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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 theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topolog ...
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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
. In essence, a sequence is a function 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 (mathematics), function is the Set (mathematics), set into which all of the output of the function is constrained to fall. It is the set in the notation . The term Range of a funct ...
of this function is usually some
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
. 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 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 particular, the two conditions are equivalent for
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s. The concept of a net, first introduced by E. H. Moore 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, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
linearly ordered 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 (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
. 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 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 (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s. A related notion, that of the filter, 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 ...
.


Definitions

Any function whose domain is a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
is called a . 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, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
. A is a non-empty set A together with a
preorder 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 ...
, 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s \N together with the usual integer comparison \,\leq\, preorder form the archetypical 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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
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 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 ...
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 (mathematics), set. A poset consists of a set toget ...
s. Moreover, directed sets are allowed to have greatest elements 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 define ...
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 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 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 (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
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 that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of
sequential space 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, deform ...
s and Fréchet–Urysohn spaces). Nets are intimately related to filters, 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 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 ultrafilters, 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 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 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 x in a topological space, let N_x denote the set of all
neighbourhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
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 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 ...
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. A subnet of a sequence is necessarily a sequence. For an example, let X = \R^n and let x_i = 0 for every i \in \N, so that x_ = (0)_ : \N \to X is the constant zero sequence. Let I = \ be directed by the usual order \,\leq\, and let s_r = 0 for each r \in R. Define \varphi : I \to \N by letting \varphi(r) = \lceil r \rceil 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 r. The map \varphi : I \to \N is an order morphism whose image is cofinal in its codomain and \left(x_ \circ \varphi\right)(r) = x_ = 0 = s_r holds for every r \in R. This shows that \left(s_\right)_ = x_ \circ \varphi is a subnet of the sequence x_ (where this subnet is not a subsequence of x_ 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 Element (mathematics), elements to be countable set, countable. The uncountability of a set is closely related to its cardinal number: a se ...
).


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, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
U of x, the net x_ is eventually in U, in which case, this net is then also said to and to . 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 system of a point x does indeed converge to x according to this definition. Notation 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 \;~~ \text ~~\; \lim_ x_a = x \;~~ \text ~~\; \lim_ x_a = x where an equals sign is used in place of the arrow \to. 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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
, 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, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality (mathematics), equality in some well-defined sense. In an equation, it ...
= is no longer guaranteed to denote a
transitive relation 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 ...
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 x = y being false. Bases and subbases Given a subbase \mathcal for the topology on X (where note that every base 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. Convergence in metric spaces Suppose (X, d) is a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
(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 X is endowed with the metric topology. If x \in X is a point and x_ = \left(x_i\right)_ is a net, then x_ \to x in (X, d) if and only if d\left(x, x_\right) \to 0 in \R, where d\left(x, x_\right) := \left(d\left(x, x_a\right)\right)_ is a net of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 (X, \, \cdot\, ) is a
normed space 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 ...
(or a seminormed space) then x_ \to x in (X, \, \cdot\, ) if and only if \left\, x - x_\right\, \to 0 in \R, where \left\, x - x_\right\, := \left(\left\, x - x_a\right\, \right)_. Convergence in topological subspaces If the set S := \ \cup \left\ 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 Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relat ...
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 depends on this topological subspace S consisting of x 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 x_.


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 Set (mathematics), 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 notatio ...
X := \prod_ X_i of the spaces \left(X_i\right)_ 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 i \in I, the canonical projection to X_i is denoted by \begin \pi_i :\;&& \prod_ X_j &&\;\to\;& X_i \\ .3ex && \left(x_j\right)_ &&\;\mapsto\;& x_i \\ \end 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, function composition is an operation that takes two function (mathematics), functions and , and produces a function such that . In this operation, the function is function application, applied to the result of applying the ...
: 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 \prod_ X_i, then f_ \to L \text \prod_i X_i \quad \text \quad \text\;i \in I, \;\pi_i\left(f_\right) := \left( \pi_i\left(f_a\right) \right)_ \;\to\; \pi_i(L) = L_i\; \text \;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 ...
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 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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
. 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, a surjective function (also known as surjection, or onto function) is a Function (mathematics), function that every element can be mapped from element so that . In other words, every element of the function's codomain is the Im ...
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 space, compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov (whose surname som ...
, 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.


