In mathematics, a filter on a set $X$ is a

of a filter $\backslash mathcal$ and that $\backslash mathcal$ is a of $\backslash mathcal$ if $\backslash mathcal$ is a filter and $\backslash mathcal\; \backslash subseteq\; \backslash mathcal$ where for filters, $\backslash mathcal\; \backslash subseteq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$
* 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, $\backslash mathcal\; \backslash leq\; \backslash mathcal$ can also be written $\backslash mathcal\; \backslash vdash\; \backslash mathcal$ which is described by saying "$\backslash mathcal$ is subordinate to $\backslash mathcal.$" 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 $\backslash ,\backslash vdash\backslash ,$ may be helpful.
There are no prefilters on $X\; =\; \backslash varnothing$ (nor are there any nets valued in $\backslash varnothing$), which is why this article, like most authors, will automatically assume without comment that $X\; \backslash neq\; \backslash varnothing$ whenever this assumption is needed.

For set $S$ (not necessarily even a subset of $X$) there exists some set $B\; \backslash in\; \backslash mathcal$ such that $B\; \backslash cap\; S\; \backslash text\; B\; \backslash text\; \backslash varnothing.$
* If $\backslash mathcal$ satisfies this condition then so does superset $\backslash mathcal\; \backslash supseteq\; \backslash mathcal.$ For example, if $T$ is any
if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter $\backslash mathcal$ is ultra if and only if it satisfies any of the following equivalent conditions:
if it is a filter on $X$ that is ultra. Equivalently, an ultrafilter on $X$ is a filter $\backslash mathcal\; \backslash text\; X$ that satisfies any of the following equivalent conditions:
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 $\backslash \; \backslash text\; X$ is ultra if and only if $X$ is a singleton set.
The ultrafilter lemma
The following important theorem is due to

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 $\backslash mathcal\_$ is dense in $X$ (and also

family
Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideal ...

$\backslash mathcal$ of 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 o ...

s such that:
# $X\; \backslash in\; \backslash mathcal$ and $\backslash emptyset\; \backslash notin\; \backslash mathcal$
# if $A\backslash in\; \backslash mathcal$ and $B\; \backslash in\; \backslash mathcal$, then $A\backslash cap\; B\backslash in\; \backslash mathcal$
# If $A,B\backslash subset\; X,A\backslash in\; \backslash mathcal$, and $A\backslash subset\; B$, then $B\backslash in\; \backslash mathcal$
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 ...

, 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 concer ...

, 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.
Filters were introduced by Henri Cartan
Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology.
He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...

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 gi ...

, they were subsequently used by Nicolas Bourbaki
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 ...

in their book '' Topologie Générale'' as an alternative to the related notion of a net developed in 1922
Events
January
* January 7 – Dáil Éireann, the parliament of the Irish Republic, ratifies the Anglo-Irish Treaty by 64–57 votes.
* January 10 – Arthur Griffith is elected President of Dáil Éireann, the day after Éamon de Valera ...

by 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 re ...

. 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 binary ...

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 post ...

ordered by set inclusion.
Preliminaries, notation, and basic notions

In this article, upper case Roman letters like $S\; \backslash text\; X$ denote sets (but not families unless indicated otherwise) and $\backslash wp(X)$ will denote thepower 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 post ...

of $X.$ A subset of a power set is called (or simply, ) where it is if it is a subset of $\backslash wp(X).$ Families of sets will be denoted by upper case calligraphy letters such as $\backslash mathcal,\; \backslash mathcal,\; \backslash text\; \backslash mathcal.$
Whenever these assumptions are needed, then it should be assumed that $X$ is non–empty and that $\backslash mathcal,\; \backslash mathcal,$ etc. are families of sets over $X.$
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 $X$ 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 fa ...

$\backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ is
and similarly the of $\backslash mathcal$ is $\backslash mathcal^\; :=\; \backslash \; =\; \backslash bigcup\_\; \backslash wp(B).$
Throughout, $f$ is a map and $S$ is a set.
Nets and their tails
A is a set $I$ together with a preorder, which will be denoted by $\backslash ,\backslash leq\backslash ,$ (unless explicitly indicated otherwise), that makes $(I,\; \backslash leq)$ into an () ; this means that for all $i,\; j\; \backslash in\; I,$ there exists some $k\; \backslash in\; I$ such that $i\; \backslash leq\; k\; \backslash text\; j\; \backslash leq\; k.$ For any indices $i\; \backslash text\; j,$ the notation $j\; \backslash geq\; i$ is defined to mean $i\; \backslash leq\; j$ while $i\; <\; j$ is defined to mean that $i\; \backslash leq\; j$ holds but it is true that $j\; \backslash leq\; i$ (if $\backslash ,\backslash leq\backslash ,$ is antisymmetric then this is equivalent to $i\; \backslash leq\; j\; \backslash text\; i\; \backslash neq\; j$).
A is a map from a non–empty directed set into $X.$
The notation $x\_\; =\; \backslash left(x\_i\backslash right)\_$ will be used to denote a net with domain $I.$
Warning about using strict comparison
If $x\_\; =\; \backslash left(x\_i\backslash right)\_$ is a net and $i\; \backslash in\; I$ then it is possible for the set $x\_\; =\; \backslash left\backslash ,$ which is called , to be empty (for example, this happens if $i$ is an upper bound 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 ...

$I$).
In this case, the family $\backslash left\backslash $ would contain the empty set, which would prevent it from being a prefilter (defined later).
This is the (important) reason for defining $\backslash operatorname\backslash left(x\_\backslash right)$ as $\backslash left\backslash $ rather than $\backslash left\backslash $ or even $\backslash left\backslash \backslash cup\; \backslash left\backslash $ and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality $\backslash ,<\backslash ,$ may not be used interchangeably with the inequality $\backslash ,\backslash leq.$
Filters and prefilters

The following is a list of properties that a family $\backslash mathcal$ 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 $\backslash mathcal\; \backslash subseteq\; \backslash wp(X).$ Many of the properties of $\backslash mathcal$ defined above and below, such as "proper" and "directed downward," do not depend on $X,$ so mentioning the set $X$ is optional when using such terms. Definitions involving being "upward closed in $X,$" such as that of "filter on $X,$" do depend on $X$ so the set $X$ should be mentioned if it is not clear from context. $$\backslash textrm(X)\; \backslash quad=\backslash quad\; \backslash textrm(X)\; \backslash ,\backslash setminus\backslash ,\; \backslash \; \backslash quad\backslash subseteq\backslash quad\; \backslash textrm(X)\; \backslash quad\backslash subseteq\backslash quad\; \backslash textrm(X).$$ \text X containing $\backslash mathcal$ called the , and $\backslash mathcal$ is said to this filter. This filter is equal to the intersection of all filters on $X$ that are supersets of $\backslash mathcal.$ The –system generated by $\backslash mathcal,$ denoted by $\backslash pi(\backslash mathcal),$ will be a prefilter and a subset of $\backslash mathcal\_.$ Moreover, the filter generated by $\backslash mathcal$ is equal to the upward closure of $\backslash pi(\backslash mathcal),$ meaning $\backslash pi(\backslash mathcal)^\; =\; \backslash mathcal\_.$ However, $\backslash mathcal^\; =\; \backslash mathcal\_$ if $\backslash mathcal$ is a prefilter (although $\backslash mathcal^$ is always an upward closed filter base for $\backslash mathcal\_$). * A $\backslash subseteq$–smallest (meaning smallest relative to $\backslash subseteq$) filter containing a filter subbase $\backslash mathcal$ will exist only under certain circumstances. It exists, for example, if the filter subbase $\backslash mathcal$ happens to also be a prefilter. It also exists if the filter (or equivalently, the –system) generated by $\backslash mathcal$ is principal, in which case $\backslash mathcal\; \backslash cup\; \backslash $ is the unique smallest prefilter containing $\backslash mathcal.$ Otherwise, in general, a $\backslash subseteq$–smallest filter containing $\backslash mathcal$ might not exist. For this reason, some authors may refer to the –system generated by $\backslash mathcal$ as However, if a $\backslash subseteq$–smallest prefilter does exist (say it is denoted by $\backslash operatorname\; \backslash mathcal$) then contrary to usual expectations, it is necessarily equal to " the prefilter generated by $\backslash mathcal$" (that is, $\backslash operatorname\; \backslash mathcal\; \backslash neq\; \backslash pi(\backslash mathcal)$ is possible). And if the filter subbase $\backslash mathcal$ happens to also be a prefilter but not a -system then unfortunately, " the prefilter generated by this prefilter" (meaning $\backslash pi(\backslash mathcal)$) will not be $\backslash mathcal\; =\; \backslash operatorname\; \backslash mathcal$ (that is, $\backslash pi(\backslash mathcal)\; \backslash neq\; \backslash mathcal$ is possible even when $\backslash mathcal$ is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the –system generated by $\backslash mathcal$".Basic examples

