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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
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
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).
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 |
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
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 is a countable subset then [20]
If is a principal filter on then and and 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
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. [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:
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", [26] 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 [8] 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
Declare that stated as is coarser than and is finer than (or subordinate to) [28] [11] [12] [9] [20] if any of the following equivalent conditions hold:
- Definition: Every contains some Explicitly, this means that for every there is some such that (thus holds).
- 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.
- which is equivalent to ;
- ;
- which is equivalent to ;
and if in addition is upward closed, which means that then this list can be extended to include:
- [6]
- 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 them is finer than the other. [28]
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,
A non-empty family that is coarser than a filter subbase must itself be a filter subbase. [9] Every filter subbase is coarser than both the π–system that it generates and the filter that it generates. [9]
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.
The relation is reflexive and transitive, which makes it into a preorder on [35] The relation is antisymmetric but if has more than one point then it is not symmetric.
The preorder induces its canonical equivalence relation on where for all is equivalent to if any of the following equivalent conditions hold: [9] [6]
Two upward closed (in ) subsets of are equivalent if and only if they are equal. [9] 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 [9]
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 : [35]
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:
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. [9] 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. [9]
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.
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, [28]), in which case the trace of is said to be induced by . The trace is always finer than the original family; that is, 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 [28]
When prefilters mesh
Given non–empty families the family
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 [36]
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): [28] [13] [37] [38] [39] [34] ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate, ideal, closed under finite unions, downward closed, directed upward. Moreover, if is a prefilter then so are both [28] 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. [37] Otherwise it is just a prefilter on and its upward closure must be taken in to obtain a filter. The upward closure of is
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 [28]
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 [38]
If is not surjective then denote the trace of by where in this case particular case the trace satisfies:
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] [28] [39]
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] [28] [39]
and moreover, if is a prefilter then so is [13] [28]
If and if denotes the inclusion map then the trace of is equal to [28] This observation allows the results in this subsection to be applied to investigating the trace on a set.
The relation is preserved under both images and preimages of families of sets. [28] This means that for any families [39]
Moreover, the following relations always hold for any family of sets : [39]
If then [20]
Suppose is a family of one or more non–empty sets, whose product will be denoted by and for every index let
Throughout, is a topological space.
Prefilters vs. filters
With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If is a proper subset then any filter on will not be a filter on although it will be a prefilter.
One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.
A note on intuition
Suppose that is a non–principal filter on an infinite set has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any there always exists some that is a proper subset of ; this may be continued ad infinitum to get a sequence of sets in with each being a proper subset of The same is not true going "upward", for if then there is no set in that contains as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.
A family is said to converge in to a point of [8] if Explicitly, means that every neighborhood contains some as a subset (that is, ); thus the following then holds: In words, a family converges to a point or subset if and only if it is finer than the neighborhood filter at A family converging to a point may be indicated by writing [32] and saying that is a limit of if this limit is a point (and not a subset), then is also called a limit point. [40] As usual, is defined to mean that and is the only limit point of that is, if also [32] (If the notation "" did not also require that the limit point be unique then the equals sign = would no longer be guaranteed to be transitive). The set of all limit points of is denoted by [8]
In the above definitions, it suffices to check that is finer than some (or equivalently, finer than every) neighborhood base in of the point (for example, such as or when ).
Examples
If is Euclidean space and denotes the Euclidean norm (which is the distance from the origin, defined as usual), then all of the following families converge to the origin:
Although was assumed to be the Euclidean norm, the example above remains valid for any other norm on
The one and only limit point in of the free prefilter is since every open ball around the origin contains some open interval of this form. The fixed prefilter does not converges in to any point and so although does converge to the set since However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter also has kernel but does not converges (in ) to it.
The free prefilter of intervals does not converge (in ) to any point. The same is also true of the prefilter because it is equivalent to and equivalent families have the same limits. In fact, if is any prefilter in any topological space then for every More generally, because the only neighborhood of is itself (that is, ), every non-empty family (including every filter subbase) converges to
For any point its neighborhood filter always converges to More generally, any neighborhood basis at converges to A point is always a limit point of the principle ultra prefilter and of the ultrafilter that it generates. The empty family does not converge to any point.
