In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.
Filters have generalizations called prefilters (also known as filter bases) and filter subbases, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said to generate. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notion is more technically convenient. A preorder on families of sets helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it defines the notion of filter convergence, where by definition, a filter (or prefilter) converges to a point if and only if where is that point's neighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions. In addition, the relation which denotes and is expressed by saying that is subordinate to also establishes a relationship in which is to as a subsequence is to a sequence (that is, the relation which is called subordination, is for filters the analog of "is a subsequence of").
Filters were introduced by Henri Cartan in 1937^{[1]}^{[2]} and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike^{[note 1]} sequence and net convergence, filter convergence is defined entirely in terms of subsets of the topological space and so it provides a notion of convergence that is completely intrinsic to the topological space. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, in general this relationship does not extend to subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship (here it is assumed that "subnet" is defined using any of its most popular definitions, which are given in this article).
Motivation
- Archetypical example of a filter
The archetypical example of a filter is the neighborhood filter at a point in a topological space which by definition is the family of sets consisting of all neighborhoods of By definition, a neighborhood of some given point (or subset) is any subset whose topological interior contains this point (or subset); importantly, neighborhoods are not required to be open sets (those are called open neighborhoods). The fundamental properties shared by neighborhood filters, which are listed below, ultimately became the definition of a "filter." A filter on is a set of subsets of that satisfies all of the following conditions:
- Not empty: – just as since is always an (open) neighborhood of (and of anything else that it contains);
- Does not contain the empty set: – just as no neighborhood of is empty;
- Closed under finite intersections: If then – just as the intersection of any two neighborhoods of is again a neighborhood of ;
- Upward closed: If and then – just as any subset of that contains a neighborhood of will necessarily be a neighborhood of (because and by definition of "neighborhood of ").
- Generalizing sequence convergence by using sets − determining sequence convergence without the sequence
A sequence in is by definition a map from the natural numbers, which are an example of a directed set, into the space The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space. With metrizable spaces (or more generally first–countable spaces or Fréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions. But there are many spaces where sequences can not be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, which never fail to characterize topological properties.
Nets directly generalize the notion of a sequence since nets are, by definition, maps from an arbitrary directed set into the space A sequence is just a net whose domain is with the natural ordering. Nets have their own notion of convergence, which is a direct generalization of sequence convergence.
Filters generalize sequence convergence in a different way by considering only values in the range of a sequence. To see how this is done, consider a sequence in which is by definition just a map whose value at is denoted by rather than the parentheses notation that is commonly used for arbitrary functions. Knowing only the range of the sequence is not enough to describe its convergence; multiple sets are needed. It turns out that the needed sets are the following,^{[note 2]} which are called the tails of the sequence :
These sets completely determine this sequence's convergence (or non–convergence) because given any point, this sequence converges to it if and only if for every neighborhood (of this point), there is some integer such that contains all of the points This can be reworded as:
- every neighborhood must contain some set of the form as a subset.
It is the above characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence With these sets in hand, the map is no longer needed to determine convergence of this sequence (no matter what topology is placed on ). By generalizing this observation, the notion of "convergence" can be extended from maps to families of sets.
The above set of tails of a sequence is in general not a filter but it does "generate" a filter via taking its upward closure. The same is true of other important families of sets such as any neighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it generates the neighborhood filter at that point). The properties that these families share led to the notion of a filter base, also called a prefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward closure only.
- Nets vs. filters − advantages and disadvantages
Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other.^{[note 3]} Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other.^{[3]} Both filters and nets can be used to completely characterize any given topology. Nets are direct generalizations of sequences and can often be used similarly to sequences, so the learning curve for nets is typically much less steep than that for filters. However, filters, and especially ultrafilters, have many more uses outside of topology, such as in set theory, mathematical logic, model theory (e.g. ultraproducts), abstract algebra,^{[4]} order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers.
