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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 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 ...

sequence
. 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 ...

codomain
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 ...

topology
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 ...

Henri Cartan
.


Definitions

Any
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 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 \to X where A is some
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
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 \,\leq\, (unless indicated otherwise), with the property that it is also () , which means that for any a, b \in A, there exists some c \in A such that a \leq c and b \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 \N together with the usual integer comparison \,\leq\, 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 ...

sequence
because by definition, a sequence in X is just a function from \N = \ 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 \,<\, instead of the original (non-strict) preorder \,\leq; in particular, if a directed set (A, \leq) has a greatest element a \in A then there does exist any b \in A such that a < b (in contrast, there exists some b \in A such that a \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 \left(x_a\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 \,\leq. However, notation for nets varies with some authors using, for instance, angled brackets \left\langle x_a \right\rangle_ instead of parentheses. A net in X may also be written as x_ = \left(x_a\right)_, which expresses the fact that this net x_ is a function x_ : A \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 \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 ...

Topology
, 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-empty
totally 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 \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 \in N_x, let x_S be a point in S. Then \left(x_S\right) is a net. As S increases with respect to \,\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_ = \left(x_a\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 \in A such that for every b \in A with b \geq a, the point x_b \in S. A point x \in X is called a or of the net x_ in X if (and only if) :for every open
neighborhood 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 \in X then this fact may be expressed by writing any of the following: : \begin & x_ && \to\; && x && \;\;\text X \\ & x_a && \to\; && x && \;\;\text X \\ \lim_ \; & x_ && \to\; && x && \;\;\text X \\ \lim_ \; & x_a && \to\; && x && \;\;\text X \\ \lim_ _a \; & x_a && \to\; && x && \;\;\text X \\ \end where if the topological space X is clear from context then the words "in X" may be omitted. If \lim_ x_ \to x in X and if this limit in X is unique (uniqueness in X means that if y \in X is such that \lim_ x_ \to y, then necessarily x = y) then this fact may be indicated by writing :\lim_ x_ = x or \lim_ x_a = x or \lim_ x_a = x where an equals sign is used in place of the arrow \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), ...

Hausdorff space
, 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 "\lim_ x_ = x" to mean \lim_ x_ \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 ...

equals sign
= 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 \in X are distinct and if each is also a limit of x_ in X then \lim_ x_ = x and \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), ...
\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 \in X, a net x_ in X converges to x if and only if it is eventually in every neighborhood U \in \mathcal of x. This characterization extends to neighborhood subbases (and so also neighborhood bases) of the given point x. If the set S := \ \cup \left\ 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 ...

subspace topology
induced on it by X, then \lim_ x_ \to x in X if and only if \lim_ x_ \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 the
product 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 := \prod_ X_i of the spaces \left(X_i\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 \in I, the canonical projection to X_i is denoted by :\pi_i : X = \prod_ X_j \to X_i and defined by \left(x_j\right)_ \mapsto x_i. Let f_ = \left(f_a\right)_ be a net in X = \prod_ X_i directed by A and for every index i \in I, let :\pi_i\left(f_\right) ~:=~ \left( \pi_i\left(f_a\right) \right)_ denote the result of "plugging f_ into \pi_i", which results in the net \pi_i\left(f_\right) : A \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 \pi_i\left(f_\right) is equal to the composition of the net f_ : A \to X with the projection \pi_i : X \to X_i; that is, \pi_i\left(f_\right) := \pi_i \,\circ\, f_. If given L = \left(L_i\right)_ \in X, then :f_ \to L in X = \prod_i X_i if and only if for every \;i \in I, \;\pi_i\left(f_\right) := \left( \pi_i\left(f_a\right) \right)_ \;\to\; \pi_i(L) = L_i\; in \;X_i. ;Tychonoff's theorem and relation to the axiom of choice If no L \in X is given but for every i \in I, there exists some L_i \in X_i such that \pi_i\left(f_\right) \to L_i in X_i then the tuple defined by L := \left(L_i\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 ...

axiom of choice
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 \in X_i is the limit of the net \pi_i\left(f_\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), ...

Hausdorff space
. If I is infinite and X = \prod_ X_j is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections \pi_i : X\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 ...

axiom of choice
. 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 \in A there exists some b \in A such that b \geq a and f(b) \in S. A point x \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 \subseteq X, f is eventually in S or f is eventually in X \setminus S. Every constant net is an ultranet. Ultranets are closely related to
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.


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 ...

limit of a function
: 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, \ldots in a topological space X can be considered a net in X defined on \mathbb. The net is eventually in a subset S of X if there exists an N \in \mathbb such that for every integer n \geq N, the point a_n is in S. So \lim _ a_n \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 \in \mathbb there exists some integer n \geq N such that a_n \in S, that is, if and only if infinitely many elements of the sequence are in S. Thus a point y \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 \in M. We direct the set M \setminus \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 \setminus \. The net f is eventually in a subset S of X if there exists some y \in M \setminus \ such that for every x \in M \setminus \ with d(x, c) \leq d(y, c) the point f(x) is in S. So \lim_ f(x) \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 \in M \setminus \ there exists some x \in M \setminus \ with d(x, c) \leq d(y, c) such that f(x) is in S. A point y \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 a
well-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 ...
, c/math> with limit point t and a function f from ordinal-indexed sequence.


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 \subseteq X is open if and only if no net in X \setminus S converges to a point of S. It is this characterization of open subsets that allows nets to characterize
topologies 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 \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 \subseteq X is closed if and only if whenever x \in X and s_ = \left(s_a\right)_ is a net with elements in S having limit x (that is, such that s_a \in S \text a \in A and \lim_ s_ \to x \text X), then necessarily x \in S. More generally, if S \subseteq X is any subset then a point x \in X is in the closure of S if and only if there exists a net s_ = \left(s_a\right)_ in S with limit x \in X and such that s_a \in S for every index a \in A. Continuity A function f : X \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_ = \left(x_a\right)_, \lim_ x_ \to x \text X \quad \text \quad \lim_a f\left(x_a\right) \to f(x) \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_ = \left(x_a\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 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 ...
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 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), ...

Hausdorff space
, 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 \to Y and x_ = \left(x_a\right)_ is an ultranet on X, then \left(f\left(x_a\right)\right)_ is an ultranet on Y.


Cauchy nets

A Cauchy net generalizes the notion of
Cauchy 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_ = \left(x_a\right)_ is a if for every entourage V there exists c \in A such that for all a, b \geq c, \left(x_a, x_b\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

A
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 ...
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 \left(x_a\right)_ in X induces a filter base of tails \ 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 ...

analysis
, 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 \left(x_a\right)_, put :\limsup x_a = \lim_ \sup_ x_b = \inf_ \sup_ x_b. Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example, :\limsup (x_a + y_a) \leq \limsup x_a + \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