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Filter (set theory)

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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

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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

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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

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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

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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

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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

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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 is a finite set.
  4. Some element of is a singleton set.
  5. is principal at some point of which means for some
  6. does not contain the Fréchet filter on
  7. is sequential.[9]

Finer/coarser, subordination, and meshing

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The preorder that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence",[24] where "" can be interpreted as " is a subsequence of " (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of meshes with which is closely related to the preorder is used in Topology to define cluster points.

Two families of sets mesh[7] and are compatible, indicated by writing if If do not mesh then they are dissociated. If then are said to mesh if mesh, or equivalently, if the trace of which is the family does not contain the empty set, where the trace is also called the restriction of

Declare that stated as is coarser than and is finer than (or subordinate to) [10][11][12][8][9] if any of the following equivalent conditions hold:
  1. Definition: Every contains some Explicitly, this means that for every there is some such that
    • Said more briefly in plain English, if every set in is larger than some set in Here, a "larger set" means a superset.
    • In words, states exactly that is larger than some set in The equivalence of (a) and (b) follows immediately.
    • From this characterization, it follows that if are families of sets, then
  2. which is equivalent to ;
  3. ;
  4. which is equivalent to ;

and if in addition is upward closed, which means that then this list can be extended to include:

  1. [5]
    • So in this case, this definition of " is finer than " would be identical to the topological definition of "finer" had been topologies on

If an upward closed family is finer than (that is, ) but then is said to be strictly finer than and is strictly coarser than

Two families are comparable if one of these sets is finer than the other.[10]

Example: If is a subsequence of then is subordinate to in symbols: and also Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence. To see this, let be arbitrary (or equivalently, let be arbitrary) and it remains to show that this set contains some For the set to contain it is sufficient to have Since are strictly increasing integers, there exists such that and so holds, as desired. Consequently, The left hand side will be a strict/proper subset of the right hand side if (for instance) every point of is unique (that is, when is injective) and is the even-indexed subsequence because under these conditions, every tail (for every ) of the subsequence will belong to the right hand side filter but not to the left hand side filter.

For another example, if is any family then always holds and furthermore,

Assume that are families of sets that satisfy Then and and also If in addition to is a filter subbase and then is a filter subbase[8] and also mesh.[19][proof 2] More generally, if both and if the intersection of any two elements of is non–empty, then mesh.[proof 2] Every filter subbase is coarser than both the π–system that it generates and the filter that it generates.[8]

If are families such that the family is ultra, and then is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily be ultra. In particular, if is a prefilter then either both and the filter it generates are ultra or neither one is ultra. If a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra. A filter subbase that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by to be ultra. If is upward closed in then [9]

Relational properties of subordination

The relation is reflexive and transitive, which makes it into a preorder on [32] The relation is antisymmetric but if has more than one point then it is not symmetric.

Symmetry: For any So the set has more than one point if and only if the relation is not symmetric.

Antisymmetry: If but while the converse does not hold in general, it does hold if is upward closed (such as if is a filter). Two filters are equivalent if and only if they are equal, which makes the restriction of to antisymmetric. But in general, is not antisymmetric on nor on ; that is, does not necessarily imply ; not even if both are prefilters.[12] For instance, if is a prefilter but not a filter then

Equivalent families of sets

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The preorder induces its canonical equivalence relation on where for all is equivalent to if any of the following equivalent conditions hold:[8][5]

  1. The upward closures of are equal.

Two upward closed (in ) subsets of are equivalent if and only if they are equal.[8] If then necessarily and is equivalent to Every equivalence class other than contains a unique representative (that is, element of the equivalence class) that is upward closed in [8]

Properties preserved between equivalent families

Let be arbitrary and let be any family of sets. If are equivalent (which implies that ) then for each of the statements/properties listed below, either it is true of both or else it is false of both :[32]