Like sequences, nets are functions and so they have the advantages of functions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse limits, which is defined in terms of composition of maps rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space and dense subspace ^{[5]}
In contrast to nets, filters (and prefilters) are families of sets and so they have the advantages of sets. For example, if is surjective then the preimage or pullback of an arbitrary filter or prefilter is both easily defined and guaranteed to be a prefilter, whereas it is less clear how to define the pullback of an arbitrary sequence (or net) so that it is once again a sequence or net (unless is also injective and consequently a bijection, which is a stringent requirement). Because filters are composed of subsets of the very topological space that is under consideration, topological set operations (such as closure or interior) may be applied to the sets that constitute the filter. Taking the closure of all the sets in a filter is sometimes useful in Functional Analysis for instance. Theorems about images or preimages of sets under functions (e.g. continuity's definitions in terms of images or preimages of sets) may also be applied to filters. Special types of filters called ultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space In fact, the class of nets in a given set is too large to even be a set (it is a proper class); this is because nets in can have domains of any cardinality. In contrast, the collection of all filters (and of all prefilters) on is a set whose cardinality is no larger than that of Unlike nets and sequences, the notions of a "filter on " and of a "topology on " are both "intrinsic to " in the sense that both consist entirely of the subsets of and do not require any set that cannot be constructed from (such as or other directed sets, which sequences and nets require).
Preliminaries, notation, and basic notions
In this article, upper case Roman letters like and denote sets (but not families unless indicated otherwise) and will denote the powerset of A subset of a powerset 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 and 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 that are used in this article. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (e.g. 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 prefilter 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 ^{[6]}^{[7]} of a family of subsets is
- and similarly the downward closure of is
Notation and Definition | Assumptions | Name |
---|---|---|
Kernel of ^{[7]} | ||
Power set of a set ^{[7]} | ||
is a set | Trace of on ^{[8]} or the restriction of to | |
^{[9]} | Elementwise (set) intersection ( will denote the usual intersection) | |
^{[9]} | Elementwise (set) union ( will denote the usual union) | |
Elementwise (set) subtraction ( will denote the usual set subtraction) | ||
is a set | Dual of in or set subtraction^{[8]} | |
Grill of in ^{[10]} |
Throughout, is a map.
Notation and Definition | Assumptions | Name |
---|---|---|
Preimage of under ^{[14]} | ||
is an arbitrary set. | Preimage a under | |
Image of under ^{[14]} | ||
is an arbitrary set. | Image a under | |
Image of |
- Topology notation
The set of all topologies will be denoted by . Suppose is a topology on
Notation and Definition | Assumptions | Name |
---|---|---|
Set or prefilter^{[note 4]} of open neighborhoods of in | ||
Set or prefilter of open neighborhoods of in | ||
Set or filter^{[note 4]} of neighborhoods of in | ||
Set or filter of neighborhoods of in |
If then and
- 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 and For any indices and 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 and ).
- A net in ^{[15]} is a map from a non–empty directed set into
Notation and Definition | Assumptions | Name |
---|---|---|
and is a directed set | Tail or section of starting at | |
and is a net | Tail or section of starting at ^{[16]} | |
and is a net | Tail or section of starting at | |
is a net | Set or prefilter of tails/sections of Also called the eventuality filter base generated by (the tails of) If is a sequence then is called the sequential filter base instead.^{[16]} | |
is a net | (Eventuality) filter of/generated by (tails of) ^{[16]} |
- 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 (e.g. 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
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:
- Proper or nondegenerate if Otherwise, if then it is called improper^{[17]} or degenerate.
- Directed downward^{[15]} if whenever then there exists some such that
- Alternatively, directed downward (resp. directed upward) if and only if is (upward) directed with respect to the preorder (resp. ), where by definition this means that for all , there exists some "greater" such that and (resp. such that and ), which can be rewritten as (resp.). This explains the word "directed."
- If a family has a greatest element with respect to (for example, if ) then it is necessarily directed downward.
- 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.
- Upward closed or Isotone in ^{[6]} if and , or equivalently, if whenever and satisfies then 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 in 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.
- Ultrafilters(X) = Filters(X) ∩ UltraPrefilters(X) ⊆ Filters(X) ∪ UltraPrefilters(X) ⊆ Prefilters(X) ⊆ FilterSubbases(X).
A family is/is a(n):
- Ideal^{[17]}^{[18]} if is downward closed and closed under finite unions.
