In topology, a subfield of mathematics, * filters * are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called * ultrafilters * have many useful technical properties and they may often be used in place of arbitrary filters.

- Motivation
- Preliminaries, notation, and basic notions
- Filters and prefilters
- Finer/coarser, subordination, and meshing
- Summary of filter limits and cluster points definitions
- Set theoretic properties, examples, and constructions involving prefilters
- Supremum, trace, and meshing
- Products and other examples
- Images and preimages of filters and prefilters
- Limits, cluster points, and nets
- Limits and cluster points of prefilters
- Limits of functions defined as limits of prefilters
- Filters and nets
- Subordinate filters and subnets
- Topologies and prefilters
- Examples of relationships between filters and topologies
- Prefilters and topological properties
- Examples of applications of prefilters
- Convergence of nets of sets
- Uniformities and Cauchy prefilters
- Topologizing the set of prefilters
- See also
- Notes
- Citations
- References

Filters have generalizations called *prefilters* (also known as *filter bases*) and *filter subbases*, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said to *generate*. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notion is more technically convenient. A preorder ≤ on families of sets helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it defines the notion of filter convergence, where by definition, a filter (or prefilter) *converges* to a point if and only if where is that point's neighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions. In addition, the relation which denotes and is expressed by saying that *is subordinate to* also establishes a relationship in which is to as a subsequence is to a sequence (that is, the relation which is called *subordination*, is for filters the analog of "is a subsequence of").

Filters were introduced by Henri Cartan in 1937^{ [1] }^{ [2] } and subsequently used by Bourbaki in their book * Topologie Générale * as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike^{ [note 1] } sequence and net convergence, filter convergence is defined *entirely* in terms of subsets of the topological space and so it provides a notion of convergence that is completely intrinsic to the topological space. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, in general this relationship does *not* extend to subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship (here it is assumed that "subnet" is defined using any of its most popular definitions, which are given in this article).

- Archetypical example of a filter

The archetypical example of a filter is the *neighborhood filter* at a point in a topological space which by definition is the family of sets consisting of all neighborhoods of By definition, a neighborhood of some given point (or subset) is any subset whose topological interior contains this point (or subset); importantly, neighborhoods are *not* required to be open sets (those are called *open neighborhoods*). The fundamental properties shared by neighborhood filters, which are listed below, ultimately became the definition of a "filter." A *filter on * is a set of subsets of that satisfies all of the following conditions:

*Not empty*: – just as since is always an (open) neighborhood of (and of anything else that it contains);*Does not contain the empty set*: – just as no neighborhood of is empty;*Closed under finite intersections*: If then – just as the intersection of any two neighborhoods of is again a neighborhood of ;*Upward closed*: If and then – just as any subset of that contains a neighborhood of will necessarily*be*a neighborhood of (because and by definition of "neighborhood of ").

- Generalizing sequence convergence by using sets − determining sequence convergence without the sequence

A * sequence in * is by definition a map from the natural numbers, which are an example of a directed set, into the space The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space. With metrizable spaces (or more generally first–countable spaces or Fréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions. But there are many spaces where sequences can *not* be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, which *never* fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps from an arbitrary directed set into the space A sequence is just a net whose domain is with the natural ordering. Nets have their own notion of convergence, which is a direct generalization of sequence convergence.

Filters generalize sequence convergence in a different way by considering *only* values in the range of a sequence. To see how this is done, consider a sequence in which is by definition just a map whose value at is denoted by rather than the more common parentheses notation Knowing only the range of the sequence is not enough to describe its convergence; multiple sets are needed. It turns out that the needed sets are the following,^{ [note 2] } which are called the *tails* of the sequence :

These sets completely determine this sequence's convergence (or non–convergence) because given any point, this sequence converges to it if and only if for every neighborhood U (of this point), there is some integer n such that U contains all of the points This can be reworded as:

- every neighborhood U must contain some set of the form as a subset.