  1. Not empty
  2. Proper (that is, is not an element)
    • Moreover, any two degenerate families are necessarily equivalent.
  3. Filter subbase
  4. Prefilter
    • In which case generate the same filter on (that is, their upward closures in are equal).
  5. Free
  6. Principal
  7. Ultra
  8. Is equal to the trivial filter
    • In words, this means that the only subset of that is equivalent to the trivial filter is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters).
  9. Meshes with
  10. Is finer than
  11. Is coarser than
  12. Is equivalent to

Missing from the above list is the word "filter" because this property is not preserved by equivalence. However, if are filters on then they are equivalent if and only if they are equal; this characterization does not extend to prefilters.

Equivalence of prefilters and filter subbases

If is a prefilter on then the following families are always equivalent to each other:

  1. ;
  2. the π–system generated by ;
  3. the filter on generated by ;

and moreover, these three families all generate the same filter on (that is, the upward closures in of these families are equal).

In particular, every prefilter is equivalent to the filter that it generates. By transitivity, two prefilters are equivalent if and only if they generate the same filter.[8][proof 3] Every prefilter is equivalent to exactly one filter on which is the filter that it generates (that is, the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.[8]

A filter subbase that is not also a prefilter cannot be equivalent to the prefilter (or filter) that it generates. In contrast, every prefilter is equivalent to the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot. Every filter is both a π–system and a ring of sets.

Examples of determining equivalence/non–equivalence

Examples: Let and let be the set of integers (or the set ). Define the sets

All three sets are filter subbases but none are filters on and only is prefilter (in fact, is even free and closed under finite intersections). The set is fixed while is free (unless ). They satisfy but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal. This conclusion can be reached by showing that the π–systems that they generate are not equivalent. Unlike with every set in the π–system generated by contains as a subset,[note 6] which is what prevents their generated π–systems (and hence their generated filters) from being equivalent. If was instead then all three families would be free and although the sets would remain not equivalent to each other, their generated π–systems would be equivalent and consequently, they would generate the same filter on ; however, this common filter would still be strictly coarser than the filter generated by

Set theoretic properties and constructions

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Trace and meshing

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If is a prefilter (resp. filter) on then the trace of which is the family is a prefilter (resp. a filter) if and only if mesh (that is, [10]), in which case the trace of is said to be induced by . If is ultra and if mesh then the trace is ultra. If is an ultrafilter on then the trace of is a filter on if and only if

For example, suppose that is a filter on is such that Then mesh and generates a filter on that is strictly finer than [10]

When prefilters mesh

Given non–empty families the family satisfies and If is proper (resp. a prefilter, a filter subbase) then this is also true of both In order to make any meaningful deductions about from needs to be proper (that is, which is the motivation for the definition of "mesh". In this case, is a prefilter (resp. filter subbase) if and only if this is true of both Said differently, if are prefilters then they mesh if and only if is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, ):

     Two prefilters (resp. filter subbases) mesh if and only if there exists a prefilter (resp. filter subbase) such that and

If the least upper bound of two filters exists in then this least upper bound is equal to [28]

Images and preimages under functions

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Throughout, will be maps between non–empty sets.

Images of prefilters

Let Many of the properties that may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.

Explicitly, if one of the following properties is true of then it will necessarily also be true of (although possibly not on the codomain unless is surjective):[10][13][33][34][35][31]

  • Filter properties: ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate.
  • Ideal properties: ideal, closed under finite unions, downward closed, directed upward.

Moreover, if is a prefilter then so are both [10] The image under a map of an ultra set is again ultra and if is an ultra prefilter then so is

If is a filter then is a filter on the range but it is a filter on the codomain if and only if is surjective.[33] Otherwise it is just a prefilter on and its upward closure must be taken in to obtain a filter. The upward closure of is where if is upward closed in (that is, a filter) then this simplifies to:

If then taking to be the inclusion map shows that any prefilter (resp. ultra prefilter, filter subbase) on is also a prefilter (resp. ultra prefilter, filter subbase) on [10]

Preimages of prefilters

Let Under the assumption that is surjective:

      is a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of

However, if is an ultrafilter on then even if is surjective (which would make a prefilter), it is nevertheless still possible for the prefilter to be neither ultra nor a filter on [34] (see this[note 7] footnote for an example).