Basic properties
If converges to a point then the same is true of any family finer than This has many important consequences. One consequence is that the limit points of a family are the same as the limit points of its upward closure:
Given the following are equivalent for a prefilter
Because subordination is transitive, if and moreover, for every both and the maximal/ultrafilter converge to Thus every topological space induces a canonical convergence defined by At the other extreme, the neighborhood filter is the smallest (that is, coarsest) filter on that converges to that is, any filter converging to must contain as a subset. Said differently, the family of filters that converge to consists exactly of those filter on that contain as a subset. Consequently, the finer the topology on then the fewer prefilters exist that have any limit points in
A family is said to cluster at a point of if it meshes with the neighborhood filter of that is, if Explicitly, this means that and every neighborhood of In particular, a point is a cluster point or an accumulation point of a family [8] if meshes with the neighborhood filter at The set of all cluster points of is denoted by where the subscript may be dropped if not needed.
In the above definitions, it suffices to check that meshes with some (or equivalently, meshes with every) neighborhood base in of When is a prefilter then the definition of " mesh" can be characterized entirely in terms of the subordination preorder
Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every both and the principal ultrafilter cluster at If clusters to a point then the same is true of any family coarser than Consequently, the cluster points of a family are the same as the cluster points of its upward closure:
Given the following are equivalent for a prefilter :
The set of all cluster points of a prefilter satisfies
Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have any cluster points or limit points. [43]
If is a limit point of then is necessarily a limit point of any family finer than (that is, if then ). [43] In contrast, if is a cluster point of then is necessarily a cluster point of any family coarser than (that is, if mesh and then mesh).
Equivalent families and subordination
Any two equivalent families can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that if and only if and also that if and only if In essence, the preorder is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder is of such great importance in applying (pre)filters to Topology.
Limit and cluster point relationships and sufficient conditions
Every limit point of a non-degenerate family is also a cluster point; in symbols:
However, every cluster point of an ultra prefilter is a limit point. Consequently, the limit points of an ultra prefilter are the same as its cluster points: that is to say, a given point is a cluster point of an ultra prefilter if and only if converges to that point. [33] [44] Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if clusters at then is a filter subbase whose generated filter converges to
If is a filter subbase such that then In particular, any limit point of a filter subbase subordinate to is necessarily also a cluster point of If is a cluster point of a prefilter then is a prefilter subordinate to that converges to
If and if is a prefilter on then every cluster point of belongs to and any point in is a limit point of a filter on [43]
Primitive sets
A subset is called primitive [45] if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter such that is equal to which recall denotes the set of limit points of Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set of cluster points of some ultra prefilter For example, every closed singleton subset is primitive. [45] The image of a primitive subset of under a continuous map is contained in a primitive subset of [45]
Assume that are two primitive subset of If is an open subset of that intersects then for any ultrafilter such that [45] In addition, if are distinct then there exists some and some ultrafilters such that and [45]
Other results
If is a complete lattice then:[ citation needed]
Suppose is a map from a set into a topological space and If is a limit point (respectively, a cluster point) of then is called a limit point or limit (respectively, a cluster point) of with respect to [43] Explicitly, is a limit of with respect to if and only if which can be written as (by definition of this notation) and stated as tend to along [46] If the limit is unique then the arrow may be replaced with an equals sign [32] The neighborhood filter can be replaced with any family equivalent to it and the same is true of
The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if is a net then
The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under ) of particular prefilters on the domain This shows that prefilters provide a general framework into which many of the various definitions of limits fit. [41] The limits in the left–most column are defined in their usual way with their obvious definitions.
Throughout, let be a map between topological spaces, If is Hausdorff then all arrows "" in the table may be replaced with equal signs "" and "" may be replaced with "". [32]
Type of limit | if and only if | Definition in terms of prefilters [41] | Assumptions |
---|---|---|---|
⇔ | |||
⇔ | |||
or |
⇔ | ||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | is a sequence in | ||
⇔ | |||
⇔ | |||
⇔ | for a double-ended sequence | ||
⇔ | a seminormed space; for example, a Banach space |
By defining different prefilters, many other notions of limits can be defined; for example,
Divergence to infinity
Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters
So for example, if then if and only if holds. Similarly, if and only if or equivalently, if and only if
More generally, if is valued in (or some other seminormed vector space) and if then if and only if holds, where
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.
In the definitions below, the first statement is the standard definition of a limit point of a net (respectively, a cluster point of a net) and it is gradually reworded until the corresponding filter concept is reached.
A net is said to converge in to a point written and is called a limit or limit point of [47] if any of the following equivalent conditions hold:As usual, is defined to mean that and is the only limit point of that is, if also [47]
- Definition: For every there exists some such that if
- For every there exists some such that the tail of starting at is contained in (that is, such that ).
- For every there exists some such that
- that is, the prefilter converges to
A point is called a cluster or accumulation point of a net if any of the following equivalent conditions hold:
- Definition: For every and every there exists some such that
- For every and every the tail of starting at intersects (that is, ).