Named examples- The singleton set $\backslash mathcal\; =\; \backslash $ is called the or It is the unique filter on $X$ because it is a subset of every filter on $X$; however, it need not be a subset of every prefilter on $X.$
- The dual ideal $\backslash wp(X)$ is also called (despite not actually being a filter). It is the only dual ideal on $X$ that is not a filter on $X.$
- If $(X,\; \backslash tau)$ is a topological space and $x\; \backslash in\; X,$ 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 ...$\backslash mathcal(x)$ at $x$ is a filter on $X.$ By definition, a family $\backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ is called a (resp. a ) at $x\; \backslash text\; (X,\; \backslash tau)$ if and only if $\backslash mathcal$ is a prefilter (resp. $\backslash mathcal$ is a filter subbase) and the filter on $X$ that $\backslash mathcal$ generates is equal to the neighborhood filter $\backslash mathcal(x).$ The subfamily $\backslash tau(x)\; \backslash subseteq\; \backslash mathcal(x)$ of open neighborhoods is a filter base for $\backslash mathcal(x).$ Both prefilters $\backslash mathcal(x)\; \backslash text\; \backslash tau(x)$ also form a bases for topologies on $X,$ with the topology generated $\backslash tau(x)$ being coarser than $\backslash tau.$ This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets $S\; \backslash subseteq\; X.$
- $\backslash mathcal$ is an if $\backslash mathcal\; =\; \backslash operatorname\backslash left(x\_\backslash right)$ for some sequence $x\_\; =\; \backslash left(x\_i\backslash right)\_^\; \backslash text\; X.$
- $\backslash mathcal$ is an or a on $X$ if $\backslash mathcal$ is a filter on $X$ 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 $\backslash mathcal$ of all cofinite subsets of $X$ (meaning those sets whose complement in $X$ is finite) is proper if and only if $\backslash mathcal$ is infinite (or equivalently, $X$ is infinite), in which case $\backslash mathcal$ is a filter on $X$ known as the or the on $X.$ If $X$ is finite then $\backslash mathcal$ is equal to the dual ideal $\backslash wp(X),$ which is not a filter. If $X$ is infinite then the family $\backslash $ of complements of singleton sets is a filter subbase that generates the Fréchet filter on $X.$ As with any family of sets over $X$ that contains $\backslash ,$ the kernel of the Fréchet filter on $X$ is the empty set: $\backslash ker\; \backslash mathcal\; =\; \backslash varnothing.$
- The intersection of all elements in any non–empty family $\backslash mathbb\; \backslash subseteq\; \backslash operatorname(X)$ is itself a filter on $X$ called the or of $\backslash mathbb\; \backslash text\; \backslash operatorname(X),$ which is why it may be denoted by $\backslash bigwedge\_\; \backslash mathcal.$ Said differently, $\backslash ker\; \backslash mathbb\; =\; \backslash bigcap\_\; \backslash mathcal\; \backslash in\; \backslash operatorname(X).$ Because every filter on $X$ has $\backslash $ as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to $\backslash ,\backslash subseteq\backslash ,\; \backslash text\; \backslash ,\backslash leq\backslash ,$) filter contained as a subset of each member of $\backslash mathbb.$
* If $\backslash mathcal\; \backslash text\; \backslash mathcal$ are filters then their infimum in $\backslash operatorname(X)$ is the filter $\backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal.$ If $\backslash mathcal\; \backslash text\; \backslash mathcal$ are prefilters then $\backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal$ is a prefilter that is coarser (with respect to $\backslash ,\backslash leq$) than both $\backslash mathcal\; \backslash text\; \backslash mathcal$ (that is, $\backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\; \backslash leq\; \backslash mathcal$); indeed, it is one of the finest such prefilters, meaning that if $\backslash mathcal$ is a prefilter such that $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal$ then necessarily $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal.$ More generally, if $\backslash mathcal\; \backslash text\; \backslash mathcal$ are non−empty families and if $\backslash mathbb\; :=\; \backslash $ then $\backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal\; \backslash in\; \backslash mathbb$ and $\backslash mathcal\; \backslash ,(\backslash cup)\backslash ,\; \backslash mathcal$ 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 $\backslash leq$) of $\backslash mathbb.$
- Let $\backslash varnothing\; \backslash neq\; \backslash mathbb\; \backslash subseteq\; \backslash operatorname(X)$ and let $\backslash cup\; \backslash mathbb\; =\; \backslash bigcup\_\; \backslash mathcal.$ The or of $\backslash mathbb\; \backslash text\; \backslash operatorname(X),$ denoted by $\backslash bigvee\_\; \backslash mathcal,$ is the smallest (relative to $\backslash subseteq$) dual ideal on $X$ containing every element of $\backslash mathbb$ as a subset; that is, it is the smallest (relative to $\backslash subseteq$) dual ideal on $X$ containing $\backslash cup\; \backslash mathbb$ as a subset. This dual ideal is $\backslash bigvee\_\; \backslash mathcal\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\backslash right)^,$ where $\backslash pi\backslash left(\backslash cup\; \backslash mathbb\backslash right)\; :=\; \backslash left\backslash $ is the –system generated by $\backslash cup\; \backslash mathbb.$ As with any non–empty family of sets, $\backslash cup\; \backslash mathbb$ is contained in filter on $X$ if and only if it is a filter subbase, or equivalently, if and only if $\backslash bigvee\_\; \backslash mathcal\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\backslash right)^$ is a filter on $X,$ in which case this family is the smallest (relative to $\backslash subseteq$) filter on $X$ containing every element of $\backslash mathbb$ as a subset and necessarily $\backslash mathbb\; \backslash subseteq\; \backslash operatorname(X).$
- Let $\backslash varnothing\; \backslash neq\; \backslash mathbb\; \backslash subseteq\; \backslash operatorname(X)$ and let $\backslash cup\; \backslash mathbb\; =\; \backslash bigcup\_\; \backslash mathcal.$ The or of $\backslash mathbb\; \backslash text\; \backslash operatorname(X),$ denoted by $\backslash bigvee\_\; \backslash mathcal$ if it exists, is by definition the smallest (relative to $\backslash subseteq$) filter on $X$ containing every element of $\backslash mathbb$ as a subset. If it exists then necessarily $\backslash bigvee\_\; \backslash mathcal\; =\; \backslash pi\backslash left(\backslash cup\; \backslash mathbb\backslash right)^$ (as defined above) and $\backslash bigvee\_\; \backslash mathcal$ will also be equal to the intersection of all filters on $X$ containing $\backslash cup\; \backslash mathbb.$ This supremum of $\backslash mathbb\; \backslash text\; \backslash operatorname(X)$ exists if and only if the dual ideal $\backslash pi\backslash left(\backslash cup\; \backslash mathbb\backslash right)^$ is a filter on $X.$ The least upper bound of a family of filters $\backslash mathbb$ may fail to be a filter. Indeed, if $X$ contains at least 2 distinct elements then there exist filters $\backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash text\; X$ for which there does exist a filter $\backslash mathcal\; \backslash text\; X$ that contains both $\backslash mathcal\; \backslash text\; \backslash mathcal.$ If $\backslash cup\; \backslash mathbb$ is not a filter subbase then the supremum of $\backslash mathbb\; \backslash text\; \backslash operatorname(X)$ does not exist and the same is true of its supremum in $\backslash operatorname(X)$ but their supremum in the set of all dual ideals on $X$ will exist (it being the degenerate filter $\backslash wp(X)$). * If $\backslash mathcal\; \backslash text\; \backslash mathcal$ are prefilters (resp. filters on $X$) then $\backslash mathcal\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal$ is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if $\backslash mathcal\; \backslash text\; \backslash mathcal$ mesh), in which case it is coarsest prefilters (resp. coarsest filter) on $X$ (with respect to $\backslash ,\backslash leq$) that is finer (with respect to $\backslash ,\backslash leq$) than both $\backslash mathcal\; \backslash text\; \backslash mathcal;$ this means that if $\backslash mathcal$ is any prefilter (resp. any filter) such that $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal$ then necessarily $\backslash mathcal\; \backslash ,(\backslash cap)\backslash ,\; \backslash mathcal\; \backslash leq\; \backslash mathcal,$ in which case it is denoted by $\backslash mathcal\; \backslash vee\; \backslash mathcal.$
- Let $I\; \backslash text\; X$ be non−empty sets and for every $i\; \backslash in\; I$ let $\backslash mathcal\_i$ be a dual ideal on $X.$ If $\backslash mathcal$ is any dual ideal on $I$ then $\backslash bigcup\_\; \backslash ;\backslash ;\backslash bigcap\_\; \backslash ;\backslash mathcal\_i$ is a dual ideal on $X$ called or .
- The club filter of a regular 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 nu ...cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...is the filter of all sets containing a club subset of $\backslash kappa.$ It is a $\backslash kappa$-complete filter closed under diagonal intersection.

