Filters in topology


In topology, filters can be used to study topological spaces and define basic topological notions such as convergence, map (topology)|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 have many useful technical properties and they may often be used in place of arbitrary filters.
Filters have generalizations called and, 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. 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 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 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 also establishes a relationship in which is to as a subsequence is to a sequence.
Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in their book 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 sequence and net convergence, filter convergence is defined 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 converges to a point if and only if the same is true of the original filter. 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 is more convenient for the problem at hand.
However, assuming that "subnet" is defined using either of its most popular definitions, then in general, this relationship does 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, neighborhood bases, convergence, various limits of functions, continuity, compactness, sequences, the filter [|equivalent] of "subsequence", uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear.

Motivation

Archetypical example of a filter
The archetypical example of a filter is the [Neighbourhood 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 required to be open sets; those are called.
Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter."
A is a set of subsets of that satisfies all of the following conditions:

  1. :  –  just as since is always a neighborhood of ;
  2. :  –  just as no neighborhood of is empty;
  3. : If  –  just as the intersection of any two neighborhoods of is again a neighborhood of ;
  4. : If then  –  just as any subset of that includes a neighborhood of will necessarily a neighborhood of .
Generalizing sequence convergence by using sets − determining sequence convergence without the sequence
A 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, 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 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 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 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 of the sequence is not enough to characterize its convergence; multiple sets are needed.
It turns out that the needed sets are the following, which are called the of the sequence :
These sets completely determine this sequence's convergence because given any point, this sequence converges to it if and only if for every neighborhood , 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 of the sequence
Specifically, with the family of ' in hand, the ' is no longer needed to determine convergence of this sequence.
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 "" a filter via taking its . 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. The properties that these families share led to the notion of a, also called a, 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.
Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other. Both filters and nets can be used to completely category of topological spaces|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, abstract algebra, combinatorics, dynamics, order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers.
Like sequences, nets are ' and so they have the.
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.
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
In contrast to nets, filters are families of ' and so they have the.
For example, if is surjective then the 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 an arbitrary sequence so as to obtain a sequence or net in the domain. 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 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 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 ; this is because nets in can have domains of cardinality.
In contrast, the collection of all filters on is a set whose cardinality is no larger than that of [Power set|]
Similar to a topology on a filter on is "intrinsic to " in the sense that both structures consist of subsets of and neither definition requires any set that cannot be constructed from .

Preliminaries, notation, and basic notions

In this article, upper case Roman letters like and denote sets and will denote the power set of A subset of a power set is called where it is if it is a subset of Families of sets will be denoted by upper case calligraphy letters such as,, and.
Whenever these assumptions are needed, then it should be assumed that is non-empty and that etc. are families of sets over
The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.
Warning about competing definitions and notation
There are unfortunately several terms in the theory of filters that are defined differently by different authors.
These include some of the most important terms such as "filter."
While different definitions of the same term usually have significant overlap, due to the very technical nature of filters, 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 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 or in of a family of sets is
and similarly the of is
Notation and DefinitionName
of
where is a set.
or where is a set; sometimes denoted by
of a set

Throughout, is a map.
Notation and DefinitionName
of or the of under
of under
of

Topology notation
Denote the set of all topologies on a set
Suppose is any subset, and is any point.
Notation and DefinitionName
or of
or of
or of
or of

If then
Nets and their tails
A is a set together with a preorder, which will be denoted by , that makes into an ; 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 true that .
A is a map from a non-empty directed set into
The notation will be used to denote a net with domain
Notation and DefinitionName
or where is a directed set.
or
or / of Also called the generated by If is a sequence then is also called the or.
of/generated by
or 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, to be empty.
In this case, the family would contain the empty set, which would prevent it from being a prefilter.
This is the reason for defining as rather than or even and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality may not be used interchangeably with the inequality

Filters and prefilters

The following is a list of properties that a family of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that
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.
There are no prefilters on , which is why this article, like most authors, will automatically assume without comment that whenever this assumption is needed.

Ultrafilters

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.
The ultrafilter lemma
The following important theorem is due to Alfred Tarski.
A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.
Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If dealing with Hausdorff spaces, then most basic results in Topology and in functional analysis can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

Kernels

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 -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.
Classifying families by their kernels
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, 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
Characterizing fixed ultra prefilters
If a family of sets is fixed 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.
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.