- For every and every
- mesh (by definition of "mesh").
- is a cluster point of
If is a map and is a net in then [3]
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
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 mapthat is,
If is a prefilter on is a net in and the prefilter associated with is ; that is: [note 6]
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).
Proposition — If is a prefilter on and then
Recall that and that if is a net in then (1) and (2) is a cluster point of if and only if is a cluster point of By using it follows that
Partially ordered net
The domain of the canonical net is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered [48] a construction (detailed here: Filter (set theory)#Partially ordered net) 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. [3] Because the tails of this partially ordered net are identical to the tails of (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. [3] If can further be assumed that the partially ordered domain is also a dense order.
The notion of " is subordinate to " (written ) is for filters and prefilters what " is a subsequence of " is for sequences. [26] For example, if denotes the set of tails of and if denotes the set of tails of the subsequence (where ) then (which by definition means ) is true but is in general false. If is a net in a topological space and if is the neighborhood filter at a point then
If is an surjective open map, and is a prefilter on that converges to then there exist a prefilter on such that and is equivalent to (that is, ). [49]
The following results are the prefilter analogs of statements involving subsequences. [50] The condition "" which is also written is the analog of " is a subsequence of " So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."
Proposition [50] [43] — Let be a prefilter on and let
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of " subnet." [53] The first definition of a subnet ("Kelley–subnet") was introduced by John L. Kelley in 1955. [53] Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet. [53] 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. [53]
A subset of a preordered space is frequent or cofinal in if for every there exists some such that If contains a tail of then is said to be eventual in }}; explicitly, this means that there exists some such that (that is, for all satisfying ). A subset is eventual if and only if its complement is not frequent (which is termed infrequent). [53] A map between two preordered sets is order–preserving if whenever satisfy then
Definitions: Let be nets. Then [53]
- 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
- is a Kelley–subnet of or a subnet in the sense of Kelley if there exists a map such that and whenever is eventual in then is eventual in
- is an AA–subnet of or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
- If is eventual in is eventual in
- For any subset mesh, then so do
- 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. [53] In particular, if is a Willard–subnet or a Kelley–subnet of then
AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters. [53] [54] Explicitly, what is meant is that the following statement is true for AA–subnets:
If are prefilters then if and only if is an AA–subnet of
If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, as this counter-example demonstrates, 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 thus remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".
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. [53] [54]
Throughout, is a topological space.
Bases and prefilters
Let be a family of sets that covers and define for every The definition of a base for some topology can be immediately reworded as: is a base for some topology on if and only if is a filter base for every If is a topology on and then the definitions of is a basis (resp. subbase) for can be reworded as:
is a base (resp. subbase) for if and only if for every is a filter base (resp. filter subbase) that generates the neighborhood filter of at
Neighborhood filters
The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter."
Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If has its usual topology and if then any neighborhood filter base of is fixed by (in fact, it is even true that ) but is not principal since In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non–principal filter on an infinite set is not necessarily free.
The neighborhood filter of every point in topological space is fixed since its kernel contains (and possibly other points if, for instance, is not a T1 space). This is also true of any neighborhood basis at For any point in a T1 space (for example, a Hausdorff space), the kernel of the neighborhood filter of is equal to the singleton set
However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis of a point in a topological space is principal if and only if the kernel of is an open set. If in addition the space is T1 then so that this basis is principal if and only if is an open set.
Generating topologies from filters and prefilters
Suppose is not empty (and ). If is a filter on then is a topology on but the converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form where is an ultrafilter on are an even more specialized subclass of such topologies; they have the property that every proper subset is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces.
If is a prefilter (resp. filter subbase, π–system, proper) on then the same is true of both and the set of all possible unions of one or more elements of If is closed under finite intersections then the set is a topology on with both being bases for it. If the π–system covers then both are also bases for If is a topology on then is a prefilter (or equivalently, a π–system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset will be a basis for if and only if is equivalent to in which case will be a prefilter.
Neighborhoods and topologies
The neighborhood filter of a nonempty subset in a topological space is equal to the intersection of all neighborhood filters of all points in [55] A subset is open in if and only if whenever is a filter on and then
Suppose are topologies on Then is finer than (that is, ) if and only if whenever is a filter on if then [45] Consequently, if and only if for every filter and every if and only if [32] However, it is possible that while also for every filter converges to some point of if and only if converges to some point of [32]
Closure
If is a prefilter on a subset then every cluster point of belongs to [44]
If is a non-empty subset, then the following are equivalent:
The following are equivalent:
Closed sets
If is not empty then the following are equivalent:
Hausdorffness
The following are equivalent:
Compactness
As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.