- Let $X\; =\; \backslash $ and let $\backslash mathcal\; =\; \backslash ,$ which makes $\backslash mathcal$ a prefilter and a filter subbase that is not closed under finite intersections. Because $\backslash mathcal$ is a prefilter, the smallest prefilter containing $\backslash mathcal$ is $\backslash mathcal.$ The –system generated by $\backslash mathcal$ is $\backslash \; \backslash cup\; \backslash mathcal.$ In particular, the smallest prefilter containing the filter subbase $\backslash mathcal$ is equal to the set of all finite intersections of sets in $\backslash mathcal.$ The filter on $X$ generated by $\backslash mathcal$ is $\backslash mathcal^\; =\; \backslash \; =\; \backslash .$ All three of $\backslash mathcal,$ the –system $\backslash mathcal$ generates, and $\backslash mathcal^$ are examples of fixed, principal, ultra prefilters that are principal at the point $p;\; \backslash mathcal^$ is also an ultrafilter on $X.$
- Let $(X,\; \backslash tau)$ be a topological space, $\backslash mathcal\; \backslash subseteq\; \backslash wp(X),$ and define $\backslash overline\; :=\; \backslash left\backslash ,$ where $\backslash mathcal$ is necessarily finer than $\backslash overline.$ If $\backslash mathcal$ is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of $\backslash overline.$ If $\backslash mathcal$ is a filter on $X$ then $\backslash overline$ is a prefilter but not necessarily a filter on $X$ although $\backslash left(\backslash overline\backslash right)^$ is a filter on $X$ equivalent to $\backslash overline.$
- The set $\backslash mathcal$ of all dense open subsets of a (non–empty) topological space $X$ 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 $\backslash mathcal.$ If $X\; =\; \backslash R^n$ (with $1\; \backslash leq\; n\; \backslash in\; \backslash N$) then the set $\backslash mathcal\_$ of all $B\; \backslash in\; \backslash mathcal$ such that $B$ has finiteLebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...is a proper –system and free prefilter that is also aproper 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 o ...of $\backslash mathcal.$ The prefilters $\backslash mathcal\_$ and $\backslash mathcal$ are equivalent and so generate the same filter on $X.$ The prefilter $\backslash mathcal\_$ is properly contained in, and not equivalent to, the prefilter consisting of all dense subsets of $\backslash R.$ Since $X$ is aBaire 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 $\backslash mathcal\_$ is dense in $X$ (and alsocomeagre 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 called ...and non–meager) so the set of all countable intersections of elements of $\backslash mathcal\_$ is a prefilter and –system; it is also finer than, and not equivalent to, $\backslash mathcal\_.$
- ''A filter subbase with no $\backslash ,\backslash subseteq-$smallest prefilter containing it'': In general, if a filter subbase $\backslash mathcal$ is not a –system then an intersection $S\_1\; \backslash cap\; \backslash cdots\; \backslash cap\; S\_n$ of $n$ sets from $\backslash mathcal$ will usually require a description involving $n$ variables that cannot be reduced down to only two (consider, for instance $\backslash pi(\backslash mathcal)$ when $\backslash mathcal\; =\; \backslash $). This example illustrates an atypical class of a filter subbases $\backslash mathcal\_R$ where all sets in both $\backslash mathcal\_R$ and its generated –system can be described as sets of the form $B\_,$ so that in particular, no more than two variables (specifically, $r\; \backslash text\; s$) are needed to describe the generated –system. For all $r,\; s\; \backslash in\; \backslash R,$ let $$B\_\; =\; (r,\; 0)\; \backslash cup\; (s,\; \backslash infty),$$ where $B\_\; =\; B\_$ always holds so no generality is lost by adding the assumption $r\; \backslash leq\; s.$ For all real $r\; \backslash leq\; s\; \backslash text\; u\; \backslash leq\; v,$ if $s\; \backslash text\; v$ is non-negative then $B\_\; \backslash cap\; B\_\; =\; B\_.$More generally, for any real numbers satisfying $r\; \backslash leq\; s\; \backslash text\; u\; \backslash leq\; v,\; B\_\; \backslash cap\; B\_\; =\; B\_$ where $m\; :=\; \backslash min(s,v,\backslash max(r,u)).$ For every set $R$ of positive reals, letIf $R,\; S\; \backslash subseteq\; \backslash R\; \backslash text\; \backslash mathcal\_R\; \backslash cap\; \backslash mathcal\_S\; =\; \backslash mathcal\_.$ This property and the fact that $\backslash mathcal\_R$ is nonempty and proper if and only if $R\; \backslash neq\; \backslash varnothing$ actually allows for the construction of even more examples of prefilters, because if $\backslash mathcal\; \backslash subseteq\; \backslash wp(\backslash R)$ is any prefilter (resp. filter subbase, –system) then so is $\backslash left\backslash .$ $$\backslash mathcal\_R\; :=\; \backslash left\backslash \; =\; \backslash \; \backslash quad\; \backslash text\; \backslash quad\; \backslash mathcal\_R\; :=\; \backslash left\backslash \; =\; \backslash .$$ Let $X\; =\; \backslash R$ and suppose $\backslash varnothing\; \backslash neq\; R\; \backslash subseteq\; (0,\; \backslash infty)$ is not a singleton set. Then $\backslash mathcal\_R$ is a filter subbase but not a prefilter and $\backslash mathcal\_R\; =\; \backslash pi\backslash left(\backslash mathcal\_R\backslash right)$ is the –system it generates, so that $\backslash mathcal\_R^$ is the unique smallest filter in $X\; =\; \backslash R$ containing $\backslash mathcal\_R.$ However, $\backslash mathcal\_R^$ is a filter on $X$ (nor is it a prefilter because it is not directed downward, although it is a filter subbase) and $\backslash mathcal\_R^$ is a proper subset of the filter $\backslash mathcal\_R^.$ If $R,\; S\; \backslash subseteq\; (0,\; \backslash infty)$ are non−empty intervals then the filter subbases $\backslash mathcal\_R\; \backslash text\; \backslash mathcal\_S$ generate the same filter on $X$ if and only if $R\; =\; S.$ If $\backslash mathcal$ is a prefilter satisfying $\backslash mathcal\_\; \backslash subseteq\; \backslash mathcal\; \backslash subseteq\; \backslash mathcal\_$It may be shown that if $\backslash mathcal$ is any family such that $\backslash mathcal\_\; \backslash subseteq\; \backslash mathcal\; \backslash subseteq\; \backslash mathcal\_$ then $\backslash mathcal$ is a prefilter if and only if for all real $0\; <\; r\; \backslash leq\; s$ there exist real $0\; <\; u\; \backslash leq\; v$ such that $u\; \backslash leq\; r\; \backslash leq\; s\; \backslash leq\; v\; \backslash text\; B\_\; \backslash in\; \backslash mathcal.$ then for any $C\; \backslash in\; \backslash mathcal\; \backslash setminus\; \backslash mathcal\_,$ the family $\backslash mathcal\; \backslash setminus\; \backslash $ is also a prefilter satisfying $\backslash mathcal\_\; \backslash subseteq\; \backslash mathcal\; \backslash setminus\; \backslash \; \backslash subseteq\; \backslash mathcal\_.$ This shows that there cannot exist a minimal/ least (with respect to $\backslash subseteq$) prefilter that both contains $\backslash mathcal\_$ and is a subset of the –system generated by $\backslash mathcal\_.$ This remains true even if the requirement that the prefilter be a subset of $\backslash mathcal\_\; =\; \backslash pi\backslash left(\backslash mathcal\_\backslash right)$ is removed; that is, (in sharp contrast to filters) there does exist a minimal/least (with respect to $\backslash subseteq$) filter containing the filter subbase $\backslash mathcal\_.$

Ultrafilters

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article. $$\backslash begin\; \backslash textrm(X)\backslash ;\; \&=\backslash ;\; \backslash textrm(X)\; \backslash ,\backslash cap\backslash ,\; \backslash textrm(X)\backslash \backslash \; \&\backslash subseteq\backslash ;\; \backslash textrm(X)\; =\; \backslash textrm(X)\backslash \backslash \; \&\backslash subseteq\backslash ;\; \backslash textrm(X)\; \backslash \backslash \; \backslash end$$ B there exists some set $B\; \backslash in\; \backslash mathcal$ such that $B\; \backslash cap\; S\; \backslash text\; B\; \backslash text\; \backslash varnothing.$ * This characterization of "$\backslash mathcal$ is ultra" does not depend on the set $X,$ so mentioning the set $X$ is optional when using the term "ultra."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 $\backslash $ is ultra and consequently, any non–degenerate superset of $\backslash $ (such as its upward closure) is also ultra.- $\backslash mathcal$ is in $\backslash operatorname(X)$ with respect to $\backslash ,\backslash leq,\backslash ,$ which means that $$\backslash text\; \backslash mathcal\; \backslash in\; \backslash operatorname(X),\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash ;\; \backslash text\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$$
- $\backslash text\; \backslash mathcal\; \backslash in\; \backslash operatorname(X),\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash ;\; \backslash text\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$ * Although this statement is identical to that given below for ultrafilters, here $\backslash mathcal$ is merely assumed to be a prefilter; it need not be a filter.
- $\backslash mathcal^$ is ultra (and thus an ultrafilter).
- $\backslash mathcal$ is equivalent (with respect to $\backslash leq$) to some ultrafilter.

- $\backslash mathcal$ is generated by an ultra prefilter.
- For any $S\; \backslash subseteq\; X,\; S\; \backslash in\; \backslash mathcal\; \backslash text\; X\; \backslash setminus\; S\; \backslash in\; \backslash mathcal.$
- $\backslash mathcal\; \backslash cup\; (X\; \backslash setminus\; \backslash mathcal)\; =\; \backslash wp(X).$ This condition can be restated as: $\backslash wp(X)$ is partitioned by $\backslash mathcal$ and its dual $X\; \backslash setminus\; \backslash mathcal.$ * The sets $\backslash mathcal\; \backslash text\; X\; \backslash setminus\; \backslash mathcal$ are disjoint whenever $\backslash mathcal$ is a prefilter.
- $\backslash wp(X)\; \backslash setminus\; \backslash mathcal\; =\; \backslash $ is an ideal.
- For any $R,\; S\; \backslash subseteq\; X,$ if $R\; \backslash cup\; S\; =\; X$ then $R\; \backslash in\; \backslash mathcal\; \backslash text\; S\; \backslash in\; \backslash mathcal.$
- For any $R,\; S\; \backslash subseteq\; X,$ if $R\; \backslash cup\; S\; \backslash in\; \backslash mathcal$ then $R\; \backslash in\; \backslash mathcal\; \backslash text\; S\; \backslash in\; \backslash mathcal$ (a filter with this property is called a ). * This property extends to any finite union of two or more sets.
- For any $R,\; S\; \backslash subseteq\; X,$ if $R\; \backslash cup\; S\; \backslash in\; \backslash mathcal\; \backslash text\; R\; \backslash cap\; S\; =\; \backslash varnothing$ then $R\; \backslash in\; \backslash mathcal\; \backslash text\; S\; \backslash in\; \backslash mathcal.$
- $\backslash mathcal$ is a filter on $X$; meaning that if $\backslash mathcal$ is a filter on $X$ such that $\backslash mathcal\; \backslash subseteq\; \backslash mathcal$ then necessarily $\backslash mathcal\; =\; \backslash mathcal$ (this equality may be replaced by $\backslash mathcal\; \backslash subseteq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal$).
* If $\backslash mathcal$ is upward closed then $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash subseteq\; \backslash mathcal.$ So this characterization of ultrafilters as maximal filters can be restated as: $$\backslash text\; \backslash mathcal\; \backslash in\; \backslash operatorname(X),\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash ;\; \backslash text\; \backslash ;\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$$
* Because subordination $\backslash ,\backslash geq\backslash ,$ 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 $X$" 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 $u\_\; \backslash text\; X$ could be interpreted as being "maximally deep" is if all important properties related to $X$ (such as convergence for example) of any subnet is completely determined by $u\_$ in all topologies on $X.$ In this case $u\_$ 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 $X$ 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 (s ...is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

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 $\backslash mathcal$ be a filter on $X$ that is not an ultrafilter. If $S\; \backslash subseteq\; X$ is such that $S\; \backslash not\backslash in\; \backslash mathcal\; \backslash text\; \backslash \; \backslash cup\; \backslash mathcal$ has the finite intersection property (because if $F\; \backslash in\; \backslash mathcal\; \backslash text\; F\; \backslash cap\; (X\; \backslash setminus\; S)\; =\; \backslash varnothing\; \backslash text\; F\; \backslash subseteq\; S$) so that by the ultrafilter lemma, there exists some ultrafilter $\backslash mathcal\_S\; \backslash text\; X$ such that $\backslash \; \backslash cup\; \backslash mathcal\; \backslash subseteq\; \backslash mathcal\_S$ (so in particular, $S\; \backslash not\backslash in\; \backslash mathcal\_S$). Intersecting all such $\backslash mathcal\_S$ proves that $\backslash mathcal\; =\; \backslash bigcap\_\; \backslash mathcal\_S.\; \backslash blacksquare$
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) 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, norm, topology, etc.) and the linear functions defined ...

