In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
and related branches, a net or Moore–Smith sequence is a generalization of the notion of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
. In essence, a sequence is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s. The
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
of this function is usually some
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
.
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
between topological spaces
and
:
#The map
is
continuous in the topological sense;
#Given any point
in
and any sequence in
converging to
the composition of
with this sequence converges to
(continuous in the sequential sense).
While it is necessarily true that condition 1 implies condition 2 (The truth of the condition 1 ensures the truth of the conditions 2.), the reverse implication is not necessarily true if the topological spaces are not both
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
. In particular, the two conditions are equivalent for
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
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.
Life
Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, di ...
and
Herman L. Smith
Herman Lyle Smith (July 7, 1892 – 1950) was an American mathematician, the co-discoverer, with E. H. Moore, of nets, and also a discoverer of the related notion of filters independently of Henri Cartan.
Born in Pittwood, Illinois, Smith recei ...
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, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
linearly ordered
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 X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
set, a net is defined on an arbitrary
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
. 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 basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
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, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
s in behavior. The term "net" was coined by
John L. Kelley.
Nets are one of the many tools used 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 preserved under Continuous function, continuous Deformation theory, deformations, such ...
to generalize certain concepts that may not be general enough in the context of
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s. A related notion, that of the
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
, 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 mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of co ...
.
Definitions
Any
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
whose
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
is called a . If this function takes values in some set
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
where
is some
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
.
A is a non-empty set
together with a
preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
, typically automatically assumed to be denoted by
(unless indicated otherwise), with the property that it is also () , which means that for any
there exists some
such that
and
In words, this property means that given any two elements (of
), 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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s
together with the usual integer comparison
preorder form the
archetypical
The concept of an archetype (; ) appears in areas relating to behavior, historical psychology, and literary analysis.
An archetype can be any of the following:
# a statement, pattern of behavior, prototype, "first" form, or a main model that ot ...
example of a directed set. Indeed, a net whose domain is the natural numbers is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
because by definition, a sequence in
is just a function from
into
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 X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
s or even
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
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 dually, that is, it is an eleme ...
s and/or
maximal element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
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
; in particular, if a directed set
has a greatest element
then there does exist any
such that
(in contrast, there exists some
such that
).
Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences.
A net in
may be denoted by
where unless there is reason to think otherwise, it should automatically be assumed that the set
is directed and that its associated preorder is denoted by
However, notation for nets varies with some authors using, for instance, angled brackets
instead of parentheses.
A net in
may also be written as
which expresses the fact that this net
is a function
whose value at an element
in its domain is denoted by
instead of the usual parentheses notation
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 that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the filled disk or "bullet" denotes the location where arguments to the net (that is, elements
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 ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
and
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, where they are used to characterize many important
topological properties
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological spac ...
that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of
sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s and
Fréchet–Urysohn space
In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X.
Fréchet–Urysohn spaces are a speci ...
s). Nets are intimately related to
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
, 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
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some give ...
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 (or "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 proper filter on ...
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
to a directed subset of
see the linked page for a definition.
Examples of nets
Every non-empty
totally ordered set
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 X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive) ...
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 ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
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
in a topological space, let
denote the set of all
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
s containing
Then
is a directed set, where the direction is given by reverse inclusion, so that
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
is contained in
For
let
be a point in
Then
is a net. As
increases with respect to
the points
in the net are constrained to lie in decreasing neighbourhoods of
so intuitively speaking, we are led to the idea that
must tend towards
in some sense. We can make this limiting concept precise.
A subnet of a sequence is necessarily a sequence.
For an example, let
and let
for every
so that
is the constant zero sequence.
Let
be directed by the usual order
and let
for each
Define
by letting
be the
ceiling
A ceiling is an overhead interior surface that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings ...
of
The map
is an order morphism whose image is cofinal in its codomain and
holds for every
This shows that
is a subnet of the sequence
(where this subnet is not a subsequence of
because it is not even a sequence since its domain is an
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
).
Limits of nets
If
is a net from a directed set
into
and if
is a subset of
then
is said to be (or ) if there exists some
such that for every
with
the point
A point
is called a or of the net
in
if (and only if)
:for every open
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
the net
is eventually in
in which case, this net is then also said to and to .
Intuitively, convergence of this net means that the values
come and stay as close as we want to
for large enough
The example net given above on the
neighborhood system In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
of a point
does indeed converge to
according to this definition.
Notation
If the net
converges in
to a point
then this fact may be expressed by writing any of the following:
where if the topological space
is clear from context then the words "in
" may be omitted.