If is not surjective then denote the trace of by where in this case particular case the trace satisfies: and consequently also:

This last equality and the fact that the trace is a family of sets over means that to draw conclusions about the trace can be used in place of and the surjection can be used in place of For example:[13][10][35]

      is a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of

In this way, the case where is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).

Even if is an ultrafilter on if is not surjective then it is nevertheless possible that which would make degenerate as well. The next characterization shows that degeneracy is the only obstacle. If is a prefilter then the following are equivalent:[13][10][35]

  1. is a prefilter;
  2. is a prefilter;
  3. ;
  4. meshes with

and moreover, if is a prefilter then so is [13][10]

If and if denotes the inclusion map then the trace of is equal to [10] This observation allows the results in this subsection to be applied to investigating the trace on a set.

Bijections, injections, and surjections

All properties involving filters are preserved under bijections. This means that if is a bijection, then is a prefilter (resp. ultra, ultra prefilter, filter on ultrafilter on filter subbase, π–system, ideal on etc.) if and only if the same is true of [34]

A map is injective if and only if for all prefilters is equivalent to [28] The image of an ultra family of sets under an injection is again ultra.

The map is a surjection if and only if whenever is a prefilter on then the same is true of (this result does not require the ultrafilter lemma).

Subordination is preserved by images and preimages

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The relation is preserved under both images and preimages of families of sets.[10] This means that for any families [35]

Moreover, the following relations always hold for any family of sets :[35] where equality will hold if is surjective.[35] Furthermore,

If then[9] and [35] where equality will hold if is injective.[35]

Products of prefilters

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Suppose is a family of one or more non–empty sets, whose product will be denoted by and for every index let denote the canonical projection. Let be non−empty families, also indexed by such that for each The product of the families [10] is defined identically to how the basic open subsets of the product topology are defined (had all of these been topologies). That is, both the notations denote the family of all cylinder subsets such that for all but finitely many and where for any one of these finitely many exceptions (that is, for any such that necessarily ). When every is a filter subbase then the family is a filter subbase for the filter on generated by [10] If is a filter subbase then the filter on that it generates is called the filter generated by .[10] If every is a prefilter on then will be a prefilter on and moreover, this prefilter is equal to the coarsest prefilter such that for every [10] However, may fail to be a filter on even if every is a filter on [10]

Set subtraction and some examples

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Set subtracting away a subset of the kernel

If is a prefilter on then is a prefilter, where this latter set is a filter if and only if is a filter and In particular, if is a neighborhood basis at a point in a topological space having at least 2 points, then is a prefilter on This construction is used to define in terms of prefilter convergence.

Using duality between ideals and dual ideals

There is a dual relation or which is defined to mean that every is contained in some Explicitly, this means that for every , there is some such that This relation is dual to in sense that if and only if [5] The relation is closely related to the downward closure of a family in a manner similar to how is related to the upward closure family.

For an example that uses this duality, suppose is a map and Define which contains the empty set if and only if does. It is possible for to be an ultrafilter and for to be empty or not closed under finite intersections (see footnote for example).[note 8] Although does not preserve properties of filters very well, if is downward closed (resp. closed under finite unions, an ideal) then this will also be true for Using the duality between ideals and dual ideals allows for a construction of the following filter.

     Suppose is a filter on and let be its dual in If then 's dual will be a filter.

Other examples

Example: The set of all dense open subsets of a topological space is a proper π–system and 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

Example: The family of all dense open sets of having finite Lebesgue measure is a proper π–system and a free prefilter. The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense open 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,

Filters and nets

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This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology − particularly in switching from utilizing nets to utilizing filters and vice verse − and because it to make it easier to understand later why subnets (with their most commonly used definitions) are not generally equivalent with "sub–prefilters".