(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 linea ...

) 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 $\backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ then for any point $x,\; x\; \backslash not\backslash in\; \backslash ker\; \backslash mathcal\; \backslash text\; X\; \backslash setminus\; \backslash \; \backslash in\; \backslash mathcal^.$ Properties of kernels If $\backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ then $\backslash ker\; \backslash left(\backslash mathcal^\backslash right)\; =\; \backslash ker\; \backslash mathcal$ and this set is also equal to the kernel of the –system that is generated by $\backslash mathcal.$ In particular, if $\backslash mathcal$ is a filter subbase then the kernels of all of the following sets are equal: :(1) $\backslash mathcal,$ (2) the –system generated by $\backslash mathcal,$ and (3) the filter generated by $\backslash mathcal.$ If $f$ is a map then $f(\backslash ker\; \backslash mathcal)\; \backslash subseteq\; \backslash ker\; f(\backslash mathcal)$ and $f^(\backslash ker\; \backslash mathcal)\; =\; \backslash ker\; f^(\backslash mathcal).$ If $\backslash mathcal\; \backslash leq\; \backslash mathcal$ then $\backslash ker\; \backslash mathcal\; \backslash subseteq\; \backslash ker\; \backslash mathcal$ while if $\backslash mathcal$ and $\backslash mathcal$ are equivalent then $\backslash ker\; \backslash mathcal\; =\; \backslash ker\; \backslash mathcal.$ Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal; that is, if $\backslash mathcal$ and $\backslash mathcal$ are principal then they are equivalent if and only if $\backslash ker\; \backslash mathcal\; =\; \backslash ker\; \backslash mathcal.$Classifying families by their kernels

If $\backslash mathcal$ is a principal filter on $X$ then $\backslash varnothing\; \backslash neq\; \backslash ker\; \backslash mathcal\; \backslash in\; \backslash mathcal$ and $$\backslash mathcal\; =\; \backslash ^\; =\; \backslash \; =\; \backslash wp(X\; \backslash setminus\; \backslash ker\; \backslash mathcal)\; \backslash ,(\backslash cup)\backslash ,\; \backslash $$ where $\backslash $ is also the smallest prefilter that generates $\backslash mathcal.$ Family of examples: For any non–empty $C\; \backslash subseteq\; \backslash R,$ the family $\backslash mathcal\_C\; =\; \backslash $ is free but it is a filter subbase if and only if no finite union of the form $\backslash left(r\_1\; +\; C\backslash right)\; \backslash cup\; \backslash cdots\; \backslash cup\; \backslash left(r\_n\; +\; C\backslash right)$ covers $\backslash R,$ in which case the filter that it generates will also be free. In particular, $\backslash mathcal\_C$ is a filter subbase if $C$ is countable (for example, $C\; =\; \backslash Q,\; \backslash Z,$ the primes), ameager 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 called ...

in $\backslash R,$ a set of finite measure, or a bounded subset of $\backslash R.$ If $C$ is a singleton set then $\backslash mathcal\_C$ is a subbase for the Fréchet filter on $\backslash R.$
For every filter $\backslash mathcal\; \backslash text\; X$ there exists a unique pair of dual ideals $\backslash mathcal^*\; \backslash text\; \backslash mathcal^\; \backslash text\; X$ such that $\backslash mathcal^*$ is free, $\backslash mathcal^$ is principal, and $\backslash mathcal^*\; \backslash wedge\; \backslash mathcal^\; =\; \backslash mathcal,$ and $\backslash mathcal^*\; \backslash text\; \backslash mathcal^$ do not mesh (that is, $\backslash mathcal^*\; \backslash vee\; \backslash mathcal^\; =\; \backslash wp(X)$). The dual ideal $\backslash mathcal^*$ is called of $\backslash mathcal$ while $\backslash mathcal^$ is called where at least one of these dual ideals is filter. If $\backslash mathcal$ is principal then $\backslash mathcal^\; :=\; \backslash mathcal\; \backslash text\; \backslash mathcal^*\; :=\; \backslash wp(X);$ otherwise, $\backslash mathcal^\; :=\; \backslash ^$ and $\backslash mathcal^*\; :=\; \backslash mathcal\; \backslash vee\; \backslash ^$ is a free (non–degenerate) filter.
Finite prefilters and finite sets
If a filter subbase $\backslash mathcal$ is finite then it is fixed (that is, not free);
this is because $\backslash ker\; \backslash mathcal\; =\; \backslash bigcap\_\; B$ is a finite intersection and the filter subbase $\backslash mathcal$ has the finite intersection property.
A finite prefilter is necessarily principal, although it does not have to be closed under finite intersections.
If $X$ is finite then all of the conclusions above hold for any $\backslash mathcal\; \backslash subseteq\; \backslash wp(X).$
In particular, on a finite set $X,$ there are no free filter subbases (and so no free prefilters), all prefilters are principal, and all filters on $X$ are principal filters generated by their (non–empty) kernels.
The trivial filter $\backslash $ is always a finite filter on $X$ and if $X$ is infinite then it is the only finite filter because a non–trivial finite filter on a set $X$ is possible if and only if $X$ is finite.
However, on any infinite set there are non–trivial filter subbases and prefilters that are finite (although they cannot be filters).
If $X$ is a singleton set then the trivial filter $\backslash $ is the only proper subset of $\backslash wp(X)$ and moreover, this set $\backslash $ is a principal ultra prefilter and any superset $\backslash mathcal\; \backslash supseteq\; \backslash mathcal$ (where $\backslash mathcal\; \backslash subseteq\; \backslash wp(Y)\; \backslash text\; X\; \backslash subseteq\; Y$) with the finite intersection property will also be a principal ultra prefilter (even if $Y$ is infinite).
Characterizing fixed ultra prefilters

If a family of sets $\backslash mathcal$ is fixed (that is, $\backslash ker\; \backslash mathcal\; \backslash neq\; \backslash varnothing$) then $\backslash mathcal$ is ultra if and only if some element of $\backslash mathcal$ is a singleton set, in which case $\backslash mathcal$ will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter $\backslash mathcal$ is ultra if and only if $\backslash ker\; \backslash mathcal$ is a singleton set. Every filter on $X$ that is principal at a single point is an ultrafilter, and if in addition $X$ is finite, then there are no ultrafilters on $X$ 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 $\backslash ,\backslash leq\backslash ,$ 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 "$\backslash mathcal\; \backslash geq\; \backslash mathcal$" can be interpreted as "$\backslash mathcal$ is a subsequence of $\backslash mathcal$" (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 $\backslash mathcal$ meshes with $\backslash mathcal,$ which is closely related to the preorder $\backslash ,\backslash leq,$ is used in Topology to define cluster points. Two families of sets $\backslash mathcal\; \backslash text\; \backslash mathcal$ and are , indicated by writing $\backslash mathcal\; \backslash \#\; \backslash mathcal,$ if $B\; \backslash cap\; C\; \backslash neq\; \backslash varnothing\; \backslash text\; B\; \backslash in\; \backslash mathcal\; \backslash text\; C\; \backslash in\; \backslash mathcal.$ If $\backslash mathcal\; \backslash text\; \backslash mathcal$ do not mesh then they are . If $S\; \backslash subseteq\; X\; \backslash text\; \backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ then $\backslash mathcal\; \backslash text\; S$ are said to if $\backslash mathcal\; \backslash text\; \backslash $ mesh, or equivalently, if the of $\backslash mathcal\; \backslash text\; S,$ which is the family $$\backslash mathcal\backslash big\backslash vert\_S\; =\; \backslash ,$$ does not contain the empty set, where the trace is also called the of $\backslash mathcal\; \backslash text\; S.$ ''Example'': If $x\_\; =\; \backslash left(x\_\backslash right)\_^\backslash infty$ is asubsequence
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 ...