It is the above characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence With these ** sets** in hand, the

The above set of tails of a sequence is in general not a filter but it does "*generate*" a filter via taking its * upward closure *. The same is true of other important families of sets such as any neighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it generates the neighborhood filter at that point). The properties that these families share led to the notion of a *filter base*, also called a *prefilter*, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward closure *only*.

- Nets vs. filters − advantages and disadvantages

Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other.^{ [note 3] } Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other.^{ [3] } 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 (e.g. ultraproducts), abstract algebra,^{ [4] } order theory, generalized convergence spaces, and in the definition and use of hyperreal numbers.

Like sequences, nets are ** functions** and so they have the advantages of functions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse limits, which is defined in terms of composition of maps rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space and dense subspace

In contrast to nets, filters (and prefilters) are families of ** sets** and so they have the advantages of sets. For example, if is surjective then the

In this article, upper case Roman letters like S and denote sets (but not families unless indicated otherwise) and ℘(*X*) will denote the powerset of A subset of a powerset is called *a family of sets * (or simply, *a family*) where it is *over * if it is a subset of ℘(*X*). Families of sets will be denoted by upper case calligraphy letters such as and Whenever these assumptions are needed, then it should be assumed that is non–empty and that etc. are families of sets over

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

- Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter." While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (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 that are used in this article. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (e.g. 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 prefilter is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

- Sets operations

- The
*upward closure*or*isotonization*in^{ [6] }of a family of subsets is

- and similarly the
*downward closure*of is

Notation and Definition | Assumptions | Name |
---|---|---|

Kernel of ^{ [7] } | ||

℘(X) = { S : S ⊆ X } | Power set of a set ^{ [7] } | |

ℬ|_{S} = { B ∩ S : B ∈ ℬ } = ℬ (∩) { S } | is a set | Trace of on S^{ [8] } or the restriction of to S |

ℬ (∩) 𝒞 = { B ∩ C : B ∈ ℬ and C ∈ 𝒞 }^{ [9] } | Elementwise (set) intersection ( will denote the usual intersection) | |

ℬ (∪) 𝒞 = { B ∪ C : B ∈ ℬ and C ∈ 𝒞 }^{ [9] } | Elementwise (set) union ( will denote the usual union) | |

ℬ (∖) 𝒞 = { B ∖ C : B ∈ ℬ and C ∈ 𝒞 } | Elementwise (set) subtraction | |

S ∖ ℬ = { S ∖ B : B ∈ ℬ } = { S } (∖) ℬ | is a set | Dual of in S or set subtraction ^{ [8] } |

S^{↑X} = { S }^{↑X} | is a set | Upward closure or Isotonization^{ [7] } |

The preorder ≤ is defined on families of sets, say and by declaring that if and only if *mesh*^{ [8] } if *B* ∩ *C* ≠ ∅ for all and .

- for every there is some such that
*F*⊆*C*

in which case it is said that is *coarser than* is *finer than* (or *subordinate to*) ^{ [10] }^{ [11] }^{ [12] } and ℱ ⊢ 𝒞 may be written.

Notation | Definition | Name |
---|---|---|

is a map | Preimage of under ^{ [13] } | |

is a map and is a set | Preimage a S under | |

is a map | Image of under ^{ [13] } | |

is a map and is a set | Image a S under |

- Topology notation

The set of all topologies will be denoted by Top(*X*). Suppose is a topology on

Notation and Definition | Assumptions | Name |
---|---|---|

Set or prefilter^{ [note 4] }of open neighborhoods of S in | ||

Set or prefilterof open neighborhoods of in | ||

Set or filter^{ [note 4] }of neighborhoods of S in | ||

Set or filter of neighborhoods of in |

- Nets and their tails

- A
*directed set*is a set together with a preorder, which will be denoted by (unless explicitly indicated otherwise), that makes into an (*upward*)*directed set*;^{ [14] }this means that for all there exists some such that and For any indices and 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 and ).

