In this article, upper case Roman letters like denote sets (but not families unless indicated otherwise) and will denote the power set of A subset of a power set is called a family of sets (or simply, a family) where it is over if it is a subset of Families of sets will be denoted by upper case calligraphy letters such as
Whenever these assumptions are needed, then it should be assumed that is non–empty and that etc. are families of sets over
The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.
Warning about competing definitions and notation
There are unfortunately several terms in the theory of filters that are defined differently by different authors.
These include some of the most important terms such as "filter".
While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences.
When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author.
For this reason, this article will clearly state all definitions as they are used.
Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.
The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions.
Their important properties are described later.
For any two families declare that if and only if for every there exists some in which case it is said that is coarser than and that is finer than (or subordinate to)  The notation may also be used in place of
A directed set is a set together with a preorder, which will be denoted by (unless explicitly indicated otherwise), that makes into an (upward) directed set; this means that for all there exists some such that For any indices the notation is defined to mean while is defined to mean that holds but it is not true that (if is antisymmetric then this is equivalent to ).
A net in  is a map from a non–empty directed set into
The notation will be used to denote a net with domain
Set or prefilter of tails/sections of Also called the eventuality filter base generated by (the tails of) If is a sequence then is also called the sequential filter base.
(Eventuality) filter of/generated by (tails of) 
Tail or section of a net starting at  where is a directed set.
Warning about using strict comparison
If is a net and then it is possible for the set which is called the tail of after, to be empty (for example, this happens if is an upper bound of the directed set).
In this case, the family would contain the empty set, which would prevent it from being a prefilter (defined later).
This is the (important) reason for defining as rather than or even and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality may not be used interchangeably with the inequality
The following is a list of properties that a family of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that
The family of sets is:
Proper or nondegenerate if Otherwise, if then it is called improper or degenerate.
Directed downward if whenever then there exists some such that
This property can be characterized in terms of directedness, which explains the word "directed": A binary relation on is called (upward) directed if for any two there is some satisfying Using in place of gives the definition of directed downward whereas using instead gives the definition of directed upward. Explicitly, is directed downward (resp. directed upward) if and only if for all there exists some "greater" such that (resp. such that ) − where the "greater" element is always on the right hand side,[note 1] − which can be rewritten as (resp. as ).
If a family has a greatest element with respect to (for example, if ) then it is necessarily directed downward.
Closed under finite intersections (resp. unions) if the intersection (resp. union) of any two elements of is an element of
If is closed under finite intersections then is necessarily directed downward. The converse is generally false.
Upward closed or Isotone in  if or equivalently, if whenever and some set satisfies Similarly, is downward closed if An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
The family which is the upward closure of is the unique smallest (with respect to ) isotone family of sets over having as a subset.
Many of the properties of defined above and below, such as "proper" and "directed downward," do not depend on so mentioning the set is optional when using such terms. Definitions involving being "upward closed in " such as that of "filter on " do depend on so the set should be mentioned if it is not clear from context.
A family is/is a(n):
Ideal if is downward closed and closed under finite unions.
Dual ideal on  if is upward closed in and also closed under finite intersections. Equivalently, is a dual ideal if for all 
Explanation of the word "dual": A family is a dual ideal (resp. an ideal) on if and only if the dual of which is the family
is an ideal (resp. a dual ideal) on In other words, dual ideal means "dual of an ideal". The family should not be confused with because these two sets are not equal in general; for instance, The dual of the dual is the original family, meaning The set belongs to the dual of if and only if 
Filter on  if is a properdual ideal on That is, a filter on is a non−empty subset of that is closed under finite intersections and upward closed in Equivalently, it is a prefilter that is upward closed in In words, a filter on is a family of sets over that (1) is not empty (or equivalently, it contains ), (2) is closed under finite intersections, (3) is upward closed in and (4) does not have the empty set as an element.
Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a proper/non–degenerate dual ideal. It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always require non-degeneracy. This article uses Henri Cartan's original definition of "filter", which required non–degeneracy.
A dual filter on is a family whose dual is a filter on Equivalently, it is an ideal on that does not contain as an element.
The power set is the one and only dual ideal on that is not also a filter. Excluding from the definition of "filter" in topology has the same benefit as excluding from the definition of "prime number": it obviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-") in many important results, thereby making their statements less awkward.
Prefilter or filter base if is proper and directed downward. Equivalently, is called a prefilter if its upward closure is a filter. It can also be defined as any family that is equivalent (with respect to ) to some filter. A proper family is a prefilter if and only if  A family is a prefilter if and only if the same is true of its upward closure.
If is a prefilter then its upward closure is the unique smallest (relative to ) filter on containing and it is called the filter generated by A filter is said to be generated by a prefilter if in which is called a filter base for
Unlike a filter, a prefilter is not necessarily closed under finite intersections.