- Dual ideal on ^{[19]} if is upward closed in and also closed under finite intersections. Equivalently, is a dual ideal if for all subsets if and only if ^{[10]}
- Explanation of the word "dual": A family is a dual ideal (resp. an ideal) on if and only if the dual of in which is the family
- is an ideal (resp. a dual ideal) on The family should not be confused with where in general The dual of the dual is the original family, meaning ; and also belongs to the dual of if and only if ^{[17]}
- Filter on ^{[19]}^{[8]} 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 dual ideal.^{[20]} It is recommended that readers always check how "filter" is defined when reading mathematical literature. This article uses Henri Cartan's original definition of filter, which required propriety.
- is a filter on if and only if its dual is an ideal that does not contain as an element. If is an ideal on that satisfies then is called its dual filter on
- Prefilter or filter base^{[8]}^{[21]} if is proper and directed downward. Equivalently, is 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.^{[9]} A proper family is a prefilter if and only if ^{[9]}
- 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.
- 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
- Filter subbase^{[8]}^{[22]} and centered^{[9]} if and satisfies any of the following equivalent conditions:
- 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 and then
- The π–system generated by is proper (i.e. is not an element).
- The π–system generated by is a prefilter.
- is a subset of some prefilter.
- is a subset of some filter.
- Assuming is a filter subbase, the filter generated by is the unique smallest (relative to ) filter on containing It is equal to the intersection of all filters on that have as a subset. The π–system generated by denoted by will be a prefilter and a subset of . Moreover, the filter generated by is the upward closure of meaning .^{[9]}
- A 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 may not exist. For this reason, some authors may refer to the π–system generated by as the prefilter generated by However, as shown in an example below, if such a –smallest prefilter does exist then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by " So unfortunately, "the prefilter generated by" a prefilter may not be which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ".
- Subfilter of a filter and that is a superfilter of ^{[17]}^{[23]} if is a filter and where for filters, if and only if
- 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
- Named examples
- The singleton set is called the indiscrete or trivial filter on ^{[25]}^{[11]} 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 ^{[10]} (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 of subsets of is called a neighborhood basis (resp. a neighborhood subbasis) at for 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 and 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 in
- 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 countablly infinite set.^{[10]} The intersection of finitely many sequential filters is again sequential.^{[10]}
- 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 ^{[11]}^{[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 in 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 and ) filter contained as a subset of each member of ^{[11]}
- If and are filters then their infimum in is the filter ^{[9]} If and are prefilters then is a prefilter and one of the finest (with respect to ) prefilters coarser (with respect to ) than both and ; that is, if is a prefilter such that and then ^{[9]} More generally, if and are non−empty families and if then and is a greatest element (with respect to ) of ^{[9]}
- Let and let The supremum or least upper bound of in 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 in 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 ^{[11]} (as defined above) and will also be equal to the intersection of all filters on containing
This supremum of in 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.^{[11]} Indeed, if contains at least 2 distinct elements then there exist filters and on for which there does not exist a filter on that contains both and
If is not a filter subbase then the supremum of in 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 denegerate filter ).^{[10]}
- If and 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 and 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 and ; this means that if is any prefilter (resp. any filter) such that and then necessarily ^{[9]} in which case it is denoted by ^{[10]}
- Let and 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]}
- 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 (with ), then the set of all such that has finite Lebesgue measure is a proper π–system and prefilter that is also a proper subset of The prefilters and generate the same filter on
- This example illustrates a 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 (i.e. and ) are needed to describe the generated π-system. However, this is not typical and in general, this should not be expected of a filter subbase that is not a π-system. More often, an intersection of sets from will usually require a description involving variables that cannot be reduced down to only two (consider, for instance, if ). For all let where so no generality is lost by adding the assumption For all real and if or then ^{[note 5]} For every let and let ^{[note 6]} 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 and generate the same filter on if and only if If is a family such that then is a prefilter if and only if for all real there exist real such that and If is such a prefilter then for any the family is also a prefilter satisfying This shows that there cannot exist a minimal (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.
Ultrafilters
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:
- Ultra^{[8]}^{[30]} if and any of the following equivalent conditions are satisfied:
- For every set there exists some set such that or (or equivalently, such that equals or ).
- For every set there exists some set such that equals or
- This characterization of " is ultra" does not depend on the set so mentioning the set is optional when using the term "ultra."
- For every set (not necessarily even a subset of ) there exists some set such that equals or
- If satisfies this condition then so does every superset In particular, a set is ultra if and only if and contains as a subset some ultra family of sets.