The following are equivalent:
If is the set of all complements of compact subsets of a given topological space then is a filter on if and only if is not compact.
Theorem [57] — If is a filter on a compact space and is the set of cluster points of then every neighborhood of belongs to Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point.
Continuity
Let be a map between topological spaces
Given the following are equivalent:
The following are equivalent:
If is a prefilter on is a cluster point of is continuous, then is a cluster point in of the prefilter [45]
A subset of a topological space is dense in if and only if for every the trace of the neighborhood filter along does not contain the empty set (in which case it will be a filter on ).
Suppose is a continuous map into a Hausdorff regular space and that is a dense subset of a topological space Then has a continuous extension if and only if for every the prefilter converges to some point in Furthermore, this continuous extension will be unique whenever it exists. [58]
Products
Suppose is a non–empty family of non–empty topological spaces and that is a family of prefilters where each is a prefilter on Then the product of these prefilters (defined above) is a prefilter on the product space which as usual, is endowed with the product topology.
If then if and only if
Suppose are topological spaces, is a prefilter on having as a cluster point, and is a prefilter on having as a cluster point. Then is a cluster point of in the product space [45] However, if then there exist sequences such that both of these sequences have a cluster point in but the sequence does not have a cluster point in [45]
Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:
Proof
|
---|
Let be compact Hausdorff topological spaces. Assume that the ultrafilter lemma holds (because of the Hausdorff assumption, this proof does not need the full strength of the axiom of choice; the ultrafilter lemma suffices). Let be given the product topology (which makes a Hausdorff space) and for every let denote this product's projections. If then is compact and the proof is complete so assume Despite the fact that because the axiom of choice is not assumed, the projection maps are not guaranteed to be surjective. Let be an ultrafilter on and for every let denote the ultrafilter on generated by the ultra prefilter Because is compact and Hausdorff, the ultrafilter converges to a unique limit point (because of 's uniqueness, this definition does not require the axiom of choice). Let where satisfies for every The characterization of convergence in the product topology that was given above implies that Thus every ultrafilter on converges to some point of which implies that is compact (recall that this implication's proof only required the ultrafilter lemma). |
A uniform space is a set equipped with a filter on that has certain properties. A base or fundamental system of entourages is a prefilter on whose upward closure is a uniform space. A prefilter on a uniform space with uniformity is called a Cauchy prefilter if for every entourage there exists some that is –small, which means that A minimal Cauchy filter is a minimal element (with respect to or equivalently, to ) of the set of all Cauchy filters on Examples of minimal Cauchy filters include the neighborhood filter of any point Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point.
A uniform space is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on converges to at least one point of (replacing all instance of the word "prefilter" with "filter" results in equivalent statement). Every compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy).
Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first–countable, or even sequential. The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS) can made into a vector space and topologized in such a way that it becomes a completion of (with the assignment becoming a linear topological embedding that identifies as a dense vector subspace of this completion).
More generally, a Cauchy space is a pair consisting of a set together a family of (proper) filters, whose members are declared to be "Cauchy filters", having all of the following properties:
The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space. A map between two Cauchy spaces is called Cauchy continuous if the image of every Cauchy filter in is a Cauchy filter in Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.
Starting with nothing more than a set it is possible to topologize the set
To reduce confusion, this article will adhere to the following notational conventions:
For every let
From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of [note 9] For all
In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on was defined without using anything other than the set there were no preexisting structures or assumptions on so this topology is completely independent of everything other than (and its subsets).
The following criteria can be used for checking for points of closure and neighborhoods. If then:
It will be henceforth assumed that because otherwise and the topology is which is uninteresting.
Subspace of ultrafilters
The set of ultrafilters on (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If has the discrete topology then the map defined by sending to the principal ultrafilter at is a topological embedding whose image is a dense subset of (see the article Stone–Čech compactification for more details).
Relationships between topologies on and the Stone topology on
Every induces a canonical map defined by which sends to the neighborhood filter of If then if and only if Thus every topology can be identified with the canonical map which allows to be canonically identified as a subset of (as a side note, it is now possible to place on and thus also on the topology of pointwise convergence on so that it now makes sense to talk about things such as sequences of topologies on converging pointwise). For every the surjection is always continuous, closed, and open, but it is injective if and only if (that is, a Kolmogorov space). In particular, for every topology the map is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters).
In addition, if is a map such that (which is true of for instance), then for every the set is a neighborhood (in the subspace topology) of
Proofs
This article may be
too long to read and navigate comfortably. (November 2022) |
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
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
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).
