In

directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

.
A is a non-empty set $A$ together with a

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, more specifically in general topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. In essence, a sequence is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

whose domain is the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. The codomain
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of this function is usually some topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

.
The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map ''f'' between topological spaces ''X'' and ''Y'':
#The map ''f'' is continuous in the topological sense;
#Given any point ''x'' in ''X'', and any sequence in ''X'' converging to ''x'', the composition of ''f'' with this sequence converges to ''f''(''x'') (continuous in the sequential sense).
While it is necessarily true that condition 1 implies condition 2, the reverse implication is not necessarily true if the topological spaces are not both first-countable
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

. In particular, the two conditions are equivalent for metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s.
The concept of a net, first introduced 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
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (fro ...

and Herman L. Smith in 1922, is to generalize the notion of a sequence so that the above conditions (with "sequence" being replaced by "net" in condition 2) are in fact equivalent for all maps of topological spaces. In particular, rather than being defined on a countable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linearly ordered
Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to Proportionality (mathema ...

set, a net is defined on an arbitrary directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

. This allows for theorems similar to the assertion that the conditions 1 and 2 above are equivalent to hold in the context of topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basisIn topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point is the collection of all Neighbourhood (mathematics), neighbourhoods of the point .
Definit ...

around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s in behaviour. The term "net" was coined by John L. Kelley.
Nets are one of the many tools used in topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

to generalize certain concepts that may only be general enough in the context of metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. A related notion, that of the filter
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

, was developed in 1937 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 French mathematician Élie Cartan and the brother of composer Jean Cartan.
Life
Cartan ...

.
Definitions

Anyfunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

whose domain
Domain may refer to:
Mathematics
*Domain of a function
In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...

is a directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is called a where if this function takes values in some set $X$ then it may also be referred to as a . Elements of a net's domain are called its . Explicitly, a is a function of the form $f\; :\; A\; \backslash to\; X$ where $A$ is some preorder
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, typically automatically assumed to be denoted by $\backslash ,\backslash leq\backslash ,$ (unless indicated otherwise), with the property that it is also () , which means that for any $a,\; b\; \backslash in\; A,$ there exists some $c\; \backslash in\; A$ such that $a\; \backslash leq\; c$ and $b\; \backslash leq\; c.$
In words, this property means that given any two elements (of $A$), there is always some element that is "above" both of them (i.e. that is greater than or equal to each of them); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way.
The natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s $\backslash N$ together with the usual integer comparison $\backslash ,\backslash leq\backslash ,$ preorder form the archetypical
The concept of an archetype (; from Ancient Greek, Greek: + ) appears in areas relating to behavior, History of psychology#Emergence of German experimental psychology, historical psychology, and literary analysis. An ''archetype'' can be:
# a stat ...

example of a directed set. Indeed, a net whose domain is the natural numbers is a sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

because by definition, a sequence in $X$ is just a function from $\backslash N\; =\; \backslash $ into $X.$ It is in this way that nets are generalizations of sequences. Importantly though, unlike the natural numbers, directed sets are required to be total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

s or even partial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

s.
Moreover, directed sets are allowed to have 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 duality (order theory), dually, t ...

s and/or maximal element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s, which is the reason why when using nets, caution is advised when using the induced strict preorder $\backslash ,<\backslash ,$ instead of the original (non-strict) preorder $\backslash ,\backslash leq$; in particular, if a directed set $(A,\; \backslash leq)$ has a greatest element $a\; \backslash in\; A$ then there does exist any $b\; \backslash in\; A$ such that $a\; <\; b$ (in contrast, there exists some $b\; \backslash in\; A$ such that $a\; \backslash leq\; b$).
Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences.
A net in $X$ may be denoted by $\backslash left(x\_a\backslash right)\_,$ where unless there is reason to think otherwise, it should automatically be assumed that the set $A$ is directed and that its associated preorder is denoted by $\backslash ,\backslash leq.$
However, notation for nets varies with some authors using, for instance, angled brackets $\backslash left\backslash langle\; x\_a\; \backslash right\backslash rangle\_$ instead of parentheses.
A net in $X$ may also be written as $x\_\; =\; \backslash left(x\_a\backslash right)\_,$ which expresses the fact that this net $x\_$ is a function $x\_\; :\; A\; \backslash to\; X$ whose value at an element $a$ in its domain is denoted by $x\_a$ instead of the usual parentheses notation $x\_(a)$ that is typically used with functions (this subscript notation being taken from sequences). As in the field of algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

