mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a filter on a set is a
family
Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of subsets such that:
# and
# if and , then
# If , and , then
A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order,
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
,
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, but can also be found in
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, from which they originate. The dual notion of a filter is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
.
Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to
filters in topology
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some give ...
, they were subsequently used by Nicolas Bourbaki in their book ''
Topologie Générale
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
'' as an alternative to the related notion of a
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
E. H. Moore
Eliakim Hastings Moore (; January 26, 1862 – December 30, 1932), usually cited as E. H. Moore or E. Hastings Moore, was an American mathematician.
Life
Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, di ...
and
Herman L. Smith
Herman Lyle Smith (July 7, 1892 – 1950) was an American mathematician, the co-discoverer, with E. H. Moore, of nets, and also a discoverer of the related notion of filters independently of Henri Cartan.
Born in Pittwood, Illinois, Smith recei ...
. Order filters are generalizations of filters from sets to arbitrary
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
s. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
ordered by
set inclusion
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
.
Preliminaries, notation, and basic notions
In this article, upper case Roman letters like denote sets (but not families unless indicated otherwise) and will denote the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of A subset of a power set is called (or simply, ) where it is 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 or in of a
family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fam ...
is
and similarly the of is
Throughout, is a map and is a set.
Nets and their tails
A is a set together with a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
, which will be denoted by (unless explicitly indicated otherwise), 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 (if is antisymmetric then this is equivalent to ).
A is a map from a non–empty directed set into
The notation will be used to denote a net with domain
Warning about using strict comparison
If is a net and then it is possible for the set which is called , to be empty (for example, this happens if is an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
of the
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
).
In this case, the family would contain the empty set, which would prevent it from being a prefilter (defined later).
This is the (important) reason for defining as rather than or even and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality may not be used interchangeably with the inequality
Filters and prefilters
The following is a list of properties that a family of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that
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.
\text X containing called the , and is said to this filter. This filter is equal to the intersection of all filters on that are supersets of The –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 However, if is a prefilter (although is always an upward closed filter base for ).
* A –smallest (meaning smallest relative to ) filter 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 –system) generated by is principal, in which case is the unique smallest prefilter containing Otherwise, in general, a –smallest filter containing might not exist. For this reason, some authors may refer to the –system generated by as However, if a –smallest prefilter does exist (say it is denoted by ) then contrary to usual expectations, it is 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 -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 –system generated by ".
of a filter and that is a of if is a filter and where for filters,
* Importantly, the expression "is a filter of" is for filters the analog of "is a sequence of". So despite having the prefix "sub" in common, "is a filter of" is actually the of "is a sequence of." However, can also be written which is described by saying " is subordinate to " With this terminology, "is ordinate to" becomes for filters (and also for prefilters) the analog of "is a sequence of," which makes this one situation where using the term "subordinate" and symbol may be helpful.
There are no prefilters on (nor are there any nets valued in ), which is why this article, like most authors, will automatically assume without comment that whenever this assumption is needed.
Basic examples
Named examples
The singleton set is called the or It is the unique filter on because it is a subset of every filter on ; however, it need not be a subset of every prefilter on
The dual ideal is also called (despite not actually being a filter). It is the only dual ideal on that is not a filter on
If is a topological space and then the
neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbou ...
at is a filter on By definition, a family is called a (resp. a ) at if and only if is a prefilter (resp. is a filter subbase) and the filter on that generates is equal to the neighborhood filter The subfamily of open neighborhoods is a filter base for Both prefilters also form a bases for topologies on with the topology generated being coarser than This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets
is an if for some sequence
is an or a on if is a filter on generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily an ultrafilter. Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set. The intersection of finitely many sequential filters is again sequential.
The set of all cofinite subsets of (meaning those sets whose complement in is finite) is proper if and only if is infinite (or equivalently, is infinite), in which case is a filter on known as the or the on If is finite then is equal to the dual ideal which is not a filter. If is infinite then the family of complements of singleton sets is a filter subbase that generates the Fréchet filter on As with any family of sets over that contains the kernel of the Fréchet filter on is the empty set:
The intersection of all elements in any non–empty family is itself a filter on called the or of which is why it may be denoted by Said differently, Because every filter on has as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to ) filter contained as a subset of each member of
* If are filters then their infimum in is the filter If are prefilters then is a prefilter that is coarser (with respect to ) than both (that is, ); indeed, it is one of the finest such prefilters, meaning that if is a prefilter such that then necessarily More generally, if are non−empty families and if then and is a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
(with respect to ) of
Let and let
The or of denoted by is the smallest (relative to ) dual ideal on containing every element of as a subset; that is, it is the smallest (relative to ) dual ideal on containing as a subset.