Nets to prefilters

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A net is canonically associated with its prefilter of tails If is a map and is a net in then [36]

Prefilters to nets

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A pointed set is a pair consisting of a non–empty set and an element For any family let

Define a canonical preorder on pointed sets by declaring

If even if so this preorder is not antisymmetric and given any family of sets is partially ordered if and only if consists entirely of singleton sets. If is a maximal element of ; moreover, all maximal elements are of this form. If is a greatest element if and only if in which case is the set of all greatest elements. However, a greatest element is a maximal element if and only if so there is at most one element that is both maximal and greatest. There is a canonical map defined by

If then the tail of the assignment starting at is

Although is not, in general, a partially ordered set, it is a directed set if (and only if) is a prefilter. So the most immediate choice for the definition of "the net in induced by a prefilter " is the assignment from into

If is a prefilter on then the net associated with is the map

that is,

If is a prefilter on is a net in and the prefilter associated with is ; that is:[note 9]

This would not necessarily be true had been defined on a proper subset of For example, suppose has at least two distinct elements, is the indiscrete filter, and is arbitrary. Had instead been defined on the singleton set where the restriction of to will temporarily be denote by then the prefilter of tails associated with would be the principal prefilter rather than the original filter ; this means that the equality is false, so unlike the prefilter can not be recovered from Worse still, while is the unique minimal filter on the prefilter instead generates a maximal filter (that is, an ultrafilter) on

However, if is a net in then it is not in general true that is equal to because, for example, the domain of may be of a completely different cardinality than that of (since unlike the domain of the domain of an arbitrary net in could have any cardinality).

Ultranets and ultra prefilters

A net is called an ultranet or universal net in if for every subset is eventually in or it is eventually in ; this happens if and only if is an ultra prefilter. A prefilter is an ultra prefilter if and only if is an ultranet in

Partially ordered net

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The domain of the canonical net is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered[37] a construction that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.[36] It begins with the construction of a strict partial order (meaning a transitive and irreflexive relation) on a subset of that is similar to the lexicographical order on of the strict partial orders For any in declare that if and only if or equivalently, if and only if

The non−strict partial order associated with denoted by is defined by declaring that Unwinding these definitions gives the following characterization:

if and only if and also

which shows that is just the lexicographical order on induced by where is partially ordered by equality [note 10] Both are serial and neither possesses a greatest element or a maximal element; this remains true if they are each restricted to the subset of defined by where it will henceforth be assumed that they are. Denote the assignment from this subset by: If then just as with before, the tail of the starting at is equal to If is a prefilter on then is a net in whose domain is a partially ordered set and moreover, [36] Because the tails of are identical (since both are equal to the prefilter ), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered.[36] If the set is replaced with the positive rational numbers then the strict partial order will also be a dense order.

Subordinate filters and subnets

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The notion of " is subordinate to " (written ) is for filters and prefilters what " is a subsequence of " is for sequences.[24] For example, if denotes the set of tails of and if denotes the set of tails of the subsequence (where ) then (that is, ) is true but is in general false.

Non–equivalence of subnets and subordinate filters

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A subset of a preordered space is frequent or cofinal in if for every there exists some If contains a tail of then is said to be eventual or eventually in ; explicitly, this means that there exists some (that is, ). An eventual set is necessarily not empty. A subset is eventual if and only if its complement is not frequent (which is termed infrequent).[38] A map between two preordered sets is order–preserving if whenever

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet."[38] The first definition of a subnet was introduced by John L. Kelley in 1955.[38] Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.[38] AA–subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA–subnets were studied in great detail by Aarnes and Andenaes but they are not often used.[38]