, the filled disk or "bullet" denotes the location where arguments to the net (i.e. elements $a\; \backslash in\; A$ of the net's domain) are placed; it helps emphasize that the net is a function and also reduces the number of indices and other symbols that must be written when referring to it later.
Nets are primarily used in the fields of Analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

and Topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, where they are used to characterize many important topological propertiesIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of sequential space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

s and Fréchet–Urysohn space
In the field of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, g ...

s). Nets are intimately related to filters
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

, which are also often used in topology. Every net may be associated with a filter and every filter may be associated with a net, where the properties of these associated objects are closely tied together (see the article about Filters in topology
In topology, a subfield of mathematics, are special Family of sets, families of subsets of a set X that can be used to study topological spaces and define all basic topological notions such a convergence, Continuous map (topology), continuity, Co ...

for more details). Nets directly generalize sequences and they may often be used very similarly to sequences. Consequently, the learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especially analysts, prefer them over filters. However, filters, and especially ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (poset) ''P'' is a certain subset of ''P,'' namely a maximal filter on ''P'', that is, a proper filter on ''P'' that cannot be enlarged to a bigger pr ...

s, have some important technical advantages over nets that ultimately result in nets being encountered much less often than filters outside of the fields of Analysis and Topology.
A subnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...

is not merely the restriction of a net $f$ to a directed subset of $A;$ see the linked page for a definition.
Examples of nets

Every non-emptytotally ordered set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is directed. Therefore, every function on such a set is a net. In particular, the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net.
Another important example is as follows. Given a point $x$ in a topological space, let $N\_x$ denote the set of all neighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

s containing $x.$ Then $N\_x$ is a directed set, where the direction is given by reverse inclusion, so that $S\; \backslash geq\; T$ if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...

$S$ is contained in $T.$ For $S\; \backslash in\; N\_x,$ let $x\_S$ be a point in $S.$ Then $\backslash left(x\_S\backslash right)$ is a net. As $S$ increases with respect to $\backslash ,\backslash geq,$ the points $x\_S$ in the net are constrained to lie in decreasing neighbourhoods of $x,$ so intuitively speaking, we are led to the idea that $x\_S$ must tend towards $x$ in some sense. We can make this limiting concept precise.
Limits of nets

If $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is a net from a directed set $A$ into $X,$ and if $S$ is a subset of $X,$ then $x\_$ is said to be (or ) if there exists some $a\; \backslash in\; A$ such that for every $b\; \backslash in\; A$ with $b\; \backslash geq\; a,$ the point $x\_b\; \backslash in\; S.$ A point $x\; \backslash in\; X$ is called a or of the net $x\_$ in $X$ if (and only if) :for every openneighborhood
A neighbourhood (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...

$U$ of $x,$ the net $x\_$ is eventually in $U,$
in which case, this net is then also said to and to .
If the net $x\_$ converges in $X$ to a point $x\; \backslash in\; X$ then this fact may be expressed by writing any of the following:
:$\backslash begin\; \&\; x\_\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \backslash ;\; \&\; x\_\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \backslash ;\; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \_a\; \backslash ;\; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash end$
where if the topological space $X$ is clear from context then the words "in $X$" may be omitted.
If $\backslash lim\_\; x\_\; \backslash to\; x$ in $X$ and if this limit in $X$ is unique (uniqueness in $X$ means that if $y\; \backslash in\; X$ is such that $\backslash lim\_\; x\_\; \backslash to\; y,$ then necessarily $x\; =\; y$) then this fact may be indicated by writing
:$\backslash lim\_\; x\_\; =\; x$ or $\backslash lim\_\; x\_a\; =\; x$ or $\backslash lim\_\; x\_a\; =\; x$
where an equals sign is used in place of the arrow $\backslash to.$ In a Hausdorff space
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

, every net has at most one limit so the limit of a convergent net in a Hausdorff space is always unique.
Some authors instead use the notation "$\backslash lim\_\; x\_\; =\; x$" to mean $\backslash lim\_\; x\_\; \backslash to\; x$ with also requiring that the limit be unique; however, if this notation is defined in this way then the equals sign
The equals sign (British English
British English (BrE) is the standard dialect
A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar an ...