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 |
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
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 is a countable subset then [20]
If is a principal filter on then and and 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
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. [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:
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", [26] 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 [8] 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
Declare that stated as is coarser than and is finer than (or subordinate to) [28] [11] [12] [9] [20] if any of the following equivalent conditions hold:
- Definition: Every contains some Explicitly, this means that for every there is some such that (thus holds).
- 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.
- which is equivalent to ;
- ;
- which is equivalent to ;
and if in addition is upward closed, which means that then this list can be extended to include:
- [6]
- 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 them is finer than the other. [28]
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,
A non-empty family that is coarser than a filter subbase must itself be a filter subbase. [9] Every filter subbase is coarser than both the π–system that it generates and the filter that it generates. [9]
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.
The relation is reflexive and transitive, which makes it into a preorder on [35] The relation is antisymmetric but if has more than one point then it is not symmetric.
The preorder induces its canonical equivalence relation on where for all is equivalent to if any of the following equivalent conditions hold: [9] [6]
Two upward closed (in ) subsets of are equivalent if and only if they are equal. [9] 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 [9]
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 : [35]
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:
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. [9] 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. [9]
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.
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, [28]), in which case the trace of is said to be induced by . The trace is always finer than the original family; that is, 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 [28]
When prefilters mesh
Given non–empty families the family
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 [36]
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): [28] [13] [37] [38] [39] [34] ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate, ideal, closed under finite unions, downward closed, directed upward. Moreover, if is a prefilter then so are both [28] 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. [37] Otherwise it is just a prefilter on and its upward closure must be taken in to obtain a filter. The upward closure of is
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 [28]
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 [38]
If is not surjective then denote the trace of by where in this case particular case the trace satisfies:
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] [28] [39]
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] [28] [39]
and moreover, if is a prefilter then so is [13] [28]
If and if denotes the inclusion map then the trace of is equal to [28] This observation allows the results in this subsection to be applied to investigating the trace on a set.
The relation is preserved under both images and preimages of families of sets. [28] This means that for any families [39]
Moreover, the following relations always hold for any family of sets : [39]
If then [20]
Suppose is a family of one or more non–empty sets, whose product will be denoted by and for every index let
Throughout, is a topological space.
Prefilters vs. filters
With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If is a proper subset then any filter on will not be a filter on although it will be a prefilter.
One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to ), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.
A note on intuition
Suppose that is a non–principal filter on an infinite set has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any there always exists some that is a proper subset of ; this may be continued ad infinitum to get a sequence of sets in with each being a proper subset of The same is not true going "upward", for if then there is no set in that contains as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.
A family is said to converge in to a point of [8] if Explicitly, means that every neighborhood contains some as a subset (that is, ); thus the following then holds: In words, a family converges to a point or subset if and only if it is finer than the neighborhood filter at A family converging to a point may be indicated by writing [32] and saying that is a limit of if this limit is a point (and not a subset), then is also called a limit point. [40] As usual, is defined to mean that and is the only limit point of that is, if also [32] (If the notation "" did not also require that the limit point be unique then the equals sign = would no longer be guaranteed to be transitive). The set of all limit points of is denoted by [8]
In the above definitions, it suffices to check that is finer than some (or equivalently, finer than every) neighborhood base in of the point (for example, such as or when ).
Examples
If is Euclidean space and denotes the Euclidean norm (which is the distance from the origin, defined as usual), then all of the following families converge to the origin:
Although was assumed to be the Euclidean norm, the example above remains valid for any other norm on
The one and only limit point in of the free prefilter is since every open ball around the origin contains some open interval of this form. The fixed prefilter does not converges in to any point and so although does converge to the set since However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter also has kernel but does not converges (in ) to it.
The free prefilter of intervals does not converge (in ) to any point. The same is also true of the prefilter because it is equivalent to and equivalent families have the same limits. In fact, if is any prefilter in any topological space then for every More generally, because the only neighborhood of is itself (that is, ), every non-empty family (including every filter subbase) converges to
For any point its neighborhood filter always converges to More generally, any neighborhood basis at converges to A point is always a limit point of the principle ultra prefilter and of the ultrafilter that it generates. The empty family does not converge to any point.
