Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, 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 notions is more technically convenient. There is a certain preorder on families of sets, denoted by that 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 also 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]} 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; indeed, the category of topological spaces can be equivalently defined entirely in terms of filters. 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, assuming that "subnet" is defined using either of its most popular definitions (which are those given by Willard and by Kelley), then 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; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that of an AA–subnet.
Thus filters/prefilters and this single preorder provide a framework that seamlessly ties together fundamental topological concepts such as topological spaces (via neighborhood filters), neighborhood bases, convergence, various limits of functions, continuity, compactness, sequences (via sequential filters), the filter equivalent of "subsequence" (subordination), uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear.
Archetypical example of a filter
See also: Filter (set theory) |
The archetypical example of a filter is the neighborhood filter at a point in a topological space which is the family of sets consisting of all neighborhoods of By definition, a neighborhood of some given point is any subset whose topological interior contains this point; that is, such that Importantly, neighborhoods are not required to be open sets; those are called open neighborhoods. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." A filter on is a set of subsets of that satisfies all of the following conditions:
Generalizing sequence convergence by using sets − determining sequence convergence without the sequence
See also: Limit of a sequence and Net (mathematics) |
A sequence in is by definition a map from the natural numbers 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 the values of a sequence. To see how this is done, consider a sequence which is by definition just a function whose value at is denoted by rather than by the usual parentheses notation that is commonly used for arbitrary functions. Knowing only the image (sometimes called "the range") of the sequence is not enough to characterize 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.
Or more briefly: every neighborhood must contain some tail as a subset. It is this characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence Specifically, with the family of sets in hand, the function 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 sequences/functions 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 (which consists of all supersets of all tails). 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.
Nets versus 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.^{[2]} 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 (ultraproducts, for example), abstract algebra,^{[3]} combinatorics,^{[4]} dynamics,^{[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 functions 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 a filter on a 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 image under of an arbitrary filter or prefilter is both easily defined and guaranteed to be a prefilter on 's domain, whereas it is less clear how to pullback (unambiguously/without choice) an arbitrary sequence (or net) so as to obtain a sequence or net in the domain (unless is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets. 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 and results about images or preimages of sets under a function may also be applied to the sets that constitute a filter; an example of such a result might be one of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. 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 Similar to a topology on a filter on is "intrinsic to " in the sense that both structures consist entirely of subsets of and neither definition requires any set that cannot be constructed from (such as or other directed sets, which sequences and nets require).
Main article: Filter (set theory) |
In this article, upper case Roman letters like 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 ^{[6]}^{[7]} of a family of sets is
and similarly the downward closure of is
Notation and Definition | Name |
---|---|
Kernel of ^{[7]} | |
Dual of where is a set.^{[8]} | |
Trace of ^{[8]} or the restriction of where is a set; sometimes denoted by | |
^{[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) | |
Power set of a set ^{[7]} |
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
If and then are said to be equivalent (with respect to subordination).
Two families mesh,^{[8]} written if
Throughout, is a map.
Notation and Definition | Name |
---|---|
^{[13]} | Image of or the preimage of under |
^{[14]} | Image of under |
Image (or range) of |
Topology notation
Denote the set of all topologies on a set Suppose is any subset, and is any point.
Notation and Definition | Name |
---|---|
Set or prefilter^{[note 4]} of open neighborhoods of | |
Set or prefilter of open neighborhoods of | |
Set or filter^{[note 4]} of neighborhoods of | |
Set or filter of neighborhoods of |
If then
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
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
π-system | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
𝜆-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
δ-Ring | Never | |||||||||
𝜎-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
𝜎-Algebra (𝜎-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
Open Topology | (even arbitrary ) |
Never | ||||||||
Closed Topology | (even arbitrary ) |
Never | ||||||||
Is necessarily true of or, is closed under: |
directed downward |
finite intersections |
finite unions |
relative complements |
complements in |
countable intersections |
countable unions |
contains | contains | Finite Intersection Property |
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in |
Main article: Filter (set theory) |
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
- 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, − which can be rewritten as (resp. as ).
- 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 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):
- 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 ^{[20]}
- 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 dual of the dual is the original family, meaning ^{[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/non–degenerate dual ideal.^{[21]} 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",^{[1]}^{[22]} which required non–degeneracy.
- 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.
- Prefilter or filter base^{[8]}^{[23]} 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 to some filter.^{[9]} A proper family is a prefilter if and only if ^{[9]} 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.
- 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 to the π–system generated by it and both of these families generate the same filter on
- Filter subbase^{[8]}^{[24]} 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 then
- The π–system generated by is proper; that is,
- The π–system generated by is a prefilter.
- is a subset of some prefilter.
- is a subset of some filter.^{[10]}
- 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 ^{[9]} 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 ".
- Subfilter of a filter and that is a superfilter of ^{[17]}^{[25]} 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,"^{[26]} 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.
Named examples
Other examples
Main articles: Ultrafilter (set theory) and Ultrafilter |
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]}^{[33]} if and any of the following equivalent conditions are satisfied:
- For every set there exists some set such that (or equivalently, such that ).
- 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."
- For every set (not necessarily even a subset of ) there exists some set such that
- Ultra prefilter^{[8]}^{[33]} 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
- 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.
- is ultra (and thus an ultrafilter).
- is equivalent 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]}
- Ultrafilter on ^{[8]}^{[33]} if it is a filter on that is ultra. Equivalently, an ultrafilter on is a filter that satisfies any of the following equivalent conditions:
- is generated by an ultra prefilter.
- For any ^{[17]}
- This condition can be restated as: is partitioned by and its dual
- 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.
- 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").
The ultrafilter lemma
The following important theorem is due to Alfred Tarski (1930).^{[34]}
The ultrafilter lemma/principle/theorem^{[28]} (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.^{[28]} 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.
The kernel is useful in classifying properties of prefilters and other families of sets.
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:
If is a map then Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal.
A family of sets is:
- Free^{[7]} if or equivalently, if this can be restated as
- A filter 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 ^{[27]} if
- The principal filter at is the filter A filter is principal at if and only if
- Countably deep if whenever