This dual ideal is where is the –system generated by
As with any non–empty family of sets, is contained in filter on if and only if it is a filter subbase, or equivalently, if and only if is a filter on in which case this family is the smallest (relative to ) filter on containing every element of as a subset and necessarily
Let and let
The or of denoted by if it exists, is by definition the smallest (relative to ) filter on containing every element of as a subset.
If it exists then necessarily (as defined above) and will also be equal to the intersection of all filters on containing
This supremum of exists if and only if the dual ideal is a filter on
The least upper bound of a family of filters may fail to be a filter. Indeed, if contains at least 2 distinct elements then there exist filters for which there does exist a filter that contains both
If is not a filter subbase then the supremum of does not exist and the same is true of its supremum in but their supremum in the set of all dual ideals on will exist (it being the degenerate filter ).
* If are prefilters (resp. filters on ) then is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if mesh), in which case it is coarsest prefilters (resp. coarsest filter) on (with respect to ) that is finer (with respect to ) than both this means that if is any prefilter (resp. any filter) such that then necessarily in which case it is denoted by
Let be non−empty sets and for every let be a dual ideal on If is any dual ideal on then is a dual ideal on called or .
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
cardinal is the filter of all sets containing a club subset of It is a -complete filter closed under
diagonal intersection Diagonal intersection is a term used in mathematics, especially in set theory.
If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha<\delta\rangle
is a
.
Other examples
Let and let which makes a prefilter and a filter subbase that is not closed under finite intersections. Because is a prefilter, the smallest prefilter containing is The –system generated by is In particular, the smallest prefilter containing the filter subbase is equal to the set of all finite intersections of sets in The filter on generated by is All three of the –system generates, and are examples of fixed, principal, ultra prefilters that are principal at the point is also an ultrafilter on
Let be a topological space, and define where is necessarily finer than If is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of If is a filter on then is a prefilter but not necessarily a filter on although is a filter on equivalent to
The set of all dense open subsets of a (non–empty) topological space is a proper –system and so also a prefilter. If the space is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
, then the set of all countable intersections of dense open subsets is a –system and a prefilter that is finer than If (with ) then the set of all such that has finite Lebesgue measure is a proper –system and free prefilter that is also a
proper subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of The prefilters and are equivalent and so generate the same filter on
The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense subsets of Since is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
, every countable intersection of sets in is dense in (and also
comeagre
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
and non–meager) so the set of all countable intersections of elements of is a prefilter and –system; it is also finer than, and not equivalent to,
''A filter subbase with no smallest prefilter containing it'': In general, if a filter subbase is not a –system then an intersection of sets from will usually require a description involving variables that cannot be reduced down to only two (consider, for instance when ). This example illustrates an atypical class of a filter subbases where all sets in both and its generated –system can be described as sets of the form so that in particular, no more than two variables (specifically, ) are needed to describe the generated –system.
For all let
where always holds so no generality is lost by adding the assumption
For all real if is non-negative then More generally, for any real numbers satisfying where
For every set of positive reals, letIf This property and the fact that is nonempty and proper if and only if actually allows for the construction of even more examples of prefilters, because if is any prefilter (resp. filter subbase, –system) then so is
Let and suppose is not a singleton set. Then is a filter subbase but not a prefilter and is the –system it generates, so that is the unique smallest filter in containing However, is a filter on (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and is a proper subset of the filter
If are non−empty intervals then the filter subbases generate the same filter on if and only if
If is a prefilter satisfying It may be shown that if is any family such that then is a prefilter if and only if for all real there exist real such that then for any the family is also a prefilter satisfying This shows that there cannot exist a minimal/ least (with respect to ) prefilter that both contains and is a subset of the –system generated by This remains true even if the requirement that the prefilter be a subset of is removed; that is, (in sharp contrast to filters) there does exist a minimal/least (with respect to ) filter containing the filter subbase
Ultrafilters
There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
s. Important properties of ultrafilters are also described in that article.
