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 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 cal ...

. In essence, a sequence is a function whose domain is the natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

s. The codomain of this function is usually some topological space.
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
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 m ...

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
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 ...

linearly ordered set, a net is defined on an arbitrary directed set. 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 sets in behavior. The term "net" was coined by John L. Kelley.
Nets are one of the many tools used in topology
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 i ...

to generalize certain concepts that may not be general enough in the context of metric 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 m ...

s. A related notion, that of the filter, was developed in 1937 by Henri Cartan.
Definitions

Any function whose domain is a directed set 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 $f\; :\; A\; \backslash to\; X$ where $A$ is some directed set. A is a non-empty set $A$ together with a preorder, typically automatically assumed to be denoted by $\backslash ,\backslash leq\backslash ,$ (unless indicated otherwise), with the property that it is also () , which means that for any $a,\; b\; \backslash in\; A,$ there exists some $c\; \backslash in\; A$ such that $a\; \backslash leq\; c$ and $b\; \backslash leq\; c.$ In words, this property means that given any two elements (of $A$), there is always some element that is "above" both of them (i.e. that is greater than or equal to each of them); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Thenatural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

s $\backslash N$ together with the usual integer comparison $\backslash ,\backslash leq\backslash ,$ 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 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 cal ...

because by definition, a sequence in $X$ is just a function from $\backslash N\; =\; \backslash $ into $X.$ It is in this way that nets are generalizations of sequences. Importantly though, unlike the natural numbers, directed sets are required to be total order
In mathematics
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
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 m ...

s.
Moreover, directed sets are allowed to have greatest elements and/or maximal element
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 ...

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

, the filled disk or "bullet" denotes the location where arguments to the net (that is, elements $a\; \backslash in\; A$ of the net's domain) are placed; it helps emphasize that the net is a function and also reduces the number of indices and other symbols that must be written when referring to it later.
Nets are primarily used in the fields of Analysis and Topology
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 i ...

, 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 spaces 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-emptytotally ordered set
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 ...

is directed. Therefore, every function on such a set is a net. In particular, the natural number
In mathematics, the natural numbers are those number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...

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 neighbourhoods containing $x.$ Then $N\_x$ is a directed set, where the direction is given by reverse inclusion, so that $S\; \backslash geq\; T$ if and only if $S$ is contained in $T.$ For $S\; \backslash in\; N\_x,$ let $x\_S$ be a point in $S.$ Then $\backslash left(x\_S\backslash right)$ is a net. As $S$ increases with respect to $\backslash ,\backslash geq,$ the points $x\_S$ in the net are constrained to lie in decreasing neighbourhoods of $x,$ so intuitively speaking, we are led to the idea that $x\_S$ must tend towards $x$ in some sense. We can make this limiting concept precise.
A subnet of a sequence is necessarily a sequence.
For an example, let $X\; =\; \backslash R^n$ and let $x\_i\; =\; 0$ for every $i\; \backslash in\; \backslash N,$ so that $x\_\; =\; (0)\_\; :\; \backslash N\; \backslash to\; X$ is the constant zero sequence.
Let $I\; =\; \backslash $ be directed by the usual order $\backslash ,\backslash leq\backslash ,$ and let $s\_r\; =\; 0$ for each $r\; \backslash in\; R.$
Define $\backslash varphi\; :\; I\; \backslash to\; \backslash N$ by letting $\backslash varphi(r)\; =\; \backslash lceil\; r\; \backslash rceil$ 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 $\backslash varphi\; :\; I\; \backslash to\; \backslash N$ is an order morphism whose image is cofinal in its codomain and $\backslash left(x\_\; \backslash circ\; \backslash varphi\backslash right)(r)\; =\; x\_\; =\; 0\; =\; s\_r$ holds for every $r\; \backslash in\; R.$ This shows that $\backslash left(s\_\backslash right)\_\; =\; x\_\; \backslash circ\; \backslash varphi$ 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).
Limits of nets

If $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is a net from a directed set $A$ into $X,$ and if $S$ is a subset of $X,$ then $x\_$ is said to be (or ) if there exists some $a\; \backslash in\; A$ such that for every $b\; \backslash in\; A$ with $b\; \backslash geq\; a,$ the point $x\_b\; \backslash in\; S.$ A point $x\; \backslash in\; X$ is called a or of the net $x\_$ in $X$ if (and only if) :for every openneighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