- Ultra prefilter^{[8]}^{[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:
- is maximal in with respect to which means that if then implies
- If then implies
- is ultra (and thus an ultrafilter).
- is equivalent (with respect to ) to some ultrafilter.
- A filter subbase that is ultra is necessarily a prefilter.^{[proof 1]} A filter subbase is ultra if and only if it is a maximal filter subbase with respect to (as above).^{[31]}
- Ultrafilter on ^{[8]}^{[30]} if it is a filter on that is ultra. Equivalently, an ultrafilter on is a filter on that satisfies any of the following equivalent conditions:
- is generated by an ultra prefilter.
- For any or ^{[17]}
- This condition can be restated as: is partitioned by and its dual
- The sets and are disjoint whenever is a prefilter.
- is an ideal.^{[31]}
- For any if then or
- For any if then or (a filter with this property is called a prime filter).
- This property extends to any finite union of two or more sets.
- For any if and then either or
- is a maximal filter on ; meaning that if is a filter on such that then
- An ultra prefilter has a similar characterization in terms of maximality with respect to where in the special case of filters, if and only if
- Because is for filters the analog of "is a subnet of," (specifically, "subnet" should mean "AA-subnet," which is defined below) an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net." This idea is actually made rigorous by ultranets.
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 on is ultra if and only if is a singleton set.
- Grills and Filter-Grills
If then its grill on is the family
where may be written if is clear from context. For example, and if then If then and moreover, if is a filter subbase then ^{[10]} The grill is upward closed in if and only if which will henceforth be assumed. Moreover, so that is upward closed in if and only if
The grill of a filter on is called a filter-grill on ^{[10]} For any is a filter-grill on if and only if (1) is upward closed in and (2) for all sets and if then or The grill operation induces a bijection whose inverse is also given by ^{[10]} If then is a filter-grill on if and only if ^{[10]} or equivalently, if and only if is an ultrafilter on ^{[10]} That is, a filter on is a filter-grill if and only if it is ultra. For any non-empty is both a filter on and a filter-grill on if and only if (1) and (2) for all the following equivalences hold:
- if and only if if and only if ^{[10]}
- The ultrafilter lemma
The following important theorem is due to Alfred Tarski (1930).^{[32]}
The ultrafilter lemma/principal/theorem^{[11]} (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.^{[11]}^{[proof 2]} 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.
Free, principal, and kernels
The kernel is useful in classifying properties of prefilters and other families of sets.
If then for any point if and only if
- Properties of kernels
For any the 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 If and are principal then they are equivalent if and only if
- Classifying families of sets by their kernels
A family of sets is/is an:
- Free^{[7]} 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.
- Fixed if in which case, is said to be fixed by any point
- Any fixed family is necessarily a filter subbase.
- Principal^{[7]} if
- A proper principal family of sets is necessarily a prefilter.
- Discrete or Principal at ^{[25]} if
- The principal filter at on is the filter A filter is principal at if and only if
- Countably deep if whenever is a countable subset then ^{[10]}
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 (e.g. the primes), a meager set in a set of finite measure, or a bounded subset of If is a singleton set then is a subbasis for the Fréchet filter on
For every filter on there exists a unique pair of dual ideals and on such that is free, is principal, and and and do not mesh (i.e. ). The dual ideal is called the free part of while is called the principal part^{[10]} where at least one of these dual ideals is filter. If is principal then and ; otherwise, and is a free (non-degenerate) filter.^{[10]}
- Characterizations of fixed ultra prefilters
If a family of sets is fixed (i.e. ) 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.^{[7]}
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:
- is fixed, or equivalently, not free, meaning .
- is principal, meaning
- Some element of is a finite set.
- Some element of is a singleton set.
- is principal at some point of which means for some
- does not contain the Fréchet filter on
- is sequential.^{[10]}
- Finite prefilters and finite sets
If a filter subbase is finite then it is fixed (i.e. 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 (or 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 . This set is a principal ultra prefilter and any superset (where and ) with the finite intersection property will also be a principal ultra prefilter (even if is infinite).
Finer/coarser, subordination, and meshing
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 be 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 and mesh^{[8]} and are compatible, indicated by writing if for all