- A
*net in*^{ [14] }is a map from a non–empty directed set into

Notation and Definition | Assumptions | Name |
---|---|---|

and is a directed set | Tail or section of starting at | |

and is a net | Tail or section of starting at ^{ [15] } | |

and is a net | Tail or section of starting at | |

is a net | Set or prefilter of tails/sections of Also called the eventuality filter base generated by (the tails of) If is a sequence then is called the sequential filter base instead.^{ [15] } | |

is a net | (Eventuality) filter of/generated by (tails of) ^{ [15] } |

- 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 (e.g. 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:

Properornondegenerateif Otherwise, if then it is calledimproper^{ [16] }ordegenerate.Directed downward^{ [14] }if wheneverA,B∈ ℬ then there exists some such that

- Alternatively,
directed downward(resp.directed upward) if and only if is (upward) directed with respect to the preorder ⊇ (resp. ), where by definition this means that for allA,B∈ ℬ, there exists some "greater" such thatA⊇CandB⊇C(resp. such thatA⊆CandB⊆C), which can be rewritten as (resp.). This explains the word "directed."- 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 closedorIsotonein^{ [6] }if and ℬ = ℬ^{↑X}, or equivalently, if whenever and satisfiesB⊆C⊆Xthen . Similarly, isdownward closedif ℬ = ℬ^{↓}. An upward (respectively, downward) closed set is also called anupper setorupset(resp. alower setordown set).

- The family which is the upward closure of in is the unique
smallestisotone 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.

- Ultrafilters(
*X*) = Filters(*X*) ∩ UltraPrefilters(*X*) ⊆ Filters(*X*) ∪ UltraPrefilters(*X*) ⊆ Prefilters(*X*) ⊆ FilterSubbases(*X*).

A family is/is a(n):

Ideal^{ [16] }^{ [17] }if is downward closed and closed under finite unions.Dual idealon^{ [18] }if is upward closed in and also closed under finite intersections.

- Explanation of the word "dual": A family is a dual ideal (resp. an ideal) on if and only if the
dual of in, which is the family

X∖ ℬ = {X∖B:B∈ ℬ },- is an ideal (resp. a dual ideal) on The family
X∖ ℬ should not be confused with ℘(X) ∖ ℬ ≝ {S⊆X:S∉ ℬ }, where in generalX∖ ℬ ≠ ℘(X) ∖ ℬ. The dual of the dual is the original family, meaningX∖ (X∖ ℬ ) = ℬ; and alsoXbelongs to the dual of if and only if^{ [16] }Filteron^{ [18] }^{ [8] }if is a proper dual ideal on That is, a filter on is a non-empty subset of ℘(X) ∖ { ∅ } 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 aproperdual ideal.^{ [19] }It is recommended that readers always check how "filter" is defined when reading mathematical literature. This article uses Henri Cartan's original definition of filter, which required propriety.- is a filter on if and only if its dual
X∖ ℬ is an ideal that doesnotcontainXas an element. If is an ideal on that satisfiesX∉ ℬ thenX∖ ℬ is called itsdual filteronPrefilterorfilter base^{ [8] }^{ [20] }if is proper and directed downward. Equivalently, is a prefilter if its upward closure is a filter. It can also be defined as any family that is equivalent (with respect to ) tosomefilter.^{ [9] }A proper family is a prefilter if and only if ℬ (∩) ℬ ≤ ℬ.^{ [9] }

- If is a prefilter then its upward closure is the unique smallest (relative to ) filter on containing and it is called
the filter generated byA filter is said to begenerated bya prefilter if ℱ = ℬ^{↑X}, in which is called afilter base for.- Unlike a filter, a prefilter is
notnecessarily closed under finite intersections.π–systemif is closed under finite intersections. Every non–empty family is contained in a unique smallest π–system calledthe π–system generated bywhich is sometimes denoted by π(ℬ ). It is equal to the intersection of all π–systems containing and also to the set of all possible finite intersections of sets from :

- π(ℬ ) = {
B_{1}∩ ⋅⋅⋅ ∩B_{n}:n≥ 1 andB_{1}, ...,B_{n}∈ ℬ }.