Finer/coarser, subordination, and meshing

The preorder that is defined below is of fundamental importance for the use of prefilters in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence", where "" can be interpreted as " is a 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 and are, indicated by writing if If do not mesh then they are. If then are said to if mesh, or equivalently, if the of which is the family
does not contain the empty set, where the trace is also called the of
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 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 subset of the right hand side if every point of is unique and is the even-indexed subsequence because under these conditions, every tail 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.
Every filter subbase is coarser than both the -system that it generates and the filter that it generates.
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 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
The relation is antisymmetric but if has more than one point then it is symmetric.

Equivalent families of sets

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

  2. The upward closures of are equal.
Two upward closed subsets of are equivalent if and only if they are equal.
If then necessarily and is equivalent to
Every equivalence class other than contains a unique representative that is upward closed in
Properties preserved between equivalent families
Let be arbitrary and let be any family of sets. If are equivalent then for each of the statements/properties listed below, either it is true of or else it is false of :

  1. Not empty
  2. Proper
    • Moreover, any two degenerate families are necessarily equivalent.
  3. Filter subbase
  4. Prefilter
    • In which case generate the same filter on .
  5. Free
  6. Principal
  7. Ultra
  8. Is equal to the trivial filter
    • In words, this means that the only subset of that is equivalent to the trivial filter the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters.
  9. Meshes with
  10. Is finer than
  11. Is coarser than
  12. Is equivalent to
Missing from the above list is the word "filter" because this property is preserved by equivalence.
However, if are filters on then they are equivalent if and only if they are equal; this characterization does extend to prefilters.
Equivalence of prefilters and filter subbases
If is a prefilter on then the following families are always equivalent to each other:

  1. ;
  2. the -system generated by ;
  3. the filter on generated by ;
and moreover, these three families all generate the same filter on .
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.
Every prefilter is equivalent to exactly one filter on which is the filter that it generates.
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.
A filter subbase that is also a prefilter can be equivalent to the prefilter 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.

Set theoretic properties and constructions relevant to topology

Trace and meshing

If is a prefilter on then the trace of which is the family is a prefilter if and only if mesh, in which case the trace of is said to be.
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
When prefilters mesh
Given non-empty families the family
satisfies and
If is proper then this is also true of both
In order to make any meaningful deductions about from needs to be proper if and only if this is true of both
Said differently, if are prefilters then they mesh if and only if is a prefilter.
Generalizing gives a well known characterization of "mesh" entirely in terms of subordination :
Two prefilters mesh if and only if there exists a prefilter such that and
If the least upper bound of two filters exists in then this least upper bound is equal to

Images and preimages under functions

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 :
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
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.
Otherwise it is just a prefilter on and its upward closure must be taken in to obtain a filter.
The upward closure of is
where if is upward closed in then this simplifies to:
If then taking to be the inclusion map shows that any prefilter on is also a prefilter on
Preimages of prefilters
Let
Under the assumption that is surjective:
is a prefilter if and only if this is true of
However, if is an ultrafilter on then even if is surjective, it is nevertheless still possible for the prefilter to be neither ultra nor a filter on
If is not surjective then denote the trace of by where in this case particular case the trace satisfies:
and consequently also:
This last equality and the fact that the trace is a family of sets over means that to draw conclusions about the trace can be used in place of and the can be used in place of
For example:
is a prefilter if and only if this is true of
In this way, the case where is not surjective can be reduced down to the case of a surjective function.
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:

  1. is a prefilter;
  2. is a prefilter;
  3. ;
  4. meshes with
and moreover, if is a prefilter then so is
If and if denotes the inclusion map then the trace of is equal to This observation allows the results in this subsection to be applied to investigating the trace on a set.

Subordination is preserved by images and preimages

The relation is preserved under both images and preimages of families of sets.
This means that for families
Moreover, the following relations always hold for family of sets :
where equality will hold if is surjective.
Furthermore,
If then
and where equality will hold if is injective.

Products of prefilters

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

Convergence, limits, and cluster points

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 a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non-injective maps. 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, meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters. 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 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 and one "downward" property.
Starting with any there always exists some that is a subset of ; this may be continued ad infinitum to get a sequence of sets in with each being a subset of The same is true going "upward", for if then there is no set in that contains as a proper subset.
Thus when it comes to limiting behavior, going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters 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.
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.