If
in
and if this limit in
is unique (uniqueness in
means that if
is such that
then necessarily
) then this fact may be indicated by writing
where an equals sign is used in place of the arrow
In a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
, 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 "
" to mean
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, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between two ...
is no longer guaranteed to denote a
transitive relation
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A homog ...
ship and so no longer denotes
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
. Specifically, without the uniqueness requirement, if
are distinct and if each is also a limit of
in
then
and
could be written (using the equals sign
) despite
being false.
Bases and subbases
Given a
subbase
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. A slightly different definition is used by so ...
for the topology on
(where note that every
base for a topology is also a subbase) and given a point
a net
in
converges to
if and only if it is eventually in every neighborhood
of
This characterization extends to
neighborhood subbases (and so also
neighborhood bases) of the given point
Convergence in metric spaces
Suppose
is a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
(or a
pseudometric space
In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric ...
) and
is endowed with the
metric topology
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
.
If
is a point and
is a net, then
in
if and only if
in
where
is a net of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s.
In
plain English
Plain English (or layman's terms) are groups of words that are to be clear and easy to know. It usually avoids the use of rare words and uncommon euphemisms to explain the subject. Plain English wording is intended to be suitable for almost anyone, ...
, this characterization says that a net converges to a point in a metric space if and only if the distance between the net and the point converges to zero.
If
is a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
(or a
seminormed space) then
in
if and only if
in
where
Convergence in topological subspaces
If the set
is endowed with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced on it by
then
in
if and only if
in
In this way, the question of whether or not the net
converges to the given point
depends on this topological subspace
consisting of
and the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of (that is, the points of) the net
Limits in a Cartesian product
A net in the
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
has a limit if and only if each projection has a limit.
Symbolically, suppose that the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
of the spaces
is endowed with the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
and that for every index
the canonical projection to
is denoted by
Let
be a net in
directed by
and for every index
let
denote the result of "plugging
into
", which results in the net
It is sometimes useful to think of this definition in terms of
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
: the net
is equal to the composition of the net
with the projection
; that is,
If given
then
Tychonoff's theorem and relation to the axiom of choice
If no
is given but for every
there exists some
such that
in
then the tuple defined by
will be a limit of
in
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
might be need to be assumed in order to conclude that this tuple
exists; the axiom of choice is not needed in some situations, such as when
is finite or when every
is the limit of the net
(because then there is nothing to choose between), which happens for example, when every
is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
. If
is infinite and
is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections
are
surjective map
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
s.
The axiom of choice is equivalent to
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname sometimes is trans ...
, 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
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
.
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 ...
.
Cluster points of a net
A net
in
is said to be or a given subset
if for every
there exists some
such that
and
A point
is said to be an or of a net if for every neighborhood
of
the net is frequently in
A point
is a cluster point of a given net if and only if it has a subset that converges to
If
is a net in
then the set of all cluster points of
in
is equal to
where
for each
If
is a cluster point of some subnet of
then
is also a cluster point of
Ultranets
A net
in set
is called a or an if for every subset
is eventually in
or
is eventually in the complement
Ultranets are closely related to
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "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 proper filter on ...
s.
Every constant net is an ultranet. Every subnet of an ultranet is an ultranet. Every net has some subnet that is an ultranet.
If
is an ultranet in
and
is a function then
is an ultranet in
Given
an ultranet clusters at
if and only it converges to
Examples of limits of nets
*
Limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limi ...
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 z ...
: see below.
* Limits of nets of
Riemann sum
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s, in the definition of the
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göt ...
. 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
in a topological space
can be considered a net in
defined on
The net is eventually in a subset
of
if there exists an
such that for every integer
the point
is in
So
if and only if for every neighborhood
of
the net is eventually in
The net is frequently in a subset
of
if and only if for every
there exists some integer
such that
that is, if and only if infinitely many elements of the sequence are in
Thus a point
is a cluster point of the net if and only if every neighborhood
of
contains infinitely many elements of the sequence.
Function from a metric space to a topological space
Consider a function from a metric space
to a topological space
and a point
We direct the set
reversely according to distance from
that is, the relation is "has at least the same distance to
as", so that "large enough" with respect to the relation means "close enough to
". The function
is a net in
defined on
The net
is eventually in a subset
of
if there exists some
such that for every
with
the point
is in
So
if and only if for every neighborhood
of
is eventually in
The net
is frequently in a subset
of
if and only if for every
there exists some
with
such that
is in
A point
is a cluster point of the net
if and only if for every neighborhood
of
the net is frequently in
Function from a well-ordered set to a topological space
Consider a
well-ordered set
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-ord ...