$U$ 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 $x\; \backslash in\; X$ then this fact may be expressed by writing any of the following:
$$\backslash begin\; \&\; x\_\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \backslash ;\; \&\; x\_\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \backslash ;\; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash lim\_\; \_a\; \backslash ;\; \&\; x\_a\; \&\&\; \backslash to\backslash ;\; \&\&\; x\; \&\&\; \backslash ;\backslash ;\backslash text\; X\; \backslash \backslash \; \backslash end$$
where if the topological space $X$ is clear from context then the words "in $X$" may be omitted.
If $\backslash lim\_\; x\_\; \backslash to\; x$ in $X$ and if this limit in $X$ is unique (uniqueness in $X$ means that if $y\; \backslash in\; X$ is such that $\backslash lim\_\; x\_\; \backslash to\; y,$ then necessarily $x\; =\; y$) then this fact may be indicated by writing
$$\backslash lim\_\; x\_\; =\; x\; \backslash ;~~\; \backslash text\; ~~\backslash ;\; \backslash lim\_\; x\_a\; =\; x\; \backslash ;~~\; \backslash text\; ~~\backslash ;\; \backslash lim\_\; x\_a\; =\; x$$
where an equals sign is used in place of the arrow $\backslash to.$ In a 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 "$\backslash lim\_\; x\_\; =\; x$" to mean $\backslash lim\_\; x\_\; \backslash to\; x$ with also requiring that the limit be unique; however, if this notation is defined in this way then the equals sign
The equals sign (British English
British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain
Great Britain is an island in the North Atlantic Ocean
The Atlantic Ocean is ...

$=$ 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 $x,\; y\; \backslash in\; X$ are distinct and if each is also a limit of $x\_$ in $X$ then $\backslash lim\_\; x\_\; =\; x$ and $\backslash lim\_\; x\_\; =\; y$ could be written (using the equals sign $=$) despite $x\; =\; y$ being false.
Bases and subbases
Given a subbase $\backslash mathcal$ for the topology on $X$ (where note that every base for a topology is also a subbase) and given a point $x\; \backslash in\; X,$ a net $x\_$ in $X$ converges to $x$ if and only if it is eventually in every neighborhood $U\; \backslash in\; \backslash mathcal$ of $x.$ This characterization extends to neighborhood subbases (and so also neighborhood bases) of the given point $x.$
Convergence in metric spaces
Suppose $(X,\; d)$ is a metric 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 m ...

(or a pseudometric space) and $X$ is endowed with the metric topology.
If $x\; \backslash in\; X$ is a point and $x\_\; =\; \backslash left(x\_i\backslash right)\_$ is a net, then $x\_\; \backslash to\; x$ in $(X,\; d)$ if and only if $d\backslash left(x,\; x\_\backslash right)\; \backslash to\; 0$ in $\backslash R,$ where $d\backslash left(x,\; x\_\backslash right)\; :=\; \backslash left(d\backslash left(x,\; x\_a\backslash right)\backslash right)\_$ is a net of real number
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.
In plain English, 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 $(X,\; \backslash ,\; \backslash cdot\backslash ,\; )$ is a normed space (or a seminormed space) then $x\_\; \backslash to\; x$ in $(X,\; \backslash ,\; \backslash cdot\backslash ,\; )$ if and only if $\backslash left\backslash ,\; x\; -\; x\_\backslash right\backslash ,\; \backslash to\; 0$ in $\backslash R,$ where $\backslash left\backslash ,\; x\; -\; x\_\backslash right\backslash ,\; :=\; \backslash left(\backslash left\backslash ,\; x\; -\; x\_a\backslash right\backslash ,\; \backslash right)\_.$
Convergence in topological subspaces
If the set $S\; :=\; \backslash \; \backslash cup\; \backslash left\backslash $ is endowed with the subspace topology induced on it by $X,$ then $\backslash lim\_\; x\_\; \backslash to\; x$ in $X$ if and only if $\backslash lim\_\; x\_\; \backslash to\; x$ in $S.$ In this way, the question of whether or not the net $x\_$ converges to the given point $x$ 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-dimensi ...