Limits and convergence

A family is said to ' to a point of if Explicitly, means that every neighborhood contains some as a subset ; thus the following then holds: In words, a family converges to a point or subset if and only if it is than the neighborhood filter at
A family converging to a point may be indicated by writing and saying that is a ' of if this limit is a point, then is also called a .
As usual, is defined to mean that and is the limit point of that is, if also .
The set of all limit points of is denoted by
In the above definitions, it suffices to check that is finer than some neighborhood base in of the point.
Examples
If is Euclidean space and denotes the Euclidean norm, then all of the following families converge to the origin:
  1. the prefilter of all open balls centered at the origin, where
  2. the prefilter of all closed balls centered at the origin, where This prefilter is equivalent to the one above.
  3. the prefilter where is a union of spheres centered at the origin having progressively smaller radii. This family consists of the sets as ranges over the positive integers.
  4. any of the families above but with the radius ranging over instead of over all positive reals.
  5. * Drawing or imagining any one of these sequences of sets when has dimension suggests that intuitively, these sets "should" converge to the origin. This is the intuition that the above definition of a "convergent prefilter" make rigorous.
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 and so although does converge to the since
However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter also has kernel but does not converges to it.
The free prefilter of intervals does not converge 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, every non-empty family 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:
In particular, the limit points of a prefilter are the same as the limit points of the filter that it generates.
Another consequence is that if a family converges to a point then the same is true of the family's trace/restriction to any given subset of
If is a prefilter and then converges to a point of if and only if this is true of the trace
If a filter subbase converges to a point then so do the filter and the -system that it generates, although the converse is not guaranteed. For example, the filter subbase does not converge to in although the filter that it generates—which is equal to the principal filter generated by —does.
Given the following are equivalent for a prefilter

  1. converges to
  2. converges to
  3. There exists a family equivalent to that converges to
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 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 prefilters exist that have any limit points in

Cluster points

A family is said to ' 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 ' or an of a family 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 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:
In particular, the cluster points of a prefilter are the same as the cluster points of the filter that it generates.
Given the following are equivalent for a prefilter :

  1. clusters at
  2. The family generated by clusters at
  3. There exists a family equivalent to that clusters at
  5. for every neighborhood of
    • If is a filter on then for every neighborhood
  6. There exists a prefilter subordinate to that converges to
The set of all cluster points of a prefilter satisfies
Consequently, the set of all cluster points of prefilter is a closed subset of This also justifies the notation for the set of cluster points.
In particular, if is non-empty then since both sides are equal to

Properties and relationships

Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have cluster points or limit points.
If is a limit point of then is necessarily a limit point of any family ' than .
In contrast, if is a cluster point of then is necessarily a cluster point of any family ' than .
Equivalent families and subordination
Any two equivalent families can be used 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 filters to Topology can both be defined in terms of the subordination relation. This is why the preorder is of such great importance in applying 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:
This is because if is a limit point of then mesh, which makes a cluster point of But in general, a cluster point need not be a limit point. For instance, every point in any given non-empty subset is a cluster point of the principle prefilter but if is Hausdorff and has more than one point then this prefilter has no limit points; the same is true of the filter that this prefilter generates.
However, every cluster point of an prefilter is a limit point. Consequently, the limit points of an 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.
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
Primitive sets
A subset is called if it is the set of limit points of some ultrafilter. 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. The image of a primitive subset of under a continuous map is contained in a primitive subset of
Assume that are two primitive subset of
If is an open subset of that intersects then for any ultrafilter such that
In addition, if are distinct then there exists some and some ultrafilters such that and
Other results
If is a complete lattice then:
Suppose is a map from a set into a topological space and If is a limit point of then is called a ' or '
Explicitly, is a limit of with respect to if and only if which can be written as and stated as If the limit is unique then the arrow may be replaced with an equals sign 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
where the left hand side states that is a limit while the right hand side states that is a limit with respect to .
The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images of particular prefilters on the domain
This shows that prefilters provide a general framework into which many of the various definitions of limits fit.
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
Type of limitDefinition in terms of prefiltersAssumptions
⇔
⇔

or
⇔
⇔
⇔
⇔
⇔
⇔
⇔is a sequence in
⇔
⇔
⇔for a double-ended sequence
⇔a seminormed 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
where along if and only if and similarly, along if and only if The family can be replaced by any family equivalent to it, such as for instance and if then if and only if holds, where

Filters and nets

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.