of $x\_\; =\; \backslash left(x\_i\backslash right)\_^\backslash infty$ then $\backslash operatorname\backslash left(x\_\backslash right)$ is subordinate to $\backslash operatorname\backslash left(x\_\backslash right);$ in symbols: $\backslash operatorname\backslash left(x\_\backslash right)\; \backslash vdash\; \backslash operatorname\backslash left(x\_\backslash right)$ and also $\backslash operatorname\backslash left(x\_\backslash right)\; \backslash leq\; \backslash operatorname\backslash left(x\_\backslash right).$
Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence.
To see this, let $C\; :=\; x\_\; \backslash in\; \backslash operatorname\backslash left(x\_\backslash right)$ be arbitrary (or equivalently, let $i\; \backslash in\; \backslash N$ be arbitrary) and it remains to show that this set contains some $F\; :=\; x\_\; \backslash in\; \backslash operatorname\backslash left(x\_\backslash right).$
For the set $x\_\; =\; \backslash left\backslash $ to contain $x\_\; =\; \backslash left\backslash ,$ it is sufficient to have $i\; \backslash leq\; i\_n.$
Since $i\_1\; <\; i\_2\; <\; \backslash cdots$ are strictly increasing integers, there exists $n\; \backslash in\; \backslash N$ such that $i\_n\; \backslash geq\; i,$ and so $x\_\; \backslash supseteq\; x\_$ holds, as desired.
Consequently, $\backslash operatorname\backslash left(x\_\backslash right)\; \backslash subseteq\; \backslash operatorname\backslash left(x\_\backslash right).$
The left hand side will be a subset of the right hand side if (for instance) every point of $x\_$ is unique (that is, when $x\_\; :\; \backslash N\; \backslash to\; X$ is injective) and $x\_$ is the even-indexed subsequence $\backslash left(x\_2,\; x\_4,\; x\_6,\; \backslash ldots\backslash right)$ because under these conditions, every tail $x\_\; =\; \backslash left\backslash $ (for every $n\; \backslash in\; \backslash N$) of the subsequence will belong to the right hand side filter but not to the left hand side filter.
For another example, if $\backslash mathcal$ is any family then $\backslash varnothing\; \backslash leq\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash leq\; \backslash $ always holds and furthermore, $\backslash \; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash varnothing\; \backslash in\; \backslash mathcal.$
Assume that $\backslash mathcal\; \backslash text\; \backslash mathcal$ are families of sets that satisfy $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$ Then $\backslash ker\; \backslash mathcal\; \backslash subseteq\; \backslash ker\; \backslash mathcal,$ and $\backslash mathcal\; \backslash neq\; \backslash varnothing\; \backslash text\; \backslash mathcal\; \backslash neq\; \backslash varnothing,$ and also $\backslash varnothing\; \backslash in\; \backslash mathcal\; \backslash text\; \backslash varnothing\; \backslash in\; \backslash mathcal.$
If in addition to $\backslash mathcal\; \backslash leq\; \backslash mathcal,\; \backslash mathcal$ is a filter base and $\backslash mathcal\; \backslash neq\; \backslash varnothing,$ then $\backslash mathcal$ is a filter subbase and also $\backslash mathcal\; \backslash text\; \backslash mathcal$ mesh.To prove that $\backslash mathcal\; \backslash text\; \backslash mathcal$ mesh, let $B\; \backslash in\; \backslash mathcal\; \backslash text\; C\; \backslash in\; \backslash mathcal.$ Because $\backslash mathcal\; \backslash leq\; \backslash mathcal$ (resp. because $\backslash mathcal\; \backslash leq\; \backslash mathcal$), there exists some $F,\; G\; \backslash in\; \backslash mathcal\; \backslash text\; F\; \backslash subseteq\; B\; \backslash text\; G\; \backslash subseteq\; C$ where by assumption $F\; \backslash cap\; G\; \backslash neq\; \backslash varnothing$ so $\backslash varnothing\; \backslash neq\; G\; \backslash cap\; F\; \backslash subseteq\; B\; \backslash cap\; C.\; \backslash blacksquare$ If $\backslash mathcal$ is a filter subbase and if $\backslash varnothing\; \backslash neq\; \backslash mathcal\; \backslash leq\; \backslash mathcal,$ then taking $\backslash mathcal\; :=\; \backslash mathcal$ implies that $\backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash text\; \backslash blacksquare$ If $C\_1,\; \backslash ldots,\; C\_n\; \backslash in\; \backslash mathcal$ then there are $F\_1,\; \backslash ldots,\; F\_n\; \backslash in\; \backslash mathcal$ such that $F\_i\; \backslash subseteq\; C\_i$ and now $\backslash varnothing\; \backslash neq\; F\_1\; \backslash cap\; \backslash cdots\; F\_n\; \backslash subseteq\; C\_1\; \backslash cap\; \backslash cdots\; C\_n.$ This shows that $\backslash mathcal$ is a filter subbase. $\backslash blacksquare$
More generally, if both $\backslash varnothing\; \backslash neq\; \backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash varnothing\; \backslash neq\; \backslash mathcal\; \backslash leq\; \backslash mathcal$ and if the intersection of any two elements of $\backslash mathcal$ is non–empty, then $\backslash mathcal\; \backslash text\; \backslash mathcal$ mesh.
Every filter subbase is coarser than both the –system that it generates and the filter that it generates.
If $\backslash mathcal\; \backslash text\; \backslash mathcal$ are families such that $\backslash mathcal\; \backslash leq\; \backslash mathcal,$ the family $\backslash mathcal$ is ultra, and $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal,$ then $\backslash mathcal$ is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily ultra. In particular, if $\backslash mathcal$ is a prefilter then either both $\backslash mathcal$ and the filter $\backslash mathcal^$ 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 $\backslash mathcal$ that is not a prefilter cannot be ultra; but it is nevertheless still possible for the prefilter and filter generated by $\backslash mathcal$ to be ultra. If $S\; \backslash subseteq\; X\; \backslash text\; \backslash mathcal\; \backslash subseteq\; \backslash wp(X)$ is upward closed in $X$ then $S\; \backslash not\backslash in\; \backslash mathcal\; \backslash text\; (X\; \backslash setminus\; S)\; \backslash \#\; \backslash mathcal.$
Relational properties of subordination
The relation $\backslash ,\backslash leq\backslash ,$ is reflexive and transitive, which makes it into a preorder on $\backslash wp(\backslash wp(X)).$
The relation $\backslash ,\backslash leq\backslash ,\; \backslash text\; \backslash operatorname(X)$ is antisymmetric but if $X$ has more than one point then it is symmetric.
:
For any $\backslash mathcal\; \backslash subseteq\; \backslash wp(X),\; \backslash mathcal\; \backslash leq\; \backslash \; \backslash text\; \backslash \; =\; \backslash mathcal.$
So the set $X$ has more than one point if and only if the relation $\backslash ,\backslash leq\backslash ,\; \backslash text\; \backslash operatorname(X)$ is symmetric.
:
If $\backslash mathcal\; \backslash subseteq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal$ but while the converse does not hold in general, it does hold if $\backslash mathcal$ is upward closed (such as if $\backslash mathcal$ is a filter).
Two filters are equivalent if and only if they are equal, which makes the restriction of $\backslash ,\backslash leq\backslash ,$ to $\backslash operatorname(X)$ antisymmetric.
But in general, $\backslash ,\backslash leq\backslash ,$ is antisymmetric on $\backslash operatorname(X)$ nor on $\backslash wp(\backslash wp(X))$; that is, $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal$ does necessarily imply $\backslash mathcal\; =\; \backslash mathcal$; not even if both $\backslash mathcal\; \backslash text\; \backslash mathcal$ are prefilters. For instance, if $\backslash mathcal$ is a prefilter but not a filter then $\backslash mathcal\; \backslash leq\; \backslash mathcal^\; \backslash text\; \backslash mathcal^\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash neq\; \backslash mathcal^.$
Equivalent families of sets

The preorder $\backslash ,\backslash leq\backslash ,$ induces its canonicalequivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...

on $\backslash wp(\backslash wp(X)),$ where for all $\backslash mathcal,\; \backslash mathcal\; \backslash in\; \backslash wp(\backslash wp(X)),$ $\backslash mathcal$ is to $\backslash mathcal$ if any of the following equivalent conditions hold:
- $\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash text\; \backslash mathcal\; \backslash leq\; \backslash mathcal.$
- The upward closures of $\backslash mathcal\; \backslash text\; \backslash mathcal$ are equal.

equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...

other than $\backslash $ contains a unique representative (that is, element of the equivalence class) that is upward closed in $X.$
Properties preserved between equivalent families
Let $\backslash mathcal,\; \backslash mathcal\; \backslash in\; \backslash wp(\backslash wp(X))$ be arbitrary and let $\backslash mathcal$ be any family of sets. If $\backslash mathcal\; \backslash text\; \backslash mathcal$ are equivalent (which implies that $\backslash ker\; \backslash mathcal\; =\; \backslash ker\; \backslash mathcal$) then for each of the statements/properties listed below, either it is true of $\backslash mathcal\; \backslash text\; \backslash mathcal$ or else it is false of $\backslash mathcal\; \backslash text\; \backslash mathcal$:
- Not empty
- Proper (that is, $\backslash varnothing$ is not an element) * Moreover, any two degenerate families are necessarily equivalent.
- Filter subbase
- Prefilter * In which case $\backslash mathcal\; \backslash text\; \backslash mathcal$ generate the same filter on $X$ (that is, their upward closures in $X$ are equal).
- Free
- Principal
- Ultra
- Is equal to the trivial filter $\backslash $ * In words, this means that the only subset of $\backslash wp(X)$ 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 $\backslash mathcal$
- Is finer than $\backslash mathcal$
- Is coarser than $\backslash mathcal$
- Is equivalent to $\backslash mathcal$

- $\backslash mathcal$;
- the –system generated by $\backslash mathcal$;
- the filter on $X$ generated by $\backslash mathcal$;