Cluster points of a net

A net x_ = \left(x_a\right)_ in X is said to be or a given subset S if for every a \in A there exists some b \in A such that b \geq a and x_b \in S. A point x \in X is said to be an or of a net if for every neighborhood U of x, the net is frequently in U. A point x \in X is a cluster point of a given net if and only if it has a subset that converges to x. If x_ = \left(x_a\right)_ is a net in X then the set of all cluster points of x_ in X is equal to \bigcap_ \operatorname_X \left(x_\right) where x_ := \left\ for each a \in A. If x \in X is a cluster point of some subnet of x_ then x is also a cluster point of x_.


Ultranets

A net x_ in set X is called a or an if for every subset S \subseteq X, x_ is eventually in S or x_ is eventually in the complement X \setminus S. Ultranets are closely related to ultrafilters. Every constant net is an ultranet. Every subnet of an ultranet is an ultranet. Every net has some subnet that is an ultranet. If x_ = \left(x_a\right)_ is an ultranet in X and f : X \to Y is a function then f \circ x_ = \left(f\left(x_a\right)\right)_ is an ultranet in Y. Given x \in X, an ultranet clusters at x if and only it converges to x.


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 limit ...
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 (mathematics), function on an Interval (mathematics), interval. It was pre ...
. 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 \N. The net is eventually in a subset S of X if there exists an N \in \N 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 \N 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, a well-order (or well-ordering or well-order relation) on a Set (mathematics), 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 ...
, c/math> with limit point t and a function f from ordinal-indexed sequence.


Subnets

The analogue of "subsequence" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard, which is as follows: If x_ = \left(x_a\right)_ and s_ = \left(s_i\right)_ are nets then s_ is called a or of x_ if there exists an order-preserving map h : I \to A such that h(I) is a cofinal subset of A and s_i = x_ \quad \text i \in I. The map h : I \to A is called and an if whenever i \leq j then h(i) \leq h(j). The set h(I) being in A means that for every a \in A, there exists some b \in h(I) such that b \geq a.


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 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 limit ...
. The following set of theorems and lemmas help cement that similarity:


