Filters in topology


In topology, filters can be used to study topological spaces and define 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 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 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 .