Basic properties
If converges to a point then the same is true of any family finer than This has many important consequences. One consequence is that the limit points of a family are the same as the limit points of its upward closure:
Given the following are equivalent for a prefilter
Because subordination is transitive, if and moreover, for every both and the maximal/ultrafilter converge to Thus every topological space induces a canonical convergence defined by At the other extreme, the neighborhood filter is the smallest (that is, coarsest) filter on that converges to that is, any filter converging to must contain as a subset. Said differently, the family of filters that converge to consists exactly of those filter on that contain as a subset. Consequently, the finer the topology on then the fewer prefilters exist that have any limit points in
A family is said to cluster at a point of if it meshes with the neighborhood filter of that is, if Explicitly, this means that and every neighborhood of In particular, a point is a cluster point or an accumulation point of a family [8] if meshes with the neighborhood filter at The set of all cluster points of is denoted by where the subscript may be dropped if not needed.
In the above definitions, it suffices to check that meshes with some (or equivalently, meshes with every) neighborhood base in of When is a prefilter then the definition of " mesh" can be characterized entirely in terms of the subordination preorder
Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every both and the principal ultrafilter cluster at If clusters to a point then the same is true of any family coarser than Consequently, the cluster points of a family are the same as the cluster points of its upward closure:
Given the following are equivalent for a prefilter :
The set of all cluster points of a prefilter satisfies
Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have any cluster points or limit points. [43]
If is a limit point of then is necessarily a limit point of any family finer than (that is, if then ). [43] In contrast, if is a cluster point of then is necessarily a cluster point of any family coarser than (that is, if mesh and then mesh).
Equivalent families and subordination
Any two equivalent families can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that if and only if and also that if and only if In essence, the preorder is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder is of such great importance in applying (pre)filters to Topology.
Limit and cluster point relationships and sufficient conditions
Every limit point of a non-degenerate family is also a cluster point; in symbols:
However, every cluster point of an ultra prefilter is a limit point. Consequently, the limit points of an ultra prefilter are the same as its cluster points: that is to say, a given point is a cluster point of an ultra prefilter if and only if converges to that point. [33] [44] Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if clusters at then is a filter subbase whose generated filter converges to
If is a filter subbase such that then In particular, any limit point of a filter subbase subordinate to is necessarily also a cluster point of If is a cluster point of a prefilter then is a prefilter subordinate to that converges to
If and if is a prefilter on then every cluster point of belongs to and any point in is a limit point of a filter on [43]
Primitive sets
A subset is called primitive [45] if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter such that is equal to which recall denotes the set of limit points of Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set of cluster points of some ultra prefilter For example, every closed singleton subset is primitive. [45] The image of a primitive subset of under a continuous map is contained in a primitive subset of [45]
Assume that are two primitive subset of If is an open subset of that intersects then for any ultrafilter such that [45] In addition, if are distinct then there exists some and some ultrafilters such that and [45]
Other results
If is a complete lattice then:[ citation needed]
Suppose is a map from a set into a topological space and If is a limit point (respectively, a cluster point) of then is called a limit point or limit (respectively, a cluster point) of with respect to [43] Explicitly, is a limit of with respect to if and only if which can be written as (by definition of this notation) and stated as tend to along [46] If the limit is unique then the arrow may be replaced with an equals sign [32] The neighborhood filter can be replaced with any family equivalent to it and the same is true of
The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if is a net then
The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under ) of particular prefilters on the domain This shows that prefilters provide a general framework into which many of the various definitions of limits fit. [41] The limits in the left–most column are defined in their usual way with their obvious definitions.
Throughout, let be a map between topological spaces, If is Hausdorff then all arrows "" in the table may be replaced with equal signs "" and "" may be replaced with "". [32]
Type of limit | if and only if | Definition in terms of prefilters [41] | Assumptions |
---|---|---|---|
⇔ | |||
⇔ | |||
or |
⇔ | ||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | |||
⇔ | is a sequence in | ||
⇔ | |||
⇔ | |||
⇔ | for a double-ended sequence | ||
⇔ | a seminormed space; for example, a Banach space |
By defining different prefilters, many other notions of limits can be defined; for example,
Divergence to infinity
Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters
So for example, if then if and only if holds. Similarly, if and only if or equivalently, if and only if
More generally, if is valued in (or some other seminormed vector space) and if then if and only if holds, where
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.
In the definitions below, the first statement is the standard definition of a limit point of a net (respectively, a cluster point of a net) and it is gradually reworded until the corresponding filter concept is reached.
A net is said to converge in to a point written and is called a limit or limit point of [47] if any of the following equivalent conditions hold:As usual, is defined to mean that and is the only limit point of that is, if also [47]
- Definition: For every there exists some such that if
- For every there exists some such that the tail of starting at is contained in (that is, such that ).
- For every there exists some such that
- that is, the prefilter converges to
A point is called a cluster or accumulation point of a net if any of the following equivalent conditions hold:
- Definition: For every and every there exists some such that
- For every and every the tail of starting at intersects (that is, ).
