenever is a prefilter.
    4. is an ideal.[17]
    5. For any if then
    6. For any if then (a filter with this property is called a prime filter).
      • This property extends to any finite union of two or more sets.
    7. For any if then either
    8. is a maximal filter on ; meaning that if is a filter on such that then necessarily (this equality may be replaced by ).
      • If is upward closed then So this characterization of ultrafilters as maximal filters can be restated as:
      • Because subordination is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA–subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from " in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example),[note 5] which is an idea that is actually made rigorous by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

Any non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property. The trivial filter is ultra if and only if is a singleton set.

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).[31]

The ultrafilter lemma/principal/theorem[10] (Tarski) — Every filter on a set is a subset of some ultrafilter on

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.[10][proof 1] Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

Kernels edit

The kernel is useful in classifying properties of prefilters and other families of sets.

The kernel[5] of a family of sets is the intersection of all sets that are elements of

If then for any point

Properties of kernels

If then and this set is also equal to the kernel of the π–system that is generated by In particular, if is a filter subbase then the kernels of all of the following sets are equal:

(1) (2) the π–system generated by and (3) the filter generated by

If is a map then and If then while if and are equivalent then Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal; that is, if and are principal then they are equivalent if and only if

Classifying families by their kernels edit

A family of sets is:
  1. Free[6] if or equivalently, if this can be restated as
    • A filter on is free if and only if is infinite and contains the Fréchet filter on as a subset.
  2. Fixed if in which case, is said to be fixed by any point
    • Any fixed family is necessarily a filter subbase.
  3. Principal[6] if
    • A proper principal family of sets is necessarily a prefilter.
  4. Discrete or Principal at [25] if in which case is called its principal element.
    • The principal filter at on is the filter A filter is principal at if and only if
  5. Countably deep if whenever is a countable subset then [9]

If is a principal filter on then and

where is also the smallest prefilter that generates

Family of examples: For any non–empty the family is free but it is a filter subbase if and only if no finite union of the form covers in which case the filter that it generates will also be free. In particular, is a filter subbase if is countable (for example, the primes), a meager set in a set of finite measure, or a bounded subset of If is a singleton set then is a subbase for the Fréchet filter on

For every filter there exists a unique pair of dual ideals such that is free, is principal, and and do not mesh (that is, ). The dual ideal is called the free part of while is called the principal part[9] where at least one of these dual ideals is filter. If is principal then otherwise, and is a free (non–degenerate) filter.[9]

Finite prefilters and finite sets

If a filter subbase is finite then it is fixed (that is, not free); this is because is a finite intersection and the filter subbase has the finite intersection property. A finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.

If is finite then all of the conclusions above hold for any In particular, on a finite set there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on are principal filters generated by their (non–empty) kernels.

The trivial filter is always a finite filter on and if is infinite then it is the only finite filter because a non–trivial finite filter on a set is possible if and only if is finite. However, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters). If is a singleton set then the trivial filter is the only proper subset of and moreover, this set is a principal ultra prefilter and any superset (where ) with the finite intersection property will also be a principal ultra prefilter (even if is infinite).

Characterizing fixed ultra prefilters edit

If a family of sets is fixed (that is, ) then is ultra if and only if some element of is a singleton set, in which case will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter is ultra if and only if is a singleton set.

Every filter on that is principal at a single point is an ultrafilter, and if in addition is finite, then there are no ultrafilters on other than these.[6]

The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

Proposition — If is an ultrafilter on then the following are equivalent:

  1. is fixed, or equivalently, not free, meaning
  2. is principal, meaning
  3. Some element of