Let be nets. Then[38]
  1. is a Willard–subnet of or a subnet in the sense of Willard if there exists an order–preserving map such that is cofinal in
  2. is a Kelley–subnet of or a subnet in the sense of Kelley if there exists a map and whenever is eventually in then is eventually in
  3. is an AA–subnet of or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
    1. If is eventually in is eventually in
    2. For any subset mesh, then so do
    3. For any subset

Kelley did not require the map to be order preserving while the definition of an AA–subnet does away entirely with any map between the two nets' domains and instead focuses entirely on − the nets' common codomain. Every Willard–subnet is a Kelley–subnet and both are AA–subnets.[38] In particular, if is a Willard–subnet or a Kelley–subnet of then

  • Example: Let and let be a constant sequence, say Let and so that is a net on Then is an AA-subnet of because But is not a Willard-subnet of because there does not exist any map whose image is a cofinal subset of Nor is a Kelley-subnet of because if is any map then is a cofinal subset of but is not eventually in

AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.[38][39] Explicitly, what is meant is that the following statement is true for AA–subnets:

     If are prefilters then is an AA–subnet of

If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, the problem is that the following statement is in general false:

     False statement: If are prefilters such that is a Kelley–subnet of

Since every Willard–subnet is a Kelley–subnet, this statement remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".

  • Counter example: For all let Let which is a proper π–system, and let where both families are prefilters on the natural numbers Because is to as a subsequence is to a sequence. So ideally, should be a subnet of Let be the domain of so contains a cofinal subset that is order isomorphic to and consequently contains neither a maximal nor greatest element. Let is both a maximal and greatest element of The directed set also contains a subset that is order isomorphic to (because it contains which contains such a subset) but no such subset can be cofinal in because of the maximal element Consequently, any order–preserving map must be eventually constant (with value ) where is then a greatest element of the range Because of this, there can be no order preserving map that satisfies the conditions required for to be a Willard–subnet of (because the range of such a map cannot be cofinal in ). Suppose for the sake of contradiction that there exists a map such that is eventually in for all Because there exist such that For every because is eventually in it is necessary that In particular, if then which by definition is equivalent to which is false. Consequently, is not a Kelley–subnet of [39]

If "subnet" is defined to mean Willard–subnet or Kelley–subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley–subnets and Willard–subnets are not fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.[38][39]

See also

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Notes

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  1. ^ Indeed, in both the cases appearing on the right is precisely what makes "greater", for if are related by some binary relation (meaning that ) then whichever one of appears on the right is said to be greater than or equal to the one that appears on the left with respect to (or less verbosely, "–greater than or equal to").
  2. ^ More generally, for any real numbers satisfying where
  3. ^ If This property and the fact that is nonempty and proper if and only if actually allows for the construction of even more examples of prefilters, because if is any prefilter (resp. filter subbase, π–system) then so is
  4. ^ It may be shown that if is any family such that then is a prefilter if and only if for all real there exist real such that
  5. ^ For instance, one sense in which a net could be interpreted as being "maximally deep" is if all important properties related to (such as convergence for example) of any subnet is completely determined by in all topologies on In this case and its subnet become effectively indistinguishable (at least topologically) if one's information about them is limited to only that which can be described in solely in terms of and directly related sets (such as its subsets).
  6. ^ The π–system generated by (resp. by ) is a prefilter whose elements are finite unions of open (resp. closed) intervals having endpoints in with two of these intervals being of the forms (resp. ) where ; in the case of it is possible for one or more of these closed intervals to be singleton sets (that is, degenerate closed intervals).
  7. ^ For an example of how this failure can happen, consider the case where there exists some such that both and its complement in contains at least two distinct points.
  8. ^ Suppose has more than one point, is a constant map, and then will consist of all non–empty subsets of
  9. ^ The set equality holds more generally: if the family of sets then the family of tails of the map (defined by ) is equal to
  10. ^ Explicitly, the partial order on induced by equality refers to the diagonal which is a homogeneous relation on that makes into a partially ordered set. If this partial order is denoted by the more familiar symbol (that is, define ) then for any which shows that (and thus also ) is nothing more than a new symbol for equality on that is, The notation is used because it avoids the unnecessary introduction of a new symbol for the diagonal.