$=$ is no longer guaranteed to denote a transitive relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

ship and so no longer denotes equality. Specifically, without the uniqueness requirement, if $x,\; y\; \backslash in\; X$ are distinct and if each is also a limit of $x\_$ in $X$ then $\backslash lim\_\; x\_\; =\; x$ and $\backslash lim\_\; x\_\; =\; y$ could be written (using the equals sign $=$) despite it being true that $x\; =\; y.$
Intuitively, convergence of this net means that the values $x\_a$ come and stay as close as we want to $x$ for large enough $a.$
The example net given above on the neighborhood systemIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

of a point $x$ does indeed converge to $x$ according to this definition.
Given a subbase
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

$\backslash mathcal$ for the topology on $X$ (where note that every base
Base or BASE may refer to:
Brands and enterprises
* Base (mobile telephony provider), a Belgian mobile telecommunications operator
*Base CRM
Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...

for a topology is also a subbase) and given a point $x\; \backslash in\; X,$ a net $x\_$ in $X$ converges to $x$ if and only if it is eventually in every neighborhood $U\; \backslash in\; \backslash mathcal$ of $x.$ This characterization extends to neighborhood subbases (and so also neighborhood bases) of the given point $x.$
If the set $S\; :=\; \backslash \; \backslash cup\; \backslash left\backslash $ is endowed with the subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

induced on it by $X,$ then $\backslash lim\_\; x\_\; \backslash to\; x$ in $X$ if and only if $\backslash lim\_\; x\_\; \backslash to\; x$ in $S.$ In this way, the question of whether or not the net $x\_$ converges to the given point $x$ is depends on this topological subspace $S$ consisting of $x$ and the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of (i.e. the points of) the net $x\_.$
Limits in a Cartesian product

A net in theproduct space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

has a limit if and only if each projection has a limit.
Symbolically, suppose that the Cartesian product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

:$X\; :=\; \backslash prod\_\; X\_i$
of the spaces $\backslash left(X\_i\backslash right)\_$ is endowed with the product topology
Product may refer to:
Business
* Product (business)
In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...

and that for every index $i\; \backslash in\; I,$ the canonical projection to $X\_i$ is denoted by
:$\backslash pi\_i\; :\; X\; =\; \backslash prod\_\; X\_j\; \backslash to\; X\_i$ and defined by $\backslash left(x\_j\backslash right)\_\; \backslash mapsto\; x\_i.$
Let $f\_\; =\; \backslash left(f\_a\backslash right)\_$ be a net in $X\; =\; \backslash prod\_\; X\_i$ directed by $A$ and for every index $i\; \backslash in\; I,$ let
:$\backslash pi\_i\backslash left(f\_\backslash right)\; ~:=~\; \backslash left(\; \backslash pi\_i\backslash left(f\_a\backslash right)\; \backslash right)\_$
denote the result of "plugging $f\_$ into $\backslash pi\_i$", which results in the net $\backslash pi\_i\backslash left(f\_\backslash right)\; :\; A\; \backslash to\; X\_i.$
It is sometimes useful to think of this definition in terms of function composition
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