of (that is, the points of) the net $x\_.$
Limits in a Cartesian product

A net in the product space has a limit if and only if each projection has a limit. Symbolically, suppose that the Cartesian product $$X\; :=\; \backslash prod\_\; X\_i$$ of the spaces $\backslash left(X\_i\backslash right)\_$ is endowed with the product topology and that for every index $i\; \backslash in\; I,$ the canonical projection to $X\_i$ is denoted by $$\backslash begin\; \backslash pi\_i\; :\backslash ;\&\&\; \backslash prod\_\; X\_j\; \&\&\backslash ;\backslash to\backslash ;\&\; X\_i\; \backslash \backslash ;\; href="/html/ALL/s/.3ex.html"\; ;"title=".3ex">.3ex$$ Let $f\_\; =\; \backslash left(f\_a\backslash right)\_$ be a net in $X\; =\; \backslash prod\_\; X\_i$ directed by $A$ and for every index $i\; \backslash in\; I,$ let $$\backslash pi\_i\backslash left(f\_\backslash right)\; ~:=~\; \backslash left(\backslash pi\_i\backslash left(f\_a\backslash right)\backslash right)\_$$ denote the result of "plugging $f\_$ into $\backslash pi\_i$", which results in the net $\backslash pi\_i\backslash left(f\_\backslash right)\; :\; A\; \backslash to\; X\_i.$ It is sometimes useful to think of this definition in terms offunction composition
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 i ...

: the net $\backslash pi\_i\backslash left(f\_\backslash right)$ is equal to the composition of the net $f\_\; :\; A\; \backslash to\; X$ with the projection $\backslash pi\_i\; :\; X\; \backslash to\; X\_i$; that is, $\backslash pi\_i\backslash left(f\_\backslash right)\; :=\; \backslash pi\_i\; \backslash ,\backslash circ\backslash ,\; f\_.$
If given $L\; =\; \backslash left(L\_i\backslash right)\_\; \backslash in\; \backslash prod\_\; X\_i,$ then
$$f\_\; \backslash to\; L\; \backslash text\; \backslash prod\_i\; X\_i\; \backslash quad\; \backslash text\; \backslash quad\; \backslash text\backslash ;i\; \backslash in\; I,\; \backslash ;\backslash pi\_i\backslash left(f\_\backslash right)\; :=\; \backslash left(\; \backslash pi\_i\backslash left(f\_a\backslash right)\; \backslash right)\_\; \backslash ;\backslash to\backslash ;\; \backslash pi\_i(L)\; =\; L\_i\backslash ;\; \backslash text\; \backslash ;X\_i.$$
Tychonoff's theorem and relation to the axiom of choice
If no $L\; \backslash in\; X$ is given but for every $i\; \backslash in\; I,$ there exists some $L\_i\; \backslash in\; X\_i$ such that $\backslash pi\_i\backslash left(f\_\backslash right)\; \backslash to\; L\_i$ in $X\_i$ then the tuple defined by $L\; :=\; \backslash left(L\_i\backslash right)\_$ will be a limit of $f\_$ in $X.$
However, the axiom of choice might be need to be assumed in order to conclude that this tuple $L$ exists; the axiom of choice is not needed in some situations, such as when $I$ is finite or when every $L\_i\; \backslash in\; X\_i$ is the limit of the net $\backslash pi\_i\backslash left(f\_\backslash right)$ (because then there is nothing to choose between), which happens for example, when every $X\_i$ is a Hausdorff space. If $I$ is infinite and $X\; =\; \backslash prod\_\; X\_j$ is not empty, then the axiom of choice would (in general) still be needed to conclude that the projections $\backslash pi\_i\; :\; X\; \backslash to\; X\_i$ are surjective maps.
The axiom of choice is equivalent to Tychonoff's theorem, 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.
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 $x\_\; =\; \backslash left(x\_a\backslash right)\_$ in $X$ is said to be or a given subset $S$ if for every $a\; \backslash in\; A$ there exists some $b\; \backslash in\; A$ such that $b\; \backslash geq\; a$ and $x\_b\; \backslash in\; S.$ A point $x\; \backslash in\; X$ is said to be an or of a net if for every neighborhood $U$ of $x,$ the net is frequently in $U.$ A point $x\; \backslash in\; X$ is a cluster point of a given net if and only if it has a subset that converges to $x.$ If $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is a net in $X$ then the set of all cluster points of $x\_$ in $X$ is equal to $$\backslash bigcap\_\; \backslash operatorname\_X\; \backslash left(x\_\backslash right)$$ where $x\_\; :=\; \backslash left\backslash $ for each $a\; \backslash in\; A.$ If $x\; \backslash in\; X$ 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 $S\; \backslash subseteq\; X,$ $x\_$ is eventually in $S$ or $x\_$ is eventually in the complement $X\; \backslash setminus\; S.$ 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 $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is an ultranet in $X$ and $f\; :\; X\; \backslash to\; Y$ is a function then $f\; \backslash circ\; x\_\; =\; \backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_$ is an ultranet in $Y.$ Given $x\; \backslash in\; X,$ an ultranet clusters at $x$ if and only it converges to $x.$Examples of limits of nets