B 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 set (not necessarily even a subset of ) there exists some set such that
* If satisfies this condition then so does superset For example, if is any
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
then is ultra and consequently, any non–degenerate superset of (such as its upward closure) is also ultra.
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 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 (with respect to ) 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).
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
This condition can be restated as: is partitioned by and its dual
* The sets are disjoint whenever is a prefilter.
is an ideal.
For any if then
For any if then (a filter with this property is called a ).
* This property extends to any finite union of two or more sets.
For any if then
is a 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),For instance, one sense in which a net could be interpreted as being "maximally deep" is if all important properties related to (such as convergence for example) of any subnet is completely determined by in all topologies on In this case and its subnet become effectively indistinguishable (at least topologically) if one's information about them is limited to only that which can be described in solely in terms of and directly related sets (such as its subsets). which is an idea that is actually made rigorous by ultranets. The
ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").
Any non–degenerate family that has a singleton set as an element is ultra, in which case it will then be an ultra prefilter if and only if it also has the finite intersection property.
The trivial filter is ultra if and only if is a singleton set.
The ultrafilter lemma
The following important theorem is due to
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
(1930).
A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.Let be a filter on that is not an ultrafilter. If is such that has the finite intersection property (because if ) so that by the ultrafilter lemma, there exists some ultrafilter such that (so in particular, ). Intersecting all such proves that
Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
(in particular from Zorn's lemma) 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 (as encountered in introductory courses) in Topology (such as
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
for compact Hausdorff spaces and the
Alexander subbase theorem
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
) and in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
(such as the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
) 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.
B
If then for any point
Properties of kernels
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:
:(1) (2) the –system generated by and (3) the filter generated by
If is a map then and
If then while if and are equivalent then
Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal; that is, if and are principal then they are equivalent if and only if
Classifying families by their kernels
If is a principal filter on then and
where 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 the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
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
For every filter there exists a unique pair of dual ideals such that is free, is principal, and and do not mesh (that is, ). The dual ideal is called of while is called where at least one of these dual ideals is filter. If is principal then otherwise, and is a free (non–degenerate) filter.
Finite prefilters and finite sets
If a filter subbase is finite then it is fixed (that is, not free);
this is because is a finite intersection and the filter subbase has the finite intersection property.
A finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.
If is finite then all of the conclusions above hold for any
In particular, on a finite set there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on are principal filters generated by their (non–empty) kernels.
The trivial filter is always a finite filter on and if is infinite then it is the only finite filter because a non–trivial finite filter on a set is possible if and only if is finite.
However, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters).
If is a singleton set then the trivial filter is the only proper subset of and moreover, this set is a principal ultra prefilter and any superset (where ) with the finite intersection property will also be a principal ultra prefilter (even if is infinite).
Characterizing fixed ultra prefilters
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.
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 (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence", 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 point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s.
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
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
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 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,
Assume that are families of sets that satisfy Then and and also
If in addition to is a filter base and then is a filter subbase and also mesh.To prove that mesh, let Because (resp. because ), there exists some where by assumption so If is a filter subbase and if then taking implies that If then there are such that and now This shows that is a filter subbase.
More generally, if both and if the intersection of any two elements of is non–empty, then mesh.
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.
If a filter subbase is ultra then it is necessarily a prefilter, in which case the filter that it generates will also be ultra. A filter subbase that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by to be ultra. If is upward closed in then
Relational properties of subordination
The relation is reflexive and transitive, which makes it into a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
on
The relation is antisymmetric but if has more than one point then it is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
.
:
For any
So the set has more than one point if and only if the relation is
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
.
:
If but while the converse does not hold in general, it does hold if is upward closed (such as if is a filter).
Two filters are equivalent if and only if they are equal, which makes the restriction of to antisymmetric.
But in general, is antisymmetric on nor on ; that is, does necessarily imply ; not even if both are prefilters. For instance, if is a prefilter but not a filter then
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:
The upward closures of are equal.
Two upward closed (in ) 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, element of the equivalence class) that is upward closed in
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 or else it is false of :
Not empty
Proper (that is, is not an element)
* Moreover, any two degenerate families are necessarily equivalent.