: the net $\backslash pi\_i\backslash left(f\_\backslash right)$ is equal to the composition of the net $f\_\; :\; A\; \backslash to\; X$ with the projection $\backslash pi\_i\; :\; X\; \backslash to\; X\_i$; that is, $\backslash pi\_i\backslash left(f\_\backslash right)\; :=\; \backslash pi\_i\; \backslash ,\backslash circ\backslash ,\; f\_.$
If given $L\; =\; \backslash left(L\_i\backslash right)\_\; \backslash in\; X,$ then
:$f\_\; \backslash to\; L$ in $X\; =\; \backslash prod\_i\; X\_i$ if and only if for every $\backslash ;i\; \backslash in\; I,$ $\backslash ;\backslash pi\_i\backslash left(f\_\backslash right)\; :=\; \backslash left(\; \backslash pi\_i\backslash left(f\_a\backslash right)\; \backslash right)\_\; \backslash ;\backslash to\backslash ;\; \backslash pi\_i(L)\; =\; L\_i\backslash ;$ in $\backslash ;X\_i.$
;Tychonoff's theorem and relation to the axiom of choice
If no $L\; \backslash in\; X$ is given but for every $i\; \backslash in\; I,$ there exists some $L\_i\; \backslash in\; X\_i$ such that $\backslash pi\_i\backslash left(f\_\backslash right)\; \backslash to\; L\_i$ in $X\_i$ then the tuple defined by $L\; :=\; \backslash left(L\_i\backslash right)\_$ will be a limit of $f\_$ in $X.$
However, 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

might be need to be assumed in order to conclude that this tuple $L$ exists; the axiom of choice is not needed in some situations, such as when $I$ is finite or when every $L\_i\; \backslash in\; X\_i$ is the limit of the net $\backslash pi\_i\backslash left(f\_\backslash right)$ (because then there is nothing to choose between), which happens for example, when every $X\_i$ is a Hausdorff space
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

. If $I$ is infinite and $X\; =\; \backslash prod\_\; X\_j$ is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections $\backslash pi\_i\; :\; X\backslash to\; X\_i$ are surjective map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s.
The axiom of choice is equivalent to Tychonoff's theorem
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which states that the product of any collection of compact topological spaces is compact.
But if every compact space is also Hausdorff, then the so called "Tychonoff's theorem for compact Hausdorff spaces" can be used instead, which is equivalent to the ultrafilter lemma and so strictly weaker than 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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

.
Nets can be used to give short proofs of both version of Tychonoff's theorem by using the characterization of net convergence given above together with the fact that a space is compact if and only if every net has a convergent subnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...

.
Ultranets and cluster points of a net

Let $f$ be a net in $X$ based on the directed set $A$ and let $S$ be a subset of $X,$ then $f$ is said to be (or ) $S$ if for every $a\; \backslash in\; A$ there exists some $b\; \backslash in\; A$ such that $b\; \backslash geq\; a$ and $f(b)\; \backslash in\; S.$ A point $x\; \backslash in\; X$ is said to be an or of a net if (and only if) for every neighborhood $U$ of $x,$ the net is frequently in $U.$ A net $f$ in set $X$ is called a or an if for every subset $S\; \backslash subseteq\; X,$ $f$ is eventually in $S$ or $f$ is eventually in $X\; \backslash setminus\; S.$ Every constant net is an ultranet. Ultranets are closely related toultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (poset) ''P'' is a certain subset of ''P,'' namely a maximal filter on ''P'', that is, a proper filter on ''P'' that cannot be enlarged to a bigger pr ...

s.
Examples of limits of nets

*Limit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...

and limit of a function
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches ze ...

: see below.
* Limits of nets of Riemann sum
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, in the definition of the Riemann integral
In the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), an ...

. In this example, the directed set is the set of partitions of the interval of integration, partially ordered by inclusion.
Examples

Sequence in a topological space

A sequence $a\_1,\; a\_2,\; \backslash ldots$ in a topological space $X$ can be considered a net in $X$ defined on $\backslash mathbb.$ The net is eventually in a subset $S$ of $X$ if there exists an $N\; \backslash in\; \backslash mathbb$ such that for every integer $n\; \backslash geq\; N,$ the point $a\_n$ is in $S.$ So $\backslash lim\; \_\; a\_n\; \backslash to\; L$ if and only if for every neighborhood $V$ of $L,$ the net is eventually in $V.$ The net is frequently in a subset $S$ of $X$ if and only if for every $N\; \backslash in\; \backslash mathbb$ there exists some integer $n\; \backslash geq\; N$ such that $a\_n\; \backslash in\; S,$ that is, if and only if infinitely many elements of the sequence are in $S.$ Thus a point $y\; \backslash in\; X$ is a cluster point of the net if and only if every neighborhood $V$ of $y$ contains infinitely many elements of the sequence.Function from a metric space to a topological space