- For every and every
- mesh (by definition of "mesh").
- is a cluster point of
If is a map and is a net in then [3]
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
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 mapthat is,
If is a prefilter on is a net in and the prefilter associated with is ; that is: [note 6]
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).
Proposition — If is a prefilter on and then
Recall that and that if is a net in then (1) and (2) is a cluster point of if and only if is a cluster point of By using it follows that
Partially ordered net
The domain of the canonical net is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered [48] a construction (detailed here: Filter (set theory)#Partially ordered net) 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. [3] Because the tails of this partially ordered net are identical to the tails of (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. [3] If can further be assumed that the partially ordered domain is also a dense order.
The notion of " is subordinate to " (written ) is for filters and prefilters what " is a subsequence of " is for sequences. [26] For example, if denotes the set of tails of and if denotes the set of tails of the subsequence (where ) then (which by definition means ) is true but is in general false. If is a net in a topological space and if is the neighborhood filter at a point then
If is an surjective open map, and is a prefilter on that converges to then there exist a prefilter on such that and is equivalent to (that is, ). [49]
The following results are the prefilter analogs of statements involving subsequences. [50] The condition "" which is also written is the analog of " is a subsequence of " So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."
Proposition [50] [43] — Let be a prefilter on and let
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of " subnet." [53] The first definition of a subnet ("Kelley–subnet") was introduced by John L. Kelley in 1955. [53] Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet. [53] 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. [53]
A subset of a preordered space is frequent or cofinal in if for every there exists some such that If contains a tail of then is said to be eventual in }}; explicitly, this means that there exists some such that (that is, for all satisfying ). A subset is eventual if and only if its complement is not frequent (which is termed infrequent). [53] A map between two preordered sets is order–preserving if whenever satisfy then
Definitions: Let be nets. Then [53]
- 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
- is a Kelley–subnet of or a subnet in the sense of Kelley if there exists a map such that and whenever is eventual in then is eventual in
- is an AA–subnet of or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
- If is eventual in is eventual in
- For any subset mesh, then so do
- 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. [53] In particular, if is a Willard–subnet or a Kelley–subnet of then
AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters. [53] [54] Explicitly, what is meant is that the following statement is true for AA–subnets:
If are prefilters then if and only if is an AA–subnet of
If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, as this counter-example demonstrates, 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 thus remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".
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. [53] [54]
Throughout, is a topological space.
Bases and prefilters
Let be a family of sets that covers and define for every The definition of a base for some topology can be immediately reworded as: is a base for some topology on if and only if is a filter base for every If is a topology on and then the definitions of is a basis (resp. subbase) for can be reworded as:
is a base (resp. subbase) for if and only if for every is a filter base (resp. filter subbase) that generates the neighborhood filter of at
Neighborhood filters
The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter."
Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If has its usual topology and if then any neighborhood filter base of is fixed by (in fact, it is even true that ) but is not principal since In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non–principal filter on an infinite set is not necessarily free.
The neighborhood filter of every point in topological space is fixed since its kernel contains (and possibly other points if, for instance, is not a T1 space). This is also true of any neighborhood basis at For any point in a T1 space (for example, a Hausdorff space), the kernel of the neighborhood filter of is equal to the singleton set
However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis of a point in a topological space is principal if and only if the kernel of is an open set. If in addition the space is T1 then so that this basis is principal if and only if is an open set.
Generating topologies from filters and prefilters
Suppose is not empty (and ). If is a filter on then is a topology on but the converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form where is an ultrafilter on are an even more specialized subclass of such topologies; they have the property that every proper subset is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces.
If is a prefilter (resp. filter subbase, π–system, proper) on then the same is true of both and the set of all possible unions of one or more elements of If is closed under finite intersections then the set is a topology on with both being bases for it. If the π–system covers then both are also bases for If is a topology on then is a prefilter (or equivalently, a π–system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset will be a basis for if and only if is equivalent to in which case will be a prefilter.
Neighborhoods and topologies
The neighborhood filter of a nonempty subset in a topological space is equal to the intersection of all neighborhood filters of all points in [55] A subset is open in if and only if whenever is a filter on and then
Suppose are topologies on Then is finer than (that is, ) if and only if whenever is a filter on if then [45] Consequently, if and only if for every filter and every if and only if [32] However, it is possible that while also for every filter converges to some point of if and only if converges to some point of [32]
Closure
If is a prefilter on a subset then every cluster point of belongs to [44]
If is a non-empty subset, then the following are equivalent:
The following are equivalent:
Closed sets
If is not empty then the following are equivalent:
Hausdorffness
The following are equivalent:
Compactness
As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.