Set theoretic properties and constructions

Trace and meshing

If $\backslash mathcal$ is a prefilter (resp. filter) on $X\; \backslash text\; S\; \backslash subseteq\; X$ then the trace of $\backslash mathcal\; \backslash text\; S,$ which is the family $\backslash mathcal\backslash big\backslash vert\_S\; :=\; \backslash mathcal\; (\backslash cap)\; \backslash ,$ is a prefilter (resp. a filter) if and only if $\backslash mathcal\; \backslash text\; S$ mesh (that is, $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\; (\backslash cap)\; \backslash $), in which case the trace of $\backslash mathcal\; \backslash text\; S$ is said to be . If $\backslash mathcal$ is ultra and if $\backslash mathcal\; \backslash text\; S$ mesh then the trace $\backslash mathcal\backslash big\backslash vert\_S$ is ultra. If $\backslash mathcal$ is an ultrafilter on $X$ then the trace of $\backslash mathcal\; \backslash text\; S$ is a filter on $S$ if and only if $S\; \backslash in\; \backslash mathcal.$ For example, suppose that $\backslash mathcal$ is a filter on $X\; \backslash text\; S\; \backslash subseteq\; X$ is such that $S\; \backslash neq\; X\; \backslash text\; X\; \backslash setminus\; S\; \backslash not\backslash in\; \backslash mathcal.$ Then $\backslash mathcal\; \backslash text\; S$ mesh and $\backslash mathcal\; \backslash cup\; \backslash $ generates a filter on $X$ that is strictly finer than $\backslash mathcal.$ When prefilters mesh Given non–empty families $\backslash mathcal\; \backslash text\; \backslash mathcal,$ the family $$\backslash mathcal\; (\backslash cap)\; \backslash mathcal\; :=\; \backslash $$ satisfies $\backslash mathcal\; \backslash leq\; \backslash mathcal\; (\backslash cap)\; \backslash mathcal$ and $\backslash mathcal\; \backslash leq\; \backslash mathcal\; (\backslash cap)\; \backslash mathcal.$ If $\backslash mathcal\; (\backslash cap)\; \backslash mathcal$ is proper (resp. a prefilter, a filter subbase) then this is also true of both $\backslash mathcal\; \backslash text\; \backslash mathcal.$ In order to make any meaningful deductions about $\backslash mathcal\; (\backslash cap)\; \backslash mathcal$ from $\backslash mathcal\; \backslash text\; \backslash mathcal,\; \backslash mathcal\; (\backslash cap)\; \backslash mathcal$ needs to be proper (that is, $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\; (\backslash cap)\; \backslash mathcal,$ which is the motivation for the definition of "mesh". In this case, $\backslash mathcal\; (\backslash cap)\; \backslash mathcal$ is a prefilter (resp. filter subbase) if and only if this is true of both $\backslash mathcal\; \backslash text\; \backslash mathcal.$ Said differently, if $\backslash mathcal\; \backslash text\; \backslash mathcal$ are prefilters then they mesh if and only if $\backslash mathcal\; (\backslash cap)\; \backslash mathcal$ is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, $\backslash ,\backslash leq\backslash ,$): Two prefilters (resp. filter subbases) $\backslash mathcal\; \backslash text\; \backslash mathcal$ mesh if and only if there exists a prefilter (resp. filter subbase) $\backslash mathcal$ such that $\backslash mathcal\; \backslash leq\; \backslash mathcal$ and $\backslash mathcal\; \backslash leq\; \backslash mathcal.$ If the least upper bound of two filters $\backslash mathcal\; \backslash text\; \backslash mathcal$ exists in $\backslash operatorname(X)$ then this least upper bound is equal to $\backslash mathcal\; (\backslash cap)\; \backslash mathcal.$Images and preimages under functions

Throughout, $f\; :\; X\; \backslash to\; Y\; \backslash text\; g\; :\; Y\; \backslash to\; Z$ will be maps between non–empty sets. Images of prefilters Let $\backslash mathcal\; \backslash subseteq\; \backslash wp(Y).$ Many of the properties that $\backslash mathcal$ 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 $\backslash mathcal\; \backslash text\; Y,$ then it will necessarily also be true of $g(\backslash mathcal)\; \backslash text\; g(Y)$ (although possibly not on the codomain $Z$ unless $g$ is surjective):- Filter properties: ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate.
- Ideal properties: ideal, closed under finite unions, downward closed, directed upward.

- $f^(\backslash mathcal)$ is a prefilter;
- $\backslash mathcal\backslash big\backslash vert\_$ is a prefilter;
- $\backslash varnothing\; \backslash not\backslash in\; \backslash mathcal\backslash big\backslash vert\_$;
- $\backslash mathcal$ meshes with $f(X)$

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 ...

if and only if whenever $\backslash mathcal$ is a prefilter on $Y$ then the same is true of $f^(\backslash mathcal)\; \backslash text\; X$ (this result does not require the ultrafilter lemma).
Subordination is preserved by images and preimages

The relation $\backslash ,\backslash leq\backslash ,$ is preserved under both images and preimages of families of sets. This means that for families $\backslash mathcal\; \backslash text\; \backslash mathcal,$ $$\backslash mathcal\; \backslash leq\; \backslash mathcal\; \backslash quad\; \backslash text\; \backslash quad\; g(\backslash mathcal)\; \backslash leq\; g(\backslash mathcal)\; \backslash quad\; \backslash text\; \backslash quad\; f^(\backslash mathcal)\; \backslash leq\; f^(\backslash mathcal).$$ Moreover, the following relations always hold for family of sets $\backslash mathcal$: $$\backslash mathcal\; \backslash leq\; f\backslash left(f^(\backslash mathcal)\backslash right)$$ where equality will hold if $f$ is surjective. Furthermore, $$f^(\backslash mathcal)\; =\; f^\backslash left(f\backslash left(f^(\backslash mathcal)\backslash right)\backslash right)\; \backslash quad\; \backslash text\; \backslash quad\; g(\backslash mathcal)\; =\; g\backslash left(g^(g(\backslash mathcal))\backslash right).$$ If $\backslash mathcal\; \backslash subseteq\; \backslash wp(X)\; \backslash text\; \backslash mathcal\; \backslash subseteq\; \backslash wp(Y)$ then $$f(\backslash mathcal)\; \backslash leq\; \backslash mathcal\; \backslash quad\; \backslash text\; \backslash quad\; \backslash mathcal\; \backslash leq\; f^(\backslash mathcal)$$ and $g^(g(\backslash mathcal))\; \backslash leq\; \backslash mathcal$ where equality will hold if $g$ is injective.Products of prefilters

Suppose $X\_\; =\; \backslash left(X\_i\backslash right)\_$ is a family of one or more non–empty sets, whose product will be denoted by $\backslash prod\; X\_\; :=\; \backslash prod\_\; X\_i,$ and for every index $i\; \backslash in\; I,$ let $$\backslash Pr\_\; :\; \backslash prod\; X\_\; \backslash to\; X\_i$$ denote the canonical projection. Let $\backslash mathcal\_\; :=\; \backslash left(\backslash mathcal\_i\backslash right)\_$ be non−empty families, also indexed by $I,$ such that $\backslash mathcal\_i\; \backslash subseteq\; \backslash wp\backslash left(X\_i\backslash right)$ for each $i\; \backslash in\; I.$ The of the families $\backslash mathcal\_$ is defined identically to how the basic open subsets of theproduct 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-seemi ...

are defined (had all of these $\backslash mathcal\_i$ been topologies). That is, both the notations
$$\backslash prod\_\; \backslash mathcal\_\; =\; \backslash prod\_\; \backslash mathcal\_i$$
denote the family of all subsets $\backslash prod\_\; S\_i\; \backslash subseteq\; \backslash prod\_\; X\_$ such that $S\_i\; =\; X\_i$ for all but finitely many $i\; \backslash in\; I$ and where $S\_i\; \backslash in\; \backslash mathcal\_i$ for any one of these finitely many exceptions (that is, for any $i$ such that $S\_i\; \backslash neq\; X\_i,$ necessarily $S\_i\; \backslash in\; \backslash mathcal\_i$).
When every $\backslash mathcal\_i$ is a filter subbase then the family $\backslash bigcup\_\; \backslash Pr\_^\; \backslash left(\backslash mathcal\_i\backslash right)$ is a filter subbase for the filter on $\backslash prod\; X\_$ generated by $\backslash mathcal\_.$
If $\backslash prod\; \backslash mathcal\_$ is a filter subbase then the filter on $\backslash prod\; X\_$ that it generates is called the .
If every $\backslash mathcal\_i$ is a prefilter on $X\_i$ then $\backslash prod\; \backslash mathcal\_$ will be a prefilter on $\backslash prod\; X\_$ and moreover, this prefilter is equal to the coarsest prefilter $\backslash mathcal\; \backslash text\; \backslash prod\; X\_$ such that
$\backslash Pr\_\; (\backslash mathcal)\; =\; \backslash mathcal\_i$
for every $i\; \backslash in\; I.$
However, $\backslash prod\; \backslash mathcal\_$ may fail to be a filter on $\backslash prod\; X\_$ even if every $\backslash mathcal\_i$ is a filter on $X\_i.$
Set subtraction and some examples

Set subtracting away a subset of the kernel If $\backslash mathcal$ is a prefilter on $X,\; S\; \backslash subseteq\; \backslash ker\; \backslash mathcal,\; \backslash text\; S\; \backslash not\backslash in\; \backslash mathcal$ then $\backslash $ is a prefilter, where this latter set is a filter if and only if $\backslash mathcal$ is a filter and $S\; =\; \backslash varnothing.$ In particular, if $\backslash mathcal$ is a neighborhood basis at a point $x$ in a topological space $X$ having at least 2 points, then $\backslash $ is a prefilter on $X.$ This construction is used to define $\backslash lim\_\; f(x)\; \backslash to\; y$ 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 $B\; \backslash in\; \backslash mathcal$ some $C\; \backslash in\; \backslash mathcal.$ Explicitly, this means that for every $B\; \backslash in\; \backslash mathcal$ , there is some $C\; \backslash in\; \backslash mathcal$ such that $B\; \backslash subseteq\; C.$ This relation is dual to $\backslash ,\backslash leq\backslash ,$ in sense that $\backslash mathcal\; \backslash vartriangleleft\; \backslash mathcal$ if and only if $(X\; \backslash setminus\; \backslash mathcal)\; \backslash leq\; (X\; \backslash setminus\; \backslash mathcal).$ The relation $\backslash mathcal\; \backslash vartriangleleft\; \backslash mathcal$ is closely related to the downward closure of a family in a manner similar to how $\backslash ,\backslash leq\backslash ,$ is related to the upward closure family. For an example that uses this duality, suppose $f\; :\; X\; \backslash to\; Y$ is a map and $\backslash Xi\; \backslash subseteq\; \backslash wp(Y).$ Define $$\backslash Xi\_f\; :=\; \backslash $$ which contains the empty set if and only if $\backslash Xi$ does. It is possible for $\backslash Xi$ to be an ultrafilter and for $\backslash Xi\_f$ to be empty or not closed under finite intersections (see footnote for example).Suppose $X$ has more than one point, $f\; :\; X\; \backslash to\; Y$ is a constant map, and $\backslash Xi\; =\; \backslash $ then $\backslash Xi\_f$ will consist of all non–empty subsets of $Y.$ Although $\backslash Xi\_f$ does not preserve properties of filters very well, if $\backslash Xi$ is downward closed (resp. closed under finite unions, an ideal) then this will also be true for $\backslash Xi\_f.$ Using the duality between ideals and dual ideals allows for a construction of the following filter. Suppose $\backslash mathcal$ is a filter on $Y$ and let $\backslash Xi\; :=\; Y\; \backslash setminus\; \backslash mathcal$ be its dual in $Y.$ If $X\; \backslash not\backslash in\; \backslash Xi\_f$ then $\backslash Xi\_f$'s dual $X\; \backslash setminus\; \backslash Xi\_f$ will be a filter. Other examples Example: The set $\backslash mathcal$ of all dense open subsets of a topological space is a proper –system and a prefilter. If the space is aBaire 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 $\backslash mathcal.$
Example: The family $\backslash mathcal\_$ of all dense open sets of $X\; =\; \backslash R$ having finite Lebesgue measure is a proper –system and a free prefilter. The prefilter $\backslash mathcal\_$ is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of $\backslash R.$ Since $X$ is a 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 called ...