Consider a function from a metric space $M$ to a topological space $X,$ and a point $c\; \backslash in\; M.$ We direct the set $M\; \backslash setminus\; \backslash $reversely according to distance from $c,$ that is, the relation is "has at least the same distance to $c$ as", so that "large enough" with respect to the relation means "close enough to $c$". The function $f$ is a net in $X$ defined on $M\; \backslash setminus\; \backslash .$ The net $f$ is eventually in a subset $S$ of $X$ if there exists some $y\; \backslash in\; M\; \backslash setminus\; \backslash $ such that for every $x\; \backslash in\; M\; \backslash setminus\; \backslash $ with $d(x,\; c)\; \backslash leq\; d(y,\; c)$ the point $f(x)$ is in $S.$ So $\backslash lim\_\; f(x)\; \backslash to\; L$ if and only if for every neighborhood $V$ of $L,$ $f$ is eventually in $V.$ The net $f$ is frequently in a subset $S$ of $X$ if and only if for every $y\; \backslash in\; M\; \backslash setminus\; \backslash $ there exists some $x\; \backslash in\; M\; \backslash setminus\; \backslash $ with $d(x,\; c)\; \backslash leq\; d(y,\; c)$ such that $f(x)$ is in $S.$ A point $y\; \backslash in\; X$ is a cluster point of the net $f$ if and only if for every neighborhood $V$ of $y,$ the net is frequently in $V.$Function from a well-ordered set to a topological space

Consider awell-ordered set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$;\; href="/html/ALL/s/,\_c.html"\; ;"title=",\; c">,\; c$Properties

Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence. The following set of theorems and lemmas help cement that similarity:Characterizations of topological properties

Characterizations of topologies A subset $S\; \backslash subseteq\; X$ is open if and only if no net in $X\; \backslash setminus\; S$ converges to a point of $S.$ It is this characterization of open subsets that allows nets to characterizetopologies
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...

.
Topologies can also be characterized by closed subsets. A subset $S\; \backslash subseteq\; X$ is closed in $X$ if and only if every limit point of every net in $S$ necessarily belongs to $S.$
Explicitly, a subset $S\; \backslash subseteq\; X$ is closed if and only if whenever $x\; \backslash in\; X$ and $s\_\; =\; \backslash left(s\_a\backslash right)\_$ is a net with elements in $S$ having limit $x$ (that is, such that $s\_a\; \backslash in\; S\; \backslash text\; a\; \backslash in\; A$ and $\backslash lim\_\; s\_\; \backslash to\; x\; \backslash text\; X$), then necessarily $x\; \backslash in\; S.$
More generally, if $S\; \backslash subseteq\; X$ is any subset then a point $x\; \backslash in\; X$ is in the closure of $S$ if and only if there exists a net $s\_\; =\; \backslash left(s\_a\backslash right)\_$ in $S$ with limit $x\; \backslash in\; X$ and such that $s\_a\; \backslash in\; S$ for every index $a\; \backslash in\; A.$
Continuity
A function $f\; :\; X\; \backslash to\; Y$ between topological spaces is continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

at the point $x$ if and only if for every net $x\_\; =\; \backslash left(x\_a\backslash right)\_,$ $$\backslash lim\_\; x\_\; \backslash to\; x\; \backslash text\; X\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_a\; f\backslash left(x\_a\backslash right)\; \backslash to\; f(x)\; \backslash text\; Y.$$ This theorem is in general not true if "net" is replaced by "sequence"; it is necessary to allow for directed sets other than just the natural numbers if $X$ is not a first-countable space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

(or not a sequential space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

).
:
Compactness
A space $X$ is compact
Compact as used in politics may refer broadly to a pact
A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations
International relations (IR), international affairs (IA) or internationa ...

if and only if every net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ in $X$ has a subnet with a limit in $X.$ This can be seen as a generalization of the Bolzano–Weierstrass theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

and Heine–Borel theorem
In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
For a subset ''S'' of Euclidean space R''n'', the following two statements are equivalent:
*''S'' is closed set, closed and bounded set, bounded
*''S'' i ...