The following are equivalent:
If is the set of all complements of compact subsets of a given topological space then is a filter on if and only if is not compact.
Theorem [57] — If is a filter on a compact space and is the set of cluster points of then every neighborhood of belongs to Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point.
Continuity
Let be a map between topological spaces
Given the following are equivalent:
The following are equivalent:
If is a prefilter on is a cluster point of is continuous, then is a cluster point in of the prefilter [45]
A subset of a topological space is dense in if and only if for every the trace of the neighborhood filter along does not contain the empty set (in which case it will be a filter on ).
Suppose is a continuous map into a Hausdorff regular space and that is a dense subset of a topological space Then has a continuous extension if and only if for every the prefilter converges to some point in Furthermore, this continuous extension will be unique whenever it exists. [58]
Products
Suppose is a non–empty family of non–empty topological spaces and that is a family of prefilters where each is a prefilter on Then the product of these prefilters (defined above) is a prefilter on the product space which as usual, is endowed with the product topology.
If then if and only if
Suppose are topological spaces, is a prefilter on having as a cluster point, and is a prefilter on having as a cluster point. Then is a cluster point of in the product space [45] However, if then there exist sequences such that both of these sequences have a cluster point in but the sequence does not have a cluster point in [45]
Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:
Proof
|
---|
Let be compact Hausdorff topological spaces. Assume that the ultrafilter lemma holds (because of the Hausdorff assumption, this proof does not need the full strength of the axiom of choice; the ultrafilter lemma suffices). Let be given the product topology (which makes a Hausdorff space) and for every let denote this product's projections. If then is compact and the proof is complete so assume Despite the fact that because the axiom of choice is not assumed, the projection maps are not guaranteed to be surjective. Let be an ultrafilter on and for every let denote the ultrafilter on generated by the ultra prefilter Because is compact and Hausdorff, the ultrafilter converges to a unique limit point (because of 's uniqueness, this definition does not require the axiom of choice). Let where satisfies for every The characterization of convergence in the product topology that was given above implies that Thus every ultrafilter on converges to some point of which implies that is compact (recall that this implication's proof only required the ultrafilter lemma). |
A uniform space is a set equipped with a filter on that has certain properties. A base or fundamental system of entourages is a prefilter on whose upward closure is a uniform space. A prefilter on a uniform space with uniformity is called a Cauchy prefilter if for every entourage there exists some that is –small, which means that A minimal Cauchy filter is a minimal element (with respect to or equivalently, to ) of the set of all Cauchy filters on Examples of minimal Cauchy filters include the neighborhood filter of any point Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point.
A uniform space is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on converges to at least one point of (replacing all instance of the word "prefilter" with "filter" results in equivalent statement). Every compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy).
Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first–countable, or even sequential. The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS) can made into a vector space and topologized in such a way that it becomes a completion of (with the assignment becoming a linear topological embedding that identifies as a dense vector subspace of this completion).
More generally, a Cauchy space is a pair consisting of a set together a family of (proper) filters, whose members are declared to be "Cauchy filters", having all of the following properties:
The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space. A map between two Cauchy spaces is called Cauchy continuous if the image of every Cauchy filter in is a Cauchy filter in Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.
Starting with nothing more than a set it is possible to topologize the set
To reduce confusion, this article will adhere to the following notational conventions:
For every let
From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of [note 9] For all
In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on was defined without using anything other than the set there were no preexisting structures or assumptions on so this topology is completely independent of everything other than (and its subsets).
The following criteria can be used for checking for points of closure and neighborhoods. If then:
It will be henceforth assumed that because otherwise and the topology is which is uninteresting.
Subspace of ultrafilters
The set of ultrafilters on (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If has the discrete topology then the map defined by sending to the principal ultrafilter at is a topological embedding whose image is a dense subset of (see the article Stone–Čech compactification for more details).
Relationships between topologies on and the Stone topology on
Every induces a canonical map defined by which sends to the neighborhood filter of If then if and only if Thus every topology can be identified with the canonical map which allows to be canonically identified as a subset of (as a side note, it is now possible to place on and thus also on the topology of pointwise convergence on so that it now makes sense to talk about things such as sequences of topologies on converging pointwise). For every the surjection is always continuous, closed, and open, but it is injective if and only if (that is, a Kolmogorov space). In particular, for every topology the map is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters).
In addition, if is a map such that (which is true of for instance), then for every the set is a neighborhood (in the subspace topology) of
Proofs