Nets to prefilters

In the definitions below, the first statement is the standard definition of a limit point of a net and it is gradually reworded until the corresponding filter concept is reached.
If is a map and is a net in then

Prefilters to nets

A 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 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 is a net in and the prefilter associated with is ; that is:
This would not necessarily be true had been defined on a proper subset of
For example, suppose has at least two distinct elements, is the indiscrete filter, and is arbitrary. Had instead been defined on the singleton set where the restriction of to will temporarily be denote by then the prefilter of tails associated with would be the principal prefilter rather than the original filter ;
this means that the equality is, so unlike the prefilter can be recovered from
Worse still, while is the unique filter on the prefilter instead generates a filter on
If is a net in then it is in general true that is equal to because, for example, the domain of may be of a completely different cardinality than that of .
Ultranets and ultra prefilters
A net is called an or in if for every subset is eventually in or it is [|eventually] in ;
this happens if and only if is an ultra prefilter.
A prefilter is an ultra prefilter if and only if is an ultranet in
Partially ordered net
The domain of the canonical net is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered a construction that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970.
It begins with the construction of a strict partial order on a subset of that is similar to the lexicographical order on of the strict partial orders
For any in declare that if and only if
or equivalently, if and only if
The non−strict partial order associated with denoted by is defined by declaring that
Unwinding these definitions gives the following characterization:
which shows that is just the lexicographical order on induced by where is partially ordered by equality
Both are serial and neither possesses a greatest element or a maximal element; this remains true if they are each restricted to the subset of defined by
where it will henceforth be assumed that they are.
Denote the assignment from this subset by:
If then just as with before, the tail of the starting at is equal to
If is a prefilter on then is a net in whose domain is a partially ordered set and moreover,
Because the tails of are identical, there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed partially ordered. If the set is replaced with the positive rational numbers then the strict partial order will also be a dense order.

Subordinate filters and subnets

The notion of " is subordinate to " is for filters and prefilters what " is a subsequence of " is for sequences.
For example, if denotes the set of tails of and if denotes the set of tails of the subsequence then 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 .

Subordination analogs of results involving subsequences

The following results are the prefilter analogs of statements involving subsequences. 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."

Non-equivalence of subnets and subordinate filters

and subnets in the sense of Kelley are the most commonly used definitions of "subnet."
The first definition of a subnet was introduced by John L. Kelley in 1955.
Stephen Willard introduced in 1970 his own variant of Kelley's definition of subnet.
AA-subnets were introduced independently by Smiley, Aarnes and Andenaes, and Murdeshwar ; AA-subnets were studied in great detail by Aarnes and Andenaes but they are not often used.
A subset of a preordered space is or in if for every there exists some such that If contains a tail of then is said to be in ; explicitly, this means that there exists some such that . A subset is [|eventual] if and only if its complement is not frequent.
A map between two preordered sets is if whenever satisfy then
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.
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 subfilters.
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. In particular, as this counter-example demonstrates, the problem is that the following statement is in general 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".

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

Topologies and prefilters

Throughout, is a topological space.

Examples of relationships between filters and topologies

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 for can be reworded as:
is a base for if and only if for every is a filter base 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 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 but is 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 . This is also true of any neighborhood basis at
For any point in a T1 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 discrete.
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. 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 topologies. Topologies of the form where is an filter on are an even more specialized subclass of such topologies; they have the property that proper subset is open or closed, but never both. These spaces are, in particular, examples of door spaces.
If is a prefilter 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 -system covers then both are also bases for If is a topology on then is a prefilter if and only if it has the finite intersection property, in which case a subset will be a basis for if and only if is equivalent to in which case will be a prefilter.