Filter subbase
Prefilter
* In which case generate the same filter on (that is, their upward closures in are equal).
Free
Principal
Ultra
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 (one exception is when both families are filters).
Meshes with
Is finer than
Is coarser than
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:
;
the –system generated by ;
the filter on generated by ;
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.This is because if are prefilters on then
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.
A filter subbase that is also a prefilter can 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.
Every filter is both a –system and a
ring of sets
In mathematics, there are two different notions of a ring of sets, both referring to certain Family of sets, families of sets.
In order theory, a nonempty family of sets \mathcal is called a ring (of sets) if it is closure (mathematics), closed u ...
.
Examples of determining equivalence/non–equivalence
Examples: Let and let be the set of integers (or the set ). Define the sets
All three sets are filter subbases but none are filters on and only is prefilter (in fact, is even free and closed under finite intersections). The set is fixed while is free (unless ). They satisfy but no two of these families are equivalent; moreover, no two of the filters generated by these three filter subbases are equivalent/equal. This conclusion can be reached by showing that the –systems that they generate are not equivalent. Unlike with every set in the –system generated by contains as a subset,The –system generated by (resp. by ) is a prefilter whose elements are finite unions of open (resp. closed) intervals having endpoints in with two of these intervals being of the forms (resp. ) where ; in the case of it is possible for one or more of these closed intervals to be singleton sets (that is, degenerate closed intervals). which is what prevents their generated –systems (and hence their generated filters) from being equivalent. If was instead then all three families would be free and although the sets would remain equivalent to each other, their generated –systems would be equivalent and consequently, they would generate the same filter on ; however, this common filter would still be strictly coarser than the filter generated by
Set theoretic properties and constructions
Trace and meshing
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, ), in which case the trace of is said to be .
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 (resp. a prefilter, a filter subbase) then this is also true of both
In order to make any meaningful deductions about from needs to be proper (that is, which is the motivation for the definition of "mesh".
In this case, is a prefilter (resp. filter subbase) 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 (that is, ):
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
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 (although possibly not on the codomain unless is surjective):
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 (that is, a filter) then this simplifies to:
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
Preimages of prefilters
Let
Under the assumption that is Surjective function, surjective:
is a prefilter (resp. filter subbase, –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 (see thisFor an example of how this failure can happen, consider the case where there exists some such that both and its complement in contains at least two distinct points. footnote for an example).
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 (resp. filter subbase, –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:
is a prefilter;
is a prefilter;
;
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.
Bijections, injections, and surjections
All properties involving filters are preserved under bijections. This means that if is a bijection, then is a prefilter (resp. ultra, ultra prefilter, filter on ultrafilter on filter subbase, –system, ideal on etc.) if and only if the same is true of
A map is injective if and only if for all prefilters is equivalent to The image of an ultra family of sets under an injection is again ultra.
The map is a
surjection
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
if and only if whenever is a prefilter on then the same is true of (this result does not require the ultrafilter lemma).
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
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
are defined (had all of these been topologies). That is, both the notations
denote the family of all subsets such that for all but finitely many and where for any one of these finitely many exceptions (that is, for any such that necessarily ).
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
Set subtraction and some examples
Set subtracting away a subset of the kernel
If is a prefilter on then is a prefilter, where this latter set is a filter if and only if is a filter and In particular, if is a neighborhood basis at a point in a topological space having at least 2 points, then is a prefilter on This construction is used to define in terms of prefilter convergence.
Using duality between ideals and dual ideals
There is a dual relation or which is defined to mean that every some Explicitly, this means that for every , there is some such that This relation is dual to in sense that if and only if The relation is closely related to the downward closure of a family in a manner similar to how is related to the upward closure family.
For an example that uses this duality, suppose is a map and Define
which contains the empty set if and only if does. It is possible for to be an ultrafilter and for to be empty or not closed under finite intersections (see footnote for example).Suppose has more than one point, is a constant map, and then will consist of all non–empty subsets of Although does not preserve properties of filters very well, if is downward closed (resp. closed under finite unions, an ideal) then this will also be true for Using the duality between ideals and dual ideals allows for a construction of the following filter.
Suppose is a filter on and let be its dual in If then 's dual will be a filter.