- 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^{ [8] }^{ [21] }andcentered^{ [9] }if and satisfies any of the following equivalent conditions:

- has the
finite intersection property, which means that intersection of any finite family of (one or more) sets in is not empty; explicitly, this means that whenevern≥ 1 andB_{1}, ...,B_{n}∈ ℬ then- The π–system generated by is proper (i.e. is not an element).
- The π–system generated by is a prefilter.
- is a subset of
someprefilter.- is a subset of
somefilter.

- The
filter generated byis the unique smallest (relative to ) filter ℱ_{ℬ}on containing It is equal to the intersection of all filters on that have as a subset. The π–system generated by denoted by π(ℬ ), will be a prefilter and a subset of ℱ_{ℬ}. Moreover, the filter generated by is the upward closure of π(ℬ ), meaning π(ℬ )^{↑X}= ℱ_{ℬ}.^{ [9] }- A smallest (relative to ) prefilter containing a filter subbase will exist only under certain circumstances. For example, (1) is a prefilter, or (2) the filter (or equivalently, the π–system) generated by is principle, in which case ℬ ∪ { ker ℬ } is the unique smallest prefilter containing Otherwise, in general, a –smallest
prefilter containing may not exist. For this reason, some authors may refer to the π–system generated by asthe prefilter generated by. However, as shown in an example below, if such a –smallest prefilter does exist then it isnotnecessarily equal to the prefilter (i.e. π–system) generated by ℬ. So unfortunately, "the prefilter generated by" a prefilter may not be which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by ".Subfilterof a filter and that is asuperfilterof^{ [16] }^{ [22] }if is a filter and ℬ ⊆ ℱ where for filters, ℬ ⊆ ℱ if and only if

- Importantly, the expression "is a
filter of" is for filters the analog of "is asupersequence of". So despite having the prefix "sub" in common, "is asubsubfilter of" is actually thereverseof "is asubsequence of."- However, can also be written ℱ ⊢ ℬ 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 asubsequence of,"sub^{ [23] }which makes this one situation where using the term "subordinate" and symbol ⊢ may be helpful.

There are no prefilters on (nor are there any nets valued in ), which is why this article, like most authors, will automatically assume without comment that whenever this assumption is needed.

- Named examples

- The singleton set is called the
*trivial*or*indiscrete filter on*.^{ [24] }^{ [10] }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 - If is a topological space and then the neighborhood filter at is a filter on By definition, a family of subsets of is called a
*neighborhood basis*(resp. a*neighborhood subbasis*) at for 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 and 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*^{ [25] }if for some sequence in - is an
*elementary filter on*^{ [26] }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.^{ [27] } - 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*or*cofinite filter*on^{ [10] }^{ [24] }If is finite then is equal to the dual ideal ℘(*X*), 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 any non–empty set of filters on is itself a filter on called the
*infimum*or*greatest lower bound*of in . Since every filter on has as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to and ) filter contained as a subset of each member of .^{ [10] }- If and are filters then their infimum in is the filter
^{ [9] }If and are prefilters then is a prefilter and one of the finest (with respect to ) prefilters coarser (with respect to ) than both and ; that is, if is a prefilter such that and then^{ [9] }More generally, if and are non-empty families and if 𝕊 ≝ { 𝒮 ⊆ ℘(*X*) : 𝒮 ≤ ℬ and 𝒮 ≤ ℱ } then and is a greatest element (with respect to ) of 𝕊.^{ [9] }

- If and are filters then their infimum in is the filter
- Let be a set of filters on and let If is a filter subbase then the filter on generated by is the
*supremum*or*least upper bound*of in .^{ [10] }By definition, the supremum, if it exists, is the smallest (relative to ) filter containing each member of as a subset. If is not a filter subbase, then the supremum of in (and also in ) does not exist.- If and are prefilters (resp. filters on ) then ℬ (∩) ℱ is a prefilter (resp. a filter) if and only if it is proper (or said differently, if and only if and mesh), in which case it is one of the coarsest (with respect to ) prefilters (resp.
*the*-coarsest filters) that is finer (with respect to ) than both and ; that is, if is a prefilter (resp. filter) such that and then^{ [9] }

- If and are prefilters (resp. filters on ) then ℬ (∩) ℱ is a prefilter (resp. a filter) if and only if it is proper (or said differently, if and only if and mesh), in which case it is one of the coarsest (with respect to ) prefilters (resp.
- Let and 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*.^{ [16] }