* Limit of a sequence and limit of a function: see below. * Limits of nets of Riemann sums, in the definition of the Riemann integral. In this example, the directed set is the set of partitions of the interval of integration, partially ordered by inclusion.Examples

Sequence in a topological space

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

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

Consider a well-ordered set $;\; href="/html/ALL/s/,\_c.html"\; ;"title=",\; c">,\; c$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 $x\_\; =\; \backslash left(x\_a\backslash right)\_$ and $s\_\; =\; \backslash left(s\_i\backslash right)\_$ are nets then $s\_$ is called a or of $x\_$ if there exists an order-preserving map $h\; :\; I\; \backslash to\; A$ such that $h(I)$ is a cofinal subset of $A$ and $$s\_i\; =\; x\_\; \backslash quad\; \backslash text\; i\; \backslash in\; I.$$ The map $h\; :\; I\; \backslash to\; A$ is called and an if whenever $i\; \backslash leq\; j$ then $h(i)\; \backslash leq\; h(j).$ The set $h(I)$ being in $A$ means that for every $a\; \backslash in\; A,$ there exists some $b\; \backslash in\; h(I)$ such that $b\; \backslash geq\; a.$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. The following set of theorems and lemmas help cement that similarity:Characterizations of topological properties

Closed sets and closure A subset $S\; \backslash subseteq\; X$ is closed in $X$ if and only if every limit point of every convergent net in $S$ necessarily belongs to $S.$ Explicitly, a subset $S\; \backslash subseteq\; X$ is closed if and only if whenever $x\; \backslash in\; X$ and $s\_\; =\; \backslash left(s\_a\backslash right)\_$ is a net valued in $S$ (meaning that $s\_a\; \backslash in\; S$ for all $a\; \backslash in\; A$) such that $\backslash lim\_\; s\_\; \backslash to\; x$ in $X,$ then necessarily $x\; \backslash in\; S.$ More generally, if $S\; \backslash subseteq\; X$ is any subset then a point $x\; \backslash in\; X$ is in the closure of $S$ if and only if there exists a net $s\_\; =\; \backslash left(s\_a\backslash right)\_$ in $S$ with limit $x\; \backslash in\; X$ and such that $s\_a\; \backslash in\; S$ for every index $a\; \backslash in\; A.$ Open sets and characterizations of topologies A subset $S\; \backslash subseteq\; X$ is open if and only if no net in $X\; \backslash setminus\; S$ converges to a point of $S.$ Also, subset $S\; \backslash subseteq\; X$ 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 $f\; :\; X\; \backslash to\; Y$ between topological spaces is continuous at the point $x$ if and only if for every net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ in the domain $X,$ $$\backslash lim\_\; x\_\; \backslash to\; x\; \backslash text\; X\; \backslash quad\; \backslash text\; \backslash quad\; \backslash lim\_a\; f\backslash left(x\_a\backslash right)\; \backslash to\; f(x)\; \backslash text\; Y.$$ 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 (or not a sequential space). ($\backslash implies$) Let $f$ be continuous at point $x,$ and let $x\_\; =\; \backslash left(x\_a\backslash right)\_$ be a net such that $\backslash lim\_\; x\_\; \backslash to\; x.$ 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 of $V,$ which is denoted by $\backslash operatorname\; V,$ is an open neighborhood of $x,$ and consequently $x\_$ is eventually in $\backslash operatorname\; V.$ Therefore $\backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_$ is eventually in $f(\backslash operatorname\; V)$ and thus also eventually in $f(V)$ which is a subset of $U.$ Thus $\backslash lim\_\; \backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_\; \backslash to\; f(x),$ and this direction is proven. ($\backslash Longleftarrow$) Let $x$ be a point such that for every net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ such that$\backslash lim\_\; x\_\; \backslash to\; x,$ $\backslash lim\_\; \backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_\; \backslash to\; f(x).$ Now suppose that $f$ is not continuous at $x.$ Then there is aneighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