.
:
Cluster and limit points

The set of cluster points of a net is equal to the set of limits of its convergentsubnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...

s.
:
A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.
Other properties

In general, a net in a space $X$ can have more than one limit, but if $X$ is aHausdorff space
In topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

, the limit of a net, if it exists, is unique. Conversely, if $X$ is not Hausdorff, then there exists a net on $X$ with two distinct limits. Thus the uniqueness of the limit is to the Hausdorff condition on the space, and indeed this may be taken as the definition. This result depends on the directedness condition; a set indexed by a general preorder
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

or partial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

may have distinct limit points even in a Hausdorff space.
If $f\; :\; X\; \backslash to\; Y$ and $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is an ultranet on $X,$ then $\backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_$ is an ultranet on $Y.$
Cauchy nets

A Cauchy net generalizes the notion ofCauchy sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

to nets defined on uniform space
In the mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s..
A net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is a if for every entourage $V$ there exists $c\; \backslash in\; A$ such that for all $a,\; b\; \backslash geq\; c,$ $\backslash left(x\_a,\; x\_b\backslash right)$ is a member of $V.$ More generally, in a Cauchy space In general topology and mathematical analysis, analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as ...

, a net $x\_$ is Cauchy if the filter generated by the net is a Cauchy filter
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

.
A topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space (an Abstra ...

(TVS) is called if every Cauchy net converges to some point. A normed space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

, which is a special type of topological vector space, is a complete TVS (equivalently, a Banach space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

) if and only if every Cauchy sequence converges to some point (a property that is called ). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-normable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

) topological vector spaces.
Relation to filters

Afilter
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

is another idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence. More specifically, for every filter base
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

an can be constructed, and convergence of the filter base implies convergence of the associated net—and the other way around (for every net there is a filter base, and convergence of the net implies convergence of the filter base).R. G. Bartle, Nets and Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557. For instance, any net $\backslash left(x\_a\backslash right)\_$ in $X$ induces a filter base of tails $\backslash $ where the filter in $X$ generated by this filter base is called the net's . This correspondence allows for any theorem that can be proven with one concept to be proven with the other. For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.
Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts. He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis
Analysis is the process of breaking a complex topic or substance
Substance may refer to:
* Substance (Jainism), a term in Jain ontology to denote the base or owner of attributes
* Chemical substance, a material with a definite chemical composit ...

, while filters are most useful in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

. In any case, he shows how the two can be used in combination to prove various theorems in general topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
Limit superior

Limit superior
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and limit inferior of a net of real numbers can be defined in a similar manner as for sequences. Some authors work even with more general structures than the real line, like complete lattices.Schechter, Sections 7.43–7.47
For a net $\backslash left(x\_a\backslash right)\_,$ put
:$\backslash limsup\; x\_a\; =\; \backslash lim\_\; \backslash sup\_\; x\_b\; =\; \backslash inf\_\; \backslash sup\_\; x\_b.$
Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,
:$\backslash limsup\; (x\_a\; +\; y\_a)\; \backslash leq\; \backslash limsup\; x\_a\; +\; \backslash limsup\; y\_a,$
where equality holds whenever one of the nets is convergent.
See also

*Characterizations of the category of topological spaces In the mathematics, mathematical field of topology, a topological space is usually defined by declaring its open sets. However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For ...

* Filters in topology
In topology, a subfield of mathematics, are special Family of sets, families of subsets of a set X that can be used to study topological spaces and define all basic topological notions such a convergence, Continuous map (topology), continuity, Co ...

* Preorder
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

* Sequential space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...

Citations

References

* * * * * * * * * * General topology Articles containing proofs