π–system if is closed under finite intersections. Every non–empty family is contained in a unique smallest π–system called the π–system generated by which is sometimes denoted by It is equal to the intersection of all π–systems containing and also to the set of all possible finite intersections of sets from :
A π–system is a prefilter if and only if it is proper. Every filter is a proper π–system and every proper π–system is a prefilter but the converses do not hold in general.
A prefilter is equivalent (with respect to ) to the π–system generated by it and both of these families generate the same filter on
Filter subbase and centered if and satisfies any of the following equivalent conditions:
has the finite intersection property, which means that the intersection of any finite family of (one or more) sets in is not empty; explicitly, this means that whenever then
The π–system generated by is proper; that is,
The π–system generated by is a prefilter.
is a subset of some prefilter.
is a subset of some filter.
Assume that is a filter subbase. Then there is a unique smallest (relative to ) filter containing called the filter generated by, and is said to be a filter subbase for this filter. This filter is equal to the intersection of all filters on that are supersets of The π–system generated by denoted by will be a prefilter and a subset of Moreover, the filter generated by is equal to the upward closure of meaning  However, if and only if is a prefilter (although is always an upward closed filter subbase for ).
A –smallest (meaning smallest relative to ) prefilter containing a filter subbase will exist only under certain circumstances. It exists, for example, if the filter subbase happens to also be a prefilter. It also exists if the filter (or equivalently, the π–system) generated by is principal, in which case is the unique smallest prefilter containing Otherwise, in general, a –smallest prefilter containing might not exist. For this reason, some authors may refer to the π–system generated by as the prefilter generated by However, if a –smallest prefilter does exist (say it is denoted by ) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by " (that is, is possible). And if the filter subbase happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning ) will not be (that is, is possible even when is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ".
Subfilter of a filter and that is a superfilter of  if is a filter and where for filters,
Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of." However, can also be written which is described by saying " is subordinate to " With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of," 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.
The singleton set is called the indiscrete or trivial filter on  It is the unique minimal 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 the degenerate filter on  (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 at is a filter on By definition, a family is called a neighborhood basis (resp. a neighborhood subbase) 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 elementary prefilter if for some sequence
is an elementary filter or a sequential filter 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 not 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 Fréchet filter or the cofinite filter 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 infimum or greatest lower bound 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 (with respect to ) of 
Let and let
The supremum or least upper bound 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 π–system generated by
As with any non–empty family of sets, is contained in some 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 supremum or least upper bound 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 not 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 one of the coarsest prefilters (resp. the 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 Kowalsky's dual ideal or Kowalsky's filter.
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 π–system generated by is In particular, the smallest prefilter containing the filter subbase is not equal to the set of all finite intersections of sets in The filter on generated by is All three of the π–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 π–system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than If (with ) then the set of all such that has finite Lebesgue measure is a proper π–system and free prefilter that is also a proper subset 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, every countable intersection of sets in is dense in (and also comeagre and non–meager) so the set of all countable intersections of elements of is a prefilter and π–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 π–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 π–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 π–system.
For all let
where always holds so no generality is lost by adding the assumption
For all real if is non-negative then [note 2]
For every set of positive reals, let[note 3]
Let and suppose is not a singleton set. Then is a filter subbase but not a prefilter and is the π–system it generates, so that is the unique smallest filter in containing However, is not 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 [note 4] 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 π–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 not exist a minimal/least (with respect to ) prefilter containing the filter subbase
There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.
A non–empty family of sets is/is an:
Ultra if and any of the following equivalent conditions are satisfied:
For every set there exists some set such that (or equivalently, such that ).
For every set there exists some set such that
This characterization of " is ultra" does not depend on the set so mentioning the set is optional when using the term "ultra."
For every set (not necessarily even a subset of ) there exists some set such that
If satisfies this condition then so does every superset For example, if is any singleton set then is ultra and consequently, any non–degenerate superset of (such as its upward closure) is also ultra.
Ultra prefilter 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:
For any if then (a filter with this property is called a prime filter).
This property extends to any finite union of two or more sets.
For any if then either
is a maximal filter on ; meaning that if is a filter on such that then necessarily (this equality may be replaced by ).
If is upward closed then So this characterization of ultrafilters as maximal filters can be restated as:
Because subordination is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA–subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from " in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example),[note 5] which is an idea that is actually made rigorous by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").
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 kernel is useful in classifying properties of prefilters and other families of sets.
The kernel of a family of sets is the intersection of all sets that are elements of
If then for any point
Properties of kernels
If then and this set is also equal to the kernel of the π–system that is generated by
In particular, if is a filter subbase then the kernels of all of the following sets are equal:
(1) (2) the π–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
A family of sets is:
Free if or equivalently, if this can be restated as
A filter on is free if and only if is infinite and contains the Fréchet filter on as a subset.
Fixed if in which case, is said to be fixed by any point