and non–meager) so the set of all countable intersections of elements of $\backslash mathcal\_$ is a prefilter and –system; it is also finer than, and not equivalent to, $\backslash mathcal\_.$
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 $x\_\; =\; \backslash left(x\_i\backslash right)\_\; \backslash text\; X$ is canonically associated with its prefilter of tails $\backslash operatorname\backslash left(x\_\backslash right).$ If $f\; :\; X\; \backslash to\; Y$ is a map and $x\_$ is a net in $X$ then $\backslash operatorname\backslash left(f\backslash left(x\_\backslash right)\backslash right)\; =\; f\backslash left(\backslash operatorname\backslash left(x\_\backslash right)\backslash right).$Prefilters to nets

A is a pair $(S,\; s)$ consisting of a non–empty set $S$ and an element $s\; \backslash in\; S.$ For any family $\backslash mathcal,$ let $$\backslash operatorname(\backslash mathcal)\; :=\; \backslash left\backslash .$$ Define a canonical preorder $\backslash ,\backslash leq\backslash ,$ on pointed sets by declaring $$(R,\; r)\; \backslash leq\; (S,\; s)\; \backslash quad\; \backslash text\; \backslash quad\; R\; \backslash supseteq\; S.$$ If $s\_0,\; s\_1\; \backslash in\; S\; \backslash text\; \backslash left(S,\; s\_0\backslash right)\; \backslash leq\; \backslash left(S,\; s\_1\backslash right)\; \backslash text\; \backslash left(S,\; s\_1\backslash right)\; \backslash leq\; \backslash left(S,\; s\_0\backslash right)$ even if $s\_0\; \backslash neq\; s\_1,$ so this preorder is not antisymmetric and given any family of sets $\backslash mathcal,$ $(\backslash operatorname(\backslash mathcal),\; \backslash leq)$ ispartially 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 ...

if and only if $\backslash mathcal\; \backslash neq\; \backslash varnothing$ consists entirely of singleton sets.
If $\backslash \; \backslash in\; \backslash mathcal\; \backslash text\; (\backslash ,\; x)$ 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 defi ...

of $\backslash operatorname(\backslash mathcal)$; moreover, all maximal elements are of this form.
If $\backslash left(B,\; b\_0\backslash right)\; \backslash in\; \backslash operatorname(\backslash mathcal)\; \backslash text\; \backslash left(B,\; b\_0\backslash right)$ 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 ...

if and only if $B\; =\; \backslash ker\; \backslash mathcal,$ in which case $\backslash $ is the set of all greatest elements. However, a greatest element $(B,\; b)$ is a maximal element if and only if $B\; =\; \backslash \; =\; \backslash ker\; \backslash mathcal,$ so there is at most one element that is both maximal and greatest.
There is a canonical map $\backslash operatorname\_\; ~:~\; \backslash operatorname(\backslash mathcal)\; \backslash to\; X$ defined by $(B,\; b)\; \backslash mapsto\; b.$
If $i\_0\; =\; \backslash left(B\_0,\; b\_0\backslash right)\; \backslash in\; \backslash operatorname(\backslash mathcal)$ then the tail of the assignment $\backslash operatorname\_$ starting at $i\_0$ is $\backslash left\backslash \; =\; B\_0.$
Although $(\backslash operatorname(\backslash mathcal),\; \backslash leq)$ 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) $\backslash mathcal$ is a prefilter.
So the most immediate choice for the definition of "the net in $X$ induced by a prefilter $\backslash mathcal$" is the assignment $(B,\; b)\; \backslash mapsto\; b$ from $\backslash operatorname(\backslash mathcal)$ into $X.$
:\;&& (\operatorname(\mathcal), \leq) &&\,\to \;& X \\
&& (B, b) &&\,\mapsto\;& b \\
\end
that is, $\backslash operatorname\_(B,\; b)\; :=\; b.$
If $\backslash mathcal$ is a prefilter on $X\; \backslash text\; \backslash operatorname\_$ is a net in $X$ and the prefilter associated with $\backslash operatorname\_$ is $\backslash mathcal$; that is:The set equality $\backslash operatorname\backslash left(\backslash operatorname\_\backslash right)\; =\; \backslash mathcal$ holds more generally: if the family of sets $\backslash mathcal\; \backslash neq\; \backslash varnothing\; \backslash text\; \backslash varnothing\; \backslash not\backslash in\; \backslash mathcal$ then the family of tails of the map $\backslash operatorname(\backslash mathcal)\; \backslash to\; X$ (defined by $(B,\; b)\; \backslash mapsto\; b$) is equal to $\backslash mathcal.$
$$\backslash operatorname\backslash left(\backslash operatorname\_\backslash right)\; =\; \backslash mathcal.$$
This would not necessarily be true had $\backslash operatorname\_$ been defined on a proper subset of $\backslash operatorname(\backslash mathcal).$
For example, suppose $X$ has at least two distinct elements, $\backslash mathcal\; :=\; \backslash $ is the indiscrete filter, and $x\; \backslash in\; X$ is arbitrary. Had $\backslash operatorname\_$ instead been defined on the singleton set $D\; :=\; \backslash ,$ where the restriction of $\backslash operatorname\_$ to $D$ will temporarily be denote by $\backslash operatorname\_D\; :\; D\; \backslash to\; X,$ then the prefilter of tails associated with $\backslash operatorname\_D\; :\; D\; \backslash to\; X$ would be the principal prefilter $\backslash $ rather than the original filter $\backslash mathcal\; =\; \backslash $;
this means that the equality $\backslash operatorname\backslash left(\backslash operatorname\_D\backslash right)\; =\; \backslash mathcal$ is , so unlike $\backslash operatorname\_,$ the prefilter $\backslash mathcal$ can be recovered from $\backslash operatorname\_D.$
Worse still, while $\backslash mathcal$ is the unique filter on $X,$ the prefilter $\backslash operatorname\backslash left(\backslash operatorname\_D\backslash right)\; =\; \backslash $ instead generates a filter (that is, an ultrafilter) on $X.$
However, if $x\_\; =\; \backslash left(x\_i\backslash right)\_$ is a net in $X$ then it is in general true that $\backslash operatorname\_$ is equal to $x\_$ because, for example, the domain of $x\_$ may be of a completely different cardinality than that of $\backslash operatorname\_$ (since unlike the domain of $\backslash operatorname\_,$ the domain of an arbitrary net in $X$ could have cardinality).
Ultranets and ultra prefilters
A net $x\_\; \backslash text\; X$ is called an or in $X$ if for every subset $S\; \backslash subseteq\; X,\; x\_$ is eventually in $S$ or it is eventually in $X\; \backslash setminus\; S$;
this happens if and only if $\backslash operatorname\backslash left(x\_\backslash right)$ is an ultra prefilter.
A prefilter $\backslash mathcal\; \backslash text\; X$ is an ultra prefilter if and only if $\backslash operatorname\_$ is an ultranet in $X.$
Partially ordered net

The domain of the canonical net $\backslash operatorname\_$ 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 in 1970. It begins with the construction of astrict 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 ...

(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 t ...

) $\backslash ,<\backslash ,$ on a subset of $\backslash mathcal\; \backslash times\; \backslash N\; \backslash times\; X$ 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 $\backslash mathcal\; \backslash times\; \backslash N$ of the strict partial orders $(\backslash mathcal,\; \backslash supsetneq)\; \backslash text\; (\backslash N,\; <).$
For any $i\; =\; (B,\; m,\; b)\; \backslash text\; j\; =\; (C,\; n,\; c)$ in $\backslash mathcal\; \backslash times\; \backslash N\; \backslash times\; X,$ declare that $i\; <\; j$ if and only if
$$B\; \backslash supseteq\; C\; \backslash text\; \backslash text\; B\; \backslash neq\; C\; \backslash text\; B\; =\; C\; \backslash text\; m\; <\; n,$$
or equivalently, if and only if $\backslash text\; B\; \backslash supseteq\; C,\; \backslash text\; B\; =\; C\; \backslash text\; m\; <\; n.$
The non−strict partial order associated with $\backslash ,<,$ denoted by $\backslash ,\backslash leq,$ is defined by declaring that $i\; \backslash leq\; j\backslash ,\; \backslash text\; i\; <\; j\; \backslash text\; i\; =\; j.$
Unwinding these definitions gives the following characterization:
which shows that $\backslash ,\backslash leq\backslash ,$ 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 $\backslash mathcal\; \backslash times\; \backslash N\; \backslash times\; X$ induced by $(\backslash mathcal,\; \backslash supseteq),\; \backslash ,(\backslash N,\; \backslash leq),\; \backslash text\; (X,\; =),$ where $X$ is partially ordered by equality $\backslash ,=.\backslash ,$Explicitly, the partial order on $X$ induced by equality $\backslash ,=\backslash ,$ refers to the diagonal $\backslash Delta\; :=\; \backslash ,$ 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 $X$ that makes $(X,\; \backslash Delta)$ 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 binary ...