$U$ of $f(x)$ whose preimage under $f,$ $V,$ is not a neighborhood of $x.$ Because $f(x)\; \backslash in\; U,$ necessarily $x\; \backslash in\; V.$ Now the set of open neighborhoods of $x$ with the containment preorder is a directed set (since the intersection of every two such neighborhoods is an open neighborhood of $x$ as well).
We construct a net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ 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 $f\backslash left(x\_a\backslash right)$ 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 $b\; \backslash geq\; a\_0,$ the member of the directed set whose index is $b$ is contained within $W$; therefore $x\_b\; \backslash in\; W.$ Thus $\backslash lim\_\; x\_\; \backslash to\; x.$ and by our assumption $\backslash lim\_\; \backslash left(f\backslash left(x\_a\backslash right)\backslash right)\_\; \backslash to\; f(x).$
But $\backslash operatorname\; U$ is an open neighborhood of $f(x)$ and thus $f\backslash left(x\_a\backslash right)$ is eventually in $\backslash operatorname\; U$ and therefore also in $U,$ in contradiction to $f\backslash left(x\_a\backslash right)$ not being in $U$ for every $a.$
This is a contradiction so $f$ must be continuous at $x.$ This completes the proof.
A function $f\; :\; X\; \backslash to\; Y$ is continuous if and only if whenever $x\_\; \backslash to\; x$ in $X$ then $f\backslash left(x\_\backslash right)\; \backslash to\; f(x)$ in $Y.$
Compactness
A space $X$ is compact if and only if every net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ in $X$ has a subnet with a limit in $X.$ This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.
($\backslash implies$)
First, suppose that $X$ is compact. We will need the following observation (see finite intersection property). Let $I$ be any non-empty set and $\backslash left\backslash \_$ be a collection of closed subsets of $X$ such that $\backslash bigcap\_\; C\_i\; \backslash neq\; \backslash varnothing$ for each finite $J\; \backslash subseteq\; I.$ Then $\backslash bigcap\_\; C\_i\; \backslash neq\; \backslash varnothing$ as well. Otherwise, $\backslash left\backslash \_$ would be an open cover for $X$ with no finite subcover contrary to the compactness of $X.$
Let $x\_\; =\; \backslash left(x\_a\backslash right)\_$ be a net in $X$ directed by $A.$ For every $a\; \backslash in\; A$ define
$$E\_a\; \backslash triangleq\; \backslash left\backslash .$$
The collection $\backslash $ has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that
$$\backslash bigcap\_\; \backslash operatorname\; E\_a\; \backslash neq\; \backslash varnothing$$
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.
($\backslash Longleftarrow$)
Conversely, suppose that every net in $X$ has a convergent subnet. For the sake of contradiction, let $\backslash left\backslash $ be an open cover of $X$ with no finite subcover. Consider $D\; \backslash triangleq\; \backslash .$ Observe that $D$ is a directed set under inclusion and for each $C\backslash in\; D,$ there exists an $x\_C\; \backslash in\; X$ such that $x\_C\; \backslash notin\; U\_a$ for all $a\; \backslash in\; C.$ Consider the net $\backslash left(x\_C\backslash right)\_.$ This net cannot have a convergent subnet, because for each $x\; \backslash in\; X$ there exists $c\; \backslash in\; I$ such that $U\_c$ is a neighbourhood of $x$; however, for all $B\; \backslash supseteq\; \backslash ,$ we have that $x\_B\; \backslash notin\; U\_c.$ 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 $x\_\; =\; \backslash left(x\_a\backslash right)\_$ be a net in a topological space $X$ (where as usual $A$ automatically assumed to be a directed set) and also let $y\; \backslash in\; X.$ 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 $a\; \backslash in\; A$ is such that $x\_a\; \backslash in\; U.$ The map $h\; :\; B\; \backslash to\; A$ 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 $y\_\; =\; \backslash left(y\_b\backslash right)\_$ 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, 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 orpartial 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 m ...

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

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.. A net $x\_\; =\; \backslash left(x\_a\backslash right)\_$ is a if for every entourage $V$ there exists $c\; \backslash in\; A$ such that for all $a,\; b\; \backslash geq\; c,$ $\backslash left(x\_a,\; x\_b\backslash right)$ is a member of $V.$ More generally, in a Cauchy space, a net $x\_$ is Cauchy if the filter generated by the net is a Cauchy filter. A topological vector space (TVS) is called if every Cauchy net converges to some point. A normed space, which is a special type of topological vector space, is a complete TVS (equivalently, a Banach space) 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 $\backslash left(x\_a\backslash right)\_$ in $X$ induces a filter base of tails $\backslash left\backslash $ where the filter in $X$ generated by this filter base is called the net's . This correspondence allows for any theorem that can be proven with one concept to be proven with the other. For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases. Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts. He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis, while filters are most useful inalgebraic topology
Algebraic topology is a branch of 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. Th ...

. In any case, he shows how the two can be used in combination to prove various theorems in general topology.
Limit superior

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

* * * *Citations

References

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