Characterizations of topological properties

Closed sets and closure A subset S \subseteq X is closed in X if and only if every limit point of every convergent 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 valued in S (meaning that s_a \in S for all a \in A) such that \lim_ s_ \to x in 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. Open sets and 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. Also, subset S \subseteq X is open if and only if every net converging to an element of S is eventually contained in S. It is these characterizations of "open subset" that allow nets to characterize topologies. Topologies can also be characterized by closed subsets since a set is open if and only if its complement is closed. So the characterizations of "closed set" in terms of nets can also be used to characterize topologies. Continuity A function f : X \to Y between topological spaces is continuous at the point x if and only if for every net x_ = \left(x_a\right)_ in the domain X, \lim_ x_ \to x \text X \quad \text \quad \lim_a f\left(x_a\right) \to f(x) \text Y. In general, this the statement would not be true if the word "net" was replaced by "sequence"; that is, it is necessary to allow for directed sets other than just the natural numbers if X is not a
first-countable space 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 system#Basis, neigh ...
(or not a
sequential space 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, deform ...
). (\implies) Let f be continuous at point x, and let x_ = \left(x_a\right)_ be a net such that \lim_ x_ \to x. Then for every open neighborhood U of f(x), its preimage under f, V := f^(U), is a neighborhood of x (by the continuity of f at x). Thus the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel ...
of V, which is denoted by \operatorname V, is an open neighborhood of x, and consequently x_ is eventually in \operatorname V. Therefore \left(f\left(x_a\right)\right)_ is eventually in f(\operatorname V) and thus also eventually in f(V) which is a subset of U. Thus \lim_ \left(f\left(x_a\right)\right)_ \to f(x), and this direction is proven. (\Longleftarrow) Let x be a point such that for every net x_ = \left(x_a\right)_ such that\lim_ x_ \to x, \lim_ \left(f\left(x_a\right)\right)_ \to f(x). Now suppose that f is not continuous at x. Then there is a
neighborhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
U of f(x) whose preimage under f, V, is not a neighborhood of x. Because f(x) \in U, necessarily x \in V. Now the set of open neighborhoods of x with the
containment Containment was a Geopolitics, geopolitical strategic foreign policy pursued by the United States during the Cold War to prevent the spread of communism after the end of World War II. The name was loosely related to the term ''Cordon sanitaire ...
preorder is a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
(since the intersection of every two such neighborhoods is an open neighborhood of x as well). We construct a net x_ = \left(x_a\right)_ such that for every open neighborhood of x whose index is a, x_a is a point in this neighborhood that is not in V; that there is always such a point follows from the fact that no open neighborhood of x is included in V (because by assumption, V is not a neighborhood of x). It follows that f\left(x_a\right) is not in U. Now, for every open neighborhood W of x, this neighborhood is a member of the directed set whose index we denote a_0. For every b \geq a_0, the member of the directed set whose index is b is contained within W; therefore x_b \in W. Thus \lim_ x_ \to x. and by our assumption \lim_ \left(f\left(x_a\right)\right)_ \to f(x). But \operatorname U is an open neighborhood of f(x) and thus f\left(x_a\right) is eventually in \operatorname U and therefore also in U, in contradiction to f\left(x_a\right) not being in U for every a. This is a contradiction so f must be continuous at x. This completes the proof. A function f : X \to Y is continuous if and only if whenever x_ \to x in X then f\left(x_\right) \to f(x) in Y. Compactness A space X is compact 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, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space \R^n. The theorem states that each ...
and Heine–Borel theorem. (\implies) First, suppose that X is compact. We will need the following observation (see finite intersection property). Let I be any non-empty set and \left\_ be a collection of closed subsets of X such that \bigcap_ C_i \neq \varnothing for each finite J \subseteq I. Then \bigcap_ C_i \neq \varnothing as well. Otherwise, \left\_ would be an open cover for X with no finite subcover contrary to the compactness of X. Let x_ = \left(x_a\right)_ be a net in X directed by A. For every a \in A define E_a \triangleq \left\. The collection \ has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that \bigcap_ \operatorname E_a \neq \varnothing and this is precisely the set of cluster points of x_. By the proof given in the next section, it is equal to the set of limits of convergent subnets of x_. Thus x_ has a convergent subnet. (\Longleftarrow) Conversely, suppose that every net in X has a convergent subnet. For the sake of contradiction, let \left\ be an open cover of X with no finite subcover. Consider D \triangleq \. Observe that D is a directed set under inclusion and for each C\in D, there exists an x_C \in X such that x_C \notin U_a for all a \in C. Consider the net \left(x_C\right)_. This net cannot have a convergent subnet, because for each x \in X there exists c \in I such that U_c is a neighbourhood of x; however, for all B \supseteq \, we have that x_B \notin U_c. This is a contradiction and completes the proof.


Cluster and limit points

The set of cluster points of a net is equal to the set of limits of its convergent subnets. Let x_ = \left(x_a\right)_ be a net in a topological space X (where as usual A automatically assumed to be a directed set) and also let y \in X. If y is a limit of a subnet of x_ then y is a cluster point of x_. Conversely, assume that y is a cluster point of x_. Let B be the set of pairs (U, a) where U is an open neighborhood of y in X and a \in A is such that x_a \in U. The map h : B \to A mapping (U, a) to a is then cofinal. Moreover, giving B the product order (the neighborhoods of y are ordered by inclusion) makes it a directed set, and the net y_ = \left(y_b\right)_ defined by y_b = x_ converges to y. 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 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 neighbourhood (mathematics), neighbourhoods of each which are disjoint s ...
, 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 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 ...
or
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 (mathematics), set. A poset consists of a set toget ...
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 to nets defined on
uniform space In the mathematical field of 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 ...
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, a net x_ is Cauchy if the filter generated by the net is a Cauchy filter. A
topological vector space 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 mo ...
(TVS) is called if every Cauchy net converges to some point. A
normed space 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 ...
, which is a special type of topological vector space, is a complete TVS (equivalently, a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a Complete metric space, complete normed vector space. Thus, a Banach space is a vector space with a Metric (mathematics), metric that allows the computation ...
) 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) topological vector spaces.


Relation to filters

A filter 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 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 \left\ 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 (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...
, while filters are most useful in
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 ...
. In any case, he shows how the two can be used in combination to prove various theorems in
general topology In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topolog ...
.


Limit superior

Limit superior In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a Function (mathematics), function (see limi ...
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

* * * *


Citations


References

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