- Other examples

- Let and let ℬ = { {
*p*}, {*p*, 1, 2 }, {*p*, 1, 3 } }, 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 { {*p*, 1 } } ∪ ℬ. 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 ℬ^{↑X}= {*S*⊆*X*:*p*∈*S*} = { {*p*} ∪*T*:*T*⊆ { 1, 2, 3 } }. All three of the π–system generates, and are examples of fixed, principal, ultra prefilters that are principal at the point p; is also an ultrafilter on - A prefilter on a topological space
*X*is finer than the prefilter { cl_{'X}*B*:*B*∈ ℬ }.^{ [28] } - The set of all dense open subsets of a (non–empty) topological space is a proper π–system and so also a prefilter. If (with 1 ≤
*n*∈ ℕ), then the set ℬ_{LebFinite}of all such that has finite Lebesgue measure is a proper π–system and prefilter that is also a proper subset of The prefilters ℬ_{LebFinite}and generate the same filter on - This example illustrates a class of a filter subbases 𝒮
_{R}where all sets in both 𝒮_{R}and its generated π-system can be described as sets of the form so that in particular, no more than two variables (i.e. r and s) are needed to describe the generated π-system. However, this is not typical and in general, this should not be expected of a filter subbase that is not a π-system. More often, an intersection of n sets from will usually require a description involving n variables that cannot be reduced down to only two (consider, for instance, if 𝒮_{R}was instead ). For all , let where so no generality is lost by adding the assumption*r*≤*s*. For all real and if or then^{ [note 5] }For every*R*⊆ ℝ, let 𝒮_{R}= {*B*_{−r, r}:*r*∈*R*} and let ℬ_{R}= {*B*_{−r, s}:*r*≤*s*with*r*,*s*∈*R*}.^{ [note 6] }Let and suppose is not a singleton set. Then 𝒮_{R}is a filter subbase but not a prefilter and ℬ_{R}is the π-system it generates, so that ℬ_{R}^{↑X}is the unique smallest filter in containing 𝒮_{R}. However, 𝒮_{R}^{↑X}is*not*a filter on (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and 𝒮_{R}^{↑X}is a proper subset of the filter ℬ_{R}^{↑X}. If are non-empty intervals then the filter subbases 𝒮_{R}and 𝒮_{S}generate the same filter on if and only if If is a family such that 𝒮_{(0, ∞)}⊆ 𝒞 ⊆ ℬ_{(0, ∞)}then is a prefilter if and only if for all real there exist real such that and*B*_{−u, v}∈ 𝒞. If is such a prefilter then for any*C*∈ 𝒞 ∖ 𝒮_{(0, ∞)}, the family 𝒞 ∖ {*C*} is also a prefilter satisfying 𝒮_{(0, ∞)}⊆ 𝒞 ∖ {*C*} ⊆ ℬ_{(0, ∞)}. This shows that there cannot exist a minimal (with respect to ) prefilter that both contains 𝒮_{(0, ∞)}and is a subset of the π-system generated by 𝒮_{(0, ∞)}. This remains true even if the requirement that the prefilter be a subset of ℬ_{(0, ∞)}is removed.

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.

A non–empty family of sets is/is an:

Ultra^{ [8] }^{ [29] }if and any of the following equivalent conditions are satisfied:

- For every set there exists some set such that
B⊆SorB⊆X∖S(or equivalently, such that equals or ).- For every set
S⊆Bthere exists some set such that equals or

- This characterization of " is ultra" does not depend on the set so mentioning the set is optional when using the term "ultra."
- For
everyset (not necessarily even a subset of ) there exists some set such that equals or

- If satisfies this condition then so does
everysuperset ℱ ⊇ ℬ. In particular, a set is ultra if and only if and contains as a subset some ultra family of sets.Ultra prefilter^{ [8] }^{ [29] }if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra.