Other examples
Example: The set of all dense open subsets of a topological space is a proper –system and a prefilter. If the space is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
, then the set of all countable intersections of dense open subsets is a –system and a prefilter that is finer than
Example: The family of all dense open sets of having finite Lebesgue measure is a proper –system and a free prefilter. The prefilter is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of Since is a
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
, every countable intersection of sets in is dense in (and also
comeagre
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
and non–meager) so the set of all countable intersections of elements of is a prefilter and –system; it is also finer than, and not equivalent to,
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 − and because it to make it easier to understand later why subnets (with their most commonly used definitions) are not generally equivalent with "sub–prefilters".
Nets to prefilters
A
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
is canonically associated with its prefilter of tails
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
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
on pointed sets by declaring
If even if so this preorder is not antisymmetric and given any family of sets is
partially ordered
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
if and only if consists entirely of singleton sets.
If is a
maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
of ; moreover, all maximal elements are of this form.
If is a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
if and only if in which case is the set of all greatest elements. However, a greatest element is a maximal element if and only if so there is at most one element that is both maximal and greatest.
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
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
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
:\;&& (\operatorname(\mathcal), \leq) &&\,\to \;& X \\
&& (B, b) &&\,\mapsto\;& b \\
\end
that is,
If is a prefilter on is a net in and the prefilter associated with is ; that is:The set equality holds more generally: if the family of sets then the family of tails of the map (defined by ) is equal to
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 (that is, an ultrafilter) on
However, 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 (since unlike the domain of the domain of an arbitrary net in could have cardinality).
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 discoveredBruns G., Schmidt J.,Zur Aquivalenz von Moore-Smith-Folgen und Filtern, Math. Nachr. 13 (1955), 169-186. 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 Albert "Tommy" Wilansky (13 September 1921, St Johns, Newfoundland – 3 July 2017, Bethlehem, Pennsylvania) was a Canadian-American mathematician, known for introducing Smith numbers.
Biography
Wilansky was educated as an undergraduate at Dalhousi ...
in 1970.
It begins with the construction of a
strict partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
(meaning a transitive and
irreflexive relation
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself.
An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to ...
) on a subset of that is similar to the
lexicographical order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
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
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a ...
on induced by where is partially ordered by equality Explicitly, the partial order on induced by equality refers to the diagonal which is a
homogeneous relation
In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) relation ...
on that makes into a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
. If this partial order is denoted by the more familiar symbol (that is, define ) then for any which shows that (and thus also ) is nothing more than a new symbol for equality on that is, The notation is used because it avoids the unnecessary introduction of a new symbol for the diagonal.
Both are serial and neither possesses a
greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an eleme ...
or a
maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
; 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 (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 partially ordered. If the set is replaced with the positive rational numbers then the strict partial order will also be a
dense order In mathematics, a partial order or total order < on a
The notion of " is subordinate to " (written ) is for filters and prefilters what " is a
subsequence
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
of " is for sequences.
For example, if denotes the set of tails of and if denotes the set of tails of the subsequence (where ) then (that is, ) is true but is in general false.
Non–equivalence of subnets and subordinate filters
A subset of a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
ed space is or in if for every there exists some If contains a tail of then is said to be or ; explicitly, this means that there exists some (that is, ). An eventual set is necessarily not empty. A subset is eventual if and only if its complement is not frequent (which is termed ).
A map between two preordered sets is if whenever
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "
subnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...
."
The first definition of a subnet was introduced by John L. Kelley in 1955.
Stephen Willard introduced his own variant of Kelley's definition of subnet in 1970.
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.
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
Example: Let and let be a constant sequence, say Let and so that is a net on Then is an AA-subnet of because But is not a Willard-subnet of because there does not exist any map whose image is a cofinal subset of Nor is a Kelley-subnet of because if is any map then is a cofinal subset of but is not eventually in
AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters.
Explicitly, what is meant is that the following statement is true for AA–subnets:
If are prefilters then is an AA–subnet of
If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes . In particular, 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 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 number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s
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 (because it contains which contains such a subset) but no such subset can be cofinal in because of the maximal element
Consequently, any order–preserving map must be eventually constant (with value ) 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 (because the range of such a map cannot be cofinal in ).
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–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 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.
See also
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Notes
Proofs
Citations
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{{Set theory
General topologyOrder theorySet theory