. If this partial order $\backslash Delta$ is denoted by the more familiar symbol $\backslash ,\backslash leq\backslash ,$ (that is, define $\backslash leq\; \backslash ;:=\backslash ;\; \backslash Delta$) then for any $b,\; c\; \backslash in\; X,$ $\backslash ;b\; \backslash leq\; c\backslash ,\; \backslash text\; \backslash ,b\; =\; c,$ which shows that $\backslash ,\backslash leq\backslash ,$ (and thus also $\backslash Delta$) is nothing more than a new symbol for equality on $X;$ that is, $(X,\; \backslash Delta)\; \backslash \; =\backslash \; (X,\; =).$ The notation $(X,\; =)$ is used because it avoids the unnecessary introduction of a new symbol for the diagonal.
Both $\backslash ,<\; \backslash text\; \backslash leq\backslash ,$ 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 elem ...

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 defi ...

; this remains true if they are each restricted to the subset of $\backslash mathcal\; \backslash times\; \backslash N\; \backslash times\; X$ defined by
$$\backslash begin\; \backslash operatorname\_\; \backslash ;\&:=\backslash ;\; \backslash ,\; \backslash \backslash \; \backslash end$$
where it will henceforth be assumed that they are.
Denote the assignment $i\; =\; (B,\; m,\; b)\; \backslash mapsto\; b$ from this subset by:
$$\backslash begin\; \backslash operatorname\_\backslash \; :\backslash \; \&\&\backslash \; \backslash operatorname\_\backslash \; \&\&\backslash ,\backslash to\; \backslash ;\&\; X\; \backslash \backslash ;\; href="/html/ALL/l/.5ex.html"\; ;"title=".5ex">.5ex$$
If $i\_0\; =\; \backslash left(B\_0,\; m\_0,\; b\_0\backslash right)\; \backslash in\; \backslash operatorname\_$ then just as with $\backslash operatorname\_$ before, the tail of the $\backslash operatorname\_$ starting at $i\_0$ is equal to $B\_0.$
If $\backslash mathcal$ is a prefilter on $X$ then $\backslash operatorname\_$ is a net in $X$ whose domain $\backslash operatorname\_$ is a partially ordered set and moreover, $\backslash operatorname\backslash left(\backslash operatorname\_\backslash right)\; =\; \backslash mathcal.$
Because the tails of $\backslash operatorname\_\; \backslash text\; \backslash operatorname\_$ are identical (since both are equal to the prefilter $\backslash mathcal$), 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 $\backslash N$ is replaced with the positive rational numbers then the strict partial order $<$ will also be a dense order.
Subordinate filters and subnets

The notion of "$\backslash mathcal$ is subordinate to $\backslash mathcal$" (written $\backslash mathcal\; \backslash vdash\; \backslash mathcal$) is for filters and prefilters what "$x\_\; =\; \backslash left(x\_\backslash right)\_^$ is asubsequence
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 ...

of $x\_\; =\; \backslash left(x\_i\backslash right)\_^$" is for sequences.
For example, if $\backslash operatorname\backslash left(x\_\backslash right)\; =\; \backslash left\backslash $ denotes the set of tails of $x\_$ and if $\backslash operatorname\backslash left(x\_\backslash right)\; =\; \backslash left\backslash $ denotes the set of tails of the subsequence $x\_$ (where $x\_\; :=\; \backslash left\backslash $) then $\backslash operatorname\backslash left(x\_\backslash right)\; ~\backslash vdash~\; \backslash operatorname\backslash left(x\_\backslash right)$ (that is, $\backslash operatorname\backslash left(x\_\backslash right)\; \backslash leq\; \backslash operatorname\backslash left(x\_\backslash right)$) is true but $\backslash operatorname\backslash left(x\_\backslash right)\; ~\backslash vdash~\; \backslash operatorname\backslash left(x\_\backslash right)$ is in general false.
Non–equivalence of subnets and subordinate filters

A subset $R\; \backslash subseteq\; I$ of a preordered space $(I,\; \backslash leq)$ is or in $I$ if for every $i\; \backslash in\; I$ there exists some $r\; \backslash in\; R\; \backslash text\; i\; \backslash leq\; r.$ If $R\; \backslash subseteq\; I$ contains a tail of $I$ then $R$ is said to be or ; explicitly, this means that there exists some $i\; \backslash in\; I\; \backslash text\; I\_\; \backslash subseteq\; R$ (that is, $j\; \backslash in\; R\; \backslash text\; j\; \backslash in\; I\; \backslash text\; i\; \backslash leq\; j$). 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 $h\; :\; A\; \backslash to\; I$ between two preordered sets is if whenever $a,\; b\; \backslash in\; A\; \backslash text\; a\; \backslash leq\; b,\; \backslash text\; h(a)\; \backslash leq\; h(b).$ 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 identica ...

."
The first definition of a subnet was introduced by John L. Kelley
John L. Kelley (December 6, 1916, Kansas – November 26, 1999, Berkeley, California) was an American mathematician at the University of California, Berkeley, who worked in general topology and functional analysis.
Kelley's 1955 text, ''General ...

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 $h$ 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 $X$ − the nets' common codomain.
Every Willard–subnet is a Kelley–subnet and both are AA–subnets.
In particular, if $y\_\; =\; \backslash left(y\_a\backslash right)\_$ is a Willard–subnet or a Kelley–subnet of $x\_\; =\; \backslash left(x\_i\backslash right)\_$ then $\backslash operatorname\backslash left(x\_\backslash right)\; \backslash leq\; \backslash operatorname\backslash left(y\_\backslash right).$
- Example: Let $I\; =\; \backslash N$ and let $x\_$ be a constant sequence, say $x\_\; =\; \backslash left(0\backslash right)\_.$ Let $s\_1\; =\; 0$ and $A\; =\; \backslash $ so that $s\_\; =\; \backslash left(s\_a\backslash right)\_\; =\; \backslash left(s\_1\backslash right)$ is a net on $A.$ Then $s\_$ is an AA-subnet of $x\_$ because $\backslash operatorname\backslash left(x\_\backslash right)\; =\; \backslash \; =\; \backslash operatorname\backslash left(s\_\backslash right).$ But $s\_$ is not a Willard-subnet of $x\_$ because there does not exist any map $h\; :\; A\; \backslash to\; I$ whose image is a cofinal subset of $I\; =\; \backslash N.$ Nor is $s\_$ a Kelley-subnet of $x\_$ because if $h\; :\; A\; \backslash to\; I$ is any map then $E\; :=\; I\; \backslash setminus\; \backslash $ is a cofinal subset of $I\; =\; \backslash N$ but $h^(E)\; =\; \backslash varnothing$ is not eventually in $A.$

- : For all $n\; \backslash in\; \backslash N,$ let $B\_n\; =\; \backslash \; \backslash cup\; \backslash N\_.$ Let $\backslash mathcal\; =\; \backslash ,$ which is a proper –system, and let $\backslash mathcal\; =\; \backslash \; \backslash cup\; \backslash mathcal,$ 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 ...s $X\; :=\; \backslash N\; =\; \backslash .$ Because $\backslash mathcal\; \backslash leq\; \backslash mathcal,\; \backslash mathcal$ is to $\backslash mathcal$ as a subsequence is to a sequence. So ideally, $S\; =\; \backslash operatorname\_$ should be a subnet of $B\; =\; \backslash operatorname\_.$ Let $I\; :=\; \backslash operatorname(\backslash mathcal)$ be the domain of $\backslash operatorname\_,$ so $I$ contains a cofinal subset that is order isomorphic to $\backslash N$ and consequently contains neither a maximal nor greatest element. Let $A\; :=\; \backslash operatorname(\backslash mathcal)\; =\; \backslash \; \backslash cup\; I,\; \backslash text\; M\; :=\; (1,\; \backslash )$ is both a maximal and greatest element of $A.$ The directed set $A$ also contains a subset that is order isomorphic to $\backslash N$ (because it contains $I,$ which contains such a subset) but no such subset can be cofinal in $A$ because of the maximal element $M.$ Consequently, any order–preserving map $h\; :\; A\; \backslash to\; I$ must be eventually constant (with value $h(M)$) where $h(M)$ is then a greatest element of the range $h(A).$ Because of this, there can be no order preserving map $h\; :\; A\; \backslash to\; I$ that satisfies the conditions required for $\backslash operatorname\_$ to be a Willard–subnet of $\backslash operatorname\_$ (because the range of such a map $h$ cannot be cofinal in $I$). Suppose for the sake of contradiction that there exists a map $h\; :\; A\; \backslash to\; I$ such that $h^\backslash left(I\_\backslash right)$ is eventually in $A$ for all $i\; \backslash in\; I.$ Because $h(M)\; \backslash in\; I,$ there exist $n,\; n\_0\; \backslash in\; \backslash N$ such that $h(M)\; =\; \backslash left(n\_0,\; B\_n\backslash right)\; \backslash text\; n\_0\; \backslash in\; B\_n.$ For every $i\; \backslash in\; I,$ because $h^\backslash left(I\_\backslash right)$ is eventually in $A,$ it is necessary that $h(M)\; \backslash in\; I\_.$ In particular, if $i\; :=\; \backslash left(n\; +\; 2,\; B\_\backslash right)$ then $h(M)\; \backslash geq\; i\; =\; \backslash left(n\; +\; 2,\; B\_\backslash right),$ which by definition is equivalent to $B\_n\; \backslash subseteq\; B\_,$ which is false. Consequently, $\backslash operatorname\_$ is not a Kelley–subnet of $\backslash operatorname\_.$

See also

* * * * * * * * * * * * * * *Notes

ProofsCitations

References

* * * * * * * * * * * * * * * * * * * * * * * * (Provides an introductory review of filters in topology and in metric spaces.) * * * * * * * * * {{Set theory General topology Order theory Set theory