- A filter subbase that is ultra is necessarily a prefilter.
^{ [proof 1] }Ultrafilter on^{ [8] }^{ [29] }if it is a filter on that is ultra. Equivalently, an ultrafilter on is a filter on that satisfies any of the following equivalent conditions:

- is generated by an ultra prefilter;
- For any or
X∖S∈ ℬ.^{ [16] }- ℬ ∪ (
X∖ ℬ ) = ℘(X). This condition can be restated as: ℘(X) is partitioned by and its dualX∖ ℬ.

- The sets and
X∖ ℬ are disjoint whenever is a prefilter.- For any if
R∪S∈ ℬ then or (a filter with this property is called aprime filter).

- This property extends to any finite union of two or more sets.
- For any if then or
- For any if
R∪S∈ ℬ and theneitheror- is a
maximalfilter on ; meaning that if is a filter on such that ℬ ⊆ ℱ then ℬ = ℱ.

- An ultra prefilter has a similar characterization in terms of maximality with respect to ≤ , where in the special case of filters, if and only if ℬ ⊆ ℱ.
- Because is for filters the analog of "is a subnet of," (specifically, "subnet" should mean "AA-subnet," which is defined below) an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net." This idea is actually made rigorous by ultranets.

** The ultrafilter lemma/principle/theorem ^{ [10] }** (Tarski (1930)

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it.^{ [10] }

- Examples

- Any 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 on is ultra if and only if is a singleton set.
- A family of sets is an ultra prefilter if and only if it is equivalent (with respect to ) to some ultrafilter on in which case this ultrafilter is necessarily equal to the upward closure Consequently, a family of sets is an ultra prefilter if and only if is an ultrafilter on

The kernel is useful in classifying the properties of prefilters and other families of sets.

If then for any point if and only if

- Properties of kernels

For any the ker (ℬ^{↑X}) = ker ℬ and this set is also equal to the kernel of the π–system that it 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 *f* (ker ℬ) ⊆ ker *f* ( ℬ ) and *f* ^{–1} (ker ℬ ) = ker *f* ^{–1} (ℬ ). If then ker 𝒞 ⊆ ker ℬ while if and are equivalent then ker ℬ = ker 𝒞. If and are principal then they are equivalent if and only if ker ℬ = ker 𝒞.

- Classifying families of sets by their kernels

A family of sets is/is an:

Free^{ [7] }if ker ℬ = ∅, or equivalently, if {X∖ {x} :x∈X} ⊆ ℬ^{↑X}; this can be restated as {X∖ {x} :x∈X} ≤ ℬ.

- A filter on is free if and only if is infinite and contains the Fréchet filter on as a subset.
Fixedif ker ℬ ≠ ∅ in which case, is said to befixed byany pointx∈ ker ℬ.

- Any fixed family is necessarily a filter subbase.
Principal^{ [7] }if ker ℬ ∈ ℬ.

- A proper principal family of sets is necessarily a prefilter.
DiscreteorPrincipal at^{ [24] }if {x} = ker ℬ ∈ ℬ.

- The
principal filter at onis the filter {x}^{↑X}. A filter is principal at if and only if ℱ = {x}^{↑X}.

Family of examples: For any non–empty *C* ⊆ ℝ, the family ℬ_{C} = { ℝ ∖ (*r* + *C*) : *r* ∈ ℝ } is free but it is a filter subbase if and only if no finite union of the form (*r*_{1} + *C*) ∪ ⋅⋅⋅ ∪ (*r*_{n} + *C*) covers ℝ, in which case the filter that it generates will also be free. In particular, ℬ_{C} is a filter subbase if is countable (e.g. *C* = ℚ, ℤ, the primes), a meager set in ℝ, a set of finite measure, or a bounded subset of ℝ. If is a singleton set then ℬ_{C} is a subbasis for the Fréchet filter on ℝ.

- Characterizations of fixed ultra prefilters

If a family of sets is fixed (i.e. ker ℬ ≠ ∅) then is ultra if and only if some element of is a singleton set, in which case will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter is ultra if and only if is a singleton set.