Proofs

  1. ^ Let be a filter on that is not an ultrafilter. If is such that has the finite intersection property (because if ) so that by the ultrafilter lemma, there exists some ultrafilter such that (so in particular, ). Intersecting all such proves that
  2. ^ a b To prove that mesh, let Because (resp. because ), there exists some where by assumption so If is a filter subbase and if then taking implies that If then there are such that and now This shows that is a filter subbase.
  3. ^ This is because if are prefilters on then

Citations

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  1. ^ Jech 2006, p. 73.
  2. ^ Koutras et al. 2021.
  3. ^ a b Cartan 1937a.
  4. ^ a b Cartan 1937b.
  5. ^ a b c d e f Dolecki & Mynard 2016, pp. 27–29.
  6. ^ a b c d e f Dolecki & Mynard 2016, pp. 33–35.
  7. ^ a b c d e f g h i j Narici & Beckenstein 2011, pp. 2–7.
  8. ^ a b c d e f g h i j k l m n o p q r Császár 1978, pp. 53–65.
  9. ^ a b c d e f g h i j k l m n Dolecki & Mynard 2016, pp. 27–54.
  10. ^ a b c d e f g h i j k l m n o p q r s t u v w x y Bourbaki 1987, pp. 57–68.
  11. ^ a b Schubert 1968, pp. 48–71.
  12. ^ a b c Narici & Beckenstein 2011, pp. 3–4.
  13. ^ a b c d e Dugundji 1966, pp. 215–221.
  14. ^ Dugundji 1966, p. 215.
  15. ^ a b c Wilansky 2013, p. 5.
  16. ^ a b c Dolecki & Mynard 2016, p. 10.
  17. ^ a b c d e f g h Schechter 1996, pp. 100–130.
  18. ^ Császár 1978, pp. 82–91.
  19. ^ a b c Dugundji 1966, pp. 211–213.
  20. ^ Schechter 1996, p. 100.
  21. ^ Császár 1978, pp. 53–65, 82–91.
  22. ^ Arkhangel'skii & Ponomarev 1984, pp. 7–8.
  23. ^ Joshi 1983, p. 244.
  24. ^ a b c Dugundji 1966, p. 212.
  25. ^ a b c Wilansky 2013, pp. 44–46.
  26. ^ Castillo, Jesus M. F.; Montalvo, Francisco (January 1990), "A Counterexample in Semimetric Spaces" (PDF), Extracta Mathematicae, 5 (1): 38–40
  27. ^ Schaefer & Wolff 1999, pp. 1–11.
  28. ^ a b c Bourbaki 1987, pp. 129–133.
  29. ^ Wilansky 2008, pp. 32–35.
  30. ^ a b c Dugundji 1966, pp. 219–221.
  31. ^ a b Jech 2006, pp. 73–89.
  32. ^ a b Császár 1978, pp. 53–65, 82–91, 102–120.
  33. ^ a b Dolecki & Mynard 2016, pp. 37–39.
  34. ^ a b c Arkhangel'skii & Ponomarev 1984, pp. 20–22.
  35. ^ a b c d e f g h Császár 1978, pp. 102–120.
  36. ^ a b c d Schechter 1996, pp. 155–171.
  37. ^ Bruns G., Schmidt J., Zur Aquivalenz von Moore-Smith-Folgen und Filtern, Math. Nachr. 13 (1955), 169-186.
  38. ^ a b c d e f g h i Schechter 1996, pp. 157–168.
  39. ^ a b c Clark, Pete L. (18 October 2016). "Convergence" (PDF). math.uga.edu/. Retrieved 18 August 2020.

References

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