Topological properties and prefilters

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
A subset is open in if and only if whenever is a filter on and then
Suppose are topologies on
Then is finer than if and only if whenever is a filter on if then Consequently, if and only if for every filter and every if and only if
However, it is possible that while also for every filter converges to point of if and only if converges to point of
Closure
If is a prefilter on a subset then every cluster point of belongs to
If is a non-empty subset, then the following are equivalent:

  2. is a limit point of a prefilter on Explicitly: there exists a prefilter such that
  3. is a limit point of a filter on
  4. There exists a prefilter such that
  5. The prefilter meshes with the neighborhood filter Said differently, is a cluster point of the prefilter
  6. The prefilter meshes with some filter base for .
The following are equivalent:

  1. is a limit points of
  2. There exists a prefilter such that
Closed sets
If is not empty then the following are equivalent:

  1. is a closed subset of
  2. If is a prefilter on such that then
  3. If is a prefilter on such that is an accumulation points of then
  4. If is such that the neighborhood filter meshes with then
Hausdorffness
The following are equivalent:

  1. is a Hausdorff space.
  2. Every prefilter on converges to at most one point in
  3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.
Compactness
As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.
The following are equivalent:

  1. is a compact space.
  2. Every ultrafilter on converges to at least one point in
    • That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma.
  3. The above statement but with the word "ultrafilter" replaced by "ultra prefilter".
  4. For every filter there exists a filter such that and converges to some point of
  5. The above statement but with each instance of the word "filter" replaced by: prefilter.
  6. Every filter on has at least one cluster point in
    • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
  7. The above statement but with the word "filter" replaced by "prefilter".
  8. Alexander subbase theorem: There exists a subbase such that every cover of by sets in has a finite subcover.
    • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
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 compact.
Continuity
Let be a map between topological spaces
Given the following are equivalent:

  1. is continuous at
  2. Definition: For every neighborhood of there exists some neighborhood of such that
  4. If is a filter on such that then
  5. The above statement but with the word "filter" replaced by "prefilter".
The following are equivalent:

  1. is continuous.
  2. If is a prefilter on such that then
  3. If is a limit point of a prefilter then is a limit point of
  4. Any one of the above two statements but with the word "prefilter" replaced by "filter".
If is a prefilter on is a cluster point of is continuous, then is a cluster point in of the prefilter
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.
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.
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 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
However, if then there exist sequences such that both of these sequences have a cluster point in but the sequence does have a cluster point in
Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:
Let be compact [Hausdorff space|] topological spaces.
Assume that the ultrafilter lemma holds.
Let be given the product topology 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 .
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.

Examples of applications of prefilters

Uniformities and Cauchy prefilters

A uniform space is a set equipped with a filter on that has certain properties. A or is a prefilter on whose upward closure is a uniform space.
A prefilter on a uniform space with uniformity is called a if for every entourage there exists some that is, which means that
A is a minimal element 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 if every Cauchy prefilter on converges to at least one point of .
Every compact uniform space is complete because any Cauchy filter has a cluster point, which is necessarily also a limit point.
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 is typically constructed by using minimal Cauchy filters.
Nets are less ideal for this construction because their domains are extremely varied ; sequences cannot be used in the general case because the topology might not be metrizable, first-countable, or even sequential.
The set of all on a Hausdorff topological vector space can made into a vector space and topologized in such a way that it becomes a completion of .
More generally, a [Cauchy space|] is a pair consisting of a set together a family of filters, whose members are declared to be "", having all of the following properties:
  1. For each the discrete ultrafilter at is an element of
  2. If is a subset of a proper filter then
  3. If and if each member of intersects each member of then
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 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.

Topologizing the set of prefilters

Starting with nothing more than a set it is possible to topologize the set
of all filter bases on with the, which is named after Marshall Harvey Stone.
To reduce confusion, this article will adhere to the following notational conventions:

  • Lower case letters for elements
  • Upper case letters for subsets
  • Upper case calligraphy letters for subsets .
  • Upper case double-struck letters for subsets
For every let
where These sets will be the basic open subsets of the Stone topology.
If then
From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of
For all
where in particular, the equality shows that the family is a -system that forms a basis for a topology on called the. It is henceforth assumed that carries this topology and that any subset of carries the induced subspace topology.
In contrast to most other general constructions of topologies, this topology on was defined with using anything other than the set there were preexisting structures or assumptions on so this topology is completely independent of everything other than .
The following criteria can be used for checking for points of closure and neighborhoods.
If then:

  • : belongs to the closure of if and only if
  • : is a neighborhood of if and only if there exists some such that .
It will be henceforth assumed that because otherwise and the topology is which is uninteresting.
Subspace of ultrafilters
The set of ultrafilters on 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 .
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 .
For every the surjection is always continuous, closed, and open, but it is injective if and only if .
In particular, for every topology the map is a topological embedding.
In addition, if is a map such that , then for every the set is a neighborhood of