Every filter on that is principal at a single point is an ultrafilter, and if in addition is finite, then there are no ultrafilters on other than these.^{ [7] }

The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

**Proposition** — If is an ultrafilter on then the following are equivalent:

- is fixed, or equivalently, not free, meaning ker ℱ ≠ ∅.
- is principle, meaning ker ℱ ∈ ℱ.
- Some element of is a finite set.
- Some element of is a singleton set.
- is principle at some point of which means ker ℱ = {
*x*} ∈ ℱ for - does
*not*contain the Fréchet filter on

- Finite prefilters and finite sets

If a filter subbase is finite then it is fixed (i.e. not free); this is because ker ℬ = *B* 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 (or 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 ℘(*X*). This set is a principal ultra prefilter and any superset ℱ ⊇ ℬ (where ℱ ⊆ ℘(*Y*) and ) with the finite intersection property will also be a principal ultra prefilter (even if is infinite).

Throughout and will be any subsets of ℘(*X*).

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",^{ [23] } where "" can be interpreted as " is a subsequence of " (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also be used to define prefilter convergence in a topological space. The definition of meshes with which is closely related to the preorder , is used in Topology to define cluster points.

Two families of sets andmesh^{ [8] }and arecompatible, indicated by writing ℬ # 𝒞, if for all and . If and do not mesh then they aredissociated. If is a set (but not necessarily a family of sets) then and are said tomeshif and mesh, or equivalently, if thetraceℬ|_{S}= {B∩S:B∈ ℬ } of on S does not contain the empty set.

Declare that and ℱ ⊢ 𝒞, stated as iscoarser thanand isfiner than(orsubordinate to)^{ [10] }^{ [11] }^{ [12] }^{ [9] }if any of the following equivalent conditions hold:

- Definition: Every
containssome Explicitly, this means that for every there is some such that

- Said more briefly, if every set in is
largerthan some set in Here, a "larger set" means a superset.- for every

- In words, states exactly that is larger than some set in The equivalence of (a) and (b) follows immediately.
- From this characterization, it follows that if are families of sets, then 𝒞
_{i}≤ ℱ if and only if 𝒞_{i}≤ ℱ for all- which is equivalent to ;
- ;
- which is equivalent to ;
and if in addition is upward closed, which means that ℱ = ℱ

^{↑X}, then this list can be extended to include:

^{ [6] }

- So in this case, this definition of " is
finerthan " would be identical to the topological definition of "finer" had and been topologies onIf an upward closed family is finer than (i.e. ) but then is said to be

strictly finerthan and isstrictly coarserthan Two families and arecomparableif one of these sets is finer than the other.^{ [10] }

Proof | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Throughout this proof, "set" will mean "subset of " unless indicated otherwise. A "larger set" means a superset. This proof is written with the aim of making the proof of each implication as intuitively clear as possible. For this reason, it is written in a more conversational style and it also tries to limit assigning symbols to sets in Because of characterization (b), it would not be beneficial to attempt this with sets in Statement (a) defines where by definition, if and only if - every set in is
*larger*than some set in**(def)**
If is a set then if and only if is larger than some set in The equivalence of (b) and If If If is a set that is larger than some set in then so is every superset of By definition, consists exactly of all supersets of For this reason, by using the corollary of (b), we conclude that implies Consequently, if This proves that
*C*∈ ℱ^{↑X}⇔ is larger than some set in**(↑X def)**
Restricting to range over it follows from - 𝒞 ⊆ ℱ
^{↑X}⇔.
By definition, is upward closed if and only if ℱ = ℱ - 𝒞 ⊆ ℱ
^{↑X}⇔ 𝒞 ≤ ℱ^{↑X}, and also that ⇔ 𝒞^{↑X}≤ ℱ^{↑X}
It remains to show (c) ⇒ (a). By |

Assume that and are families of sets that satisfy Then ker ℱ ⊆ ker 𝒞, and implies and also implies If in addition to is a filter *sub*base and then is a filter subbase^{ [9] } and also and mesh.^{ [18] }^{ [proof 2] } Every filter subbase is coarser than both the π-system that it generates and the filter that it generates.^{ [9] }

If and are families such that the family is ultra, and then