In 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 essence, a sequence is a function whose domain is the natural numbers. 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 between topological spaces and :
While condition 1 always guarantees condition 2, the converse is not necessarily true if the topological spaces are not both firstcountable. In particular, the two conditions are equivalent for metric spaces. The spaces for which the converse holds are the sequential spaces.
The concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922,^{[1]} 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 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.^{[2]}^{[3]}
Nets are one of the many tools used in topology to generalize certain concepts that may not be general enough in the context of metric spaces. A related notion, that of the filter, was developed in 1937 by Henri Cartan.
Any function whose domain is a directed set is called a net. If this function takes values in some set then it may also be referred to as a net in Explicitly, a net in is a function of the form where is some directed set. Elements of a net's domain are called its indices. A directed set is a nonempty set together with a preorder, typically automatically assumed to be denoted by (unless indicated otherwise), with the property that it is also (upward) directed, 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 (that is, 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 numbers together with the usual integer comparison preorder form the archetypical example of a directed set. Indeed, a net whose domain is the natural numbers is a sequence 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 not required to be total orders or even partial orders. Moreover, directed sets are allowed to have greatest elements and/or maximal elements, which is the reason why when using nets, caution is advised when using the induced strict preorder instead of the original (nonstrict) preorder ; in particular, if a directed set has a greatest element then there does not exist any such that (in contrast, there always 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, 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 and Topology, 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 to a directed subset of see the linked page for a definition.
Every nonempty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers 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 neighbourhoods containing Then is a directed set, where the direction is given by reverse inclusion, so that if and only if 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 not necessarily a sequence.^{[4]} 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 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).
A net is said to be eventually or residually in a set if there exists some such that for every with the point And it is said to be frequently or cofinally in if for every there exists some such that and ^{[4]} A point is called a limit point (respectively, cluster point) of a net if that net is eventually (respectively, cofinally) in every neighborhood of that point.
Explicitly, a point is said to be an accumulation point or cluster point of a net if for every neighborhood of the net is frequently in ^{[4]}
A point is called a limit point or limit of the net in if (and only if)
in which case, this net is then also said to converge to/towards and to have as a limit.
Intuitively, convergence of a 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 of a point does indeed converge to according to this definition.
Notation for limits
If the net converges in to a point then this fact may be expressed by writing any of the following:
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
Bases and subbases
Given a subbase 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 (or a pseudometric space) and is endowed with the metric topology. If is a point and is a net, then in if and only if in where is a net of real numbers. 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 is a normed space (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 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 solely on this topological subspace consisting of and the image of (that is, the points of) the net
A net in the product space has a limit if and only if each projection has a limit.
Explicitly, let be topological spaces, endow their Cartesian product
Let be a net in directed by and for every index let
For any given point the net converges to in the product space if and only if for every index converges to in ^{[6]} And whenever the net clusters at in then clusters at for every index ^{[7]} However, the converse does not hold in general.^{[7]} For example, suppose and let denote the sequence that alternates between and Then and are cluster points of both and in but is not a cluster point of since the open ball of radius centered at does not contain even a single point
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 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 unique limit of the net (because then there is nothing to choose between), which happens for example, when every is a Hausdorff space. 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 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.
A point is a cluster point of a given net if and only if it has a subset that converges to ^{[8]} If is a net in then the set of all cluster points of in is equal to^{[7]}
A net in set is called a universal net or an ultranet if for every subset is eventually in or is eventually in the complement ^{[4]} Ultranets are closely related to ultrafilters.
Every constant net is an ultranet. Every subnet of an ultranet is an ultranet.^{[7]} Every net has some subnet that is an ultranet.^{[4]} If is an ultranet in and is a function then is an ultranet in ^{[4]}
Given an ultranet clusters at if and only it converges to ^{[4]}
Every limit of a sequence and limit of a function can be interpreted as a limit of a net (as described below).
The definition of the value of a Riemann integral can be interpreted as a limit of a net of Riemann sums where the net's directed set is the set of all partitions of the interval of integration, partially ordered by inclusion.
Interpret the set of all functions with prototype as the Cartesian product (by identifying a function with the tuple and conversely) and endow it with the product topology. This (product) topology on is identical to the topology of pointwise convergence. Let denote the set of all functions that are equal to everywhere except for at most finitely many points (that is, such that the set is finite). Then the constant function belongs to the closure of in that is, ^{[7]} This will be proven by constructing a net in that converges to However, there does not exist any sequence in that converges to ^{[9]} which makes this one instance where (nonsequence) nets must be used because sequences alone can not reach the desired conclusion. Compare elements of pointwise in the usual way by declaring that if and only if for all This pointwise comparison is a partial order that makes a directed set since given any their pointwise minimum belongs to and satisfies and This partial order turns the identity map (defined by ) into an valued net. This net converges pointwise to in which implies that belongs to the closure of in
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.
Fix a point in a metric space that has at least two point (such as with the Euclidean metric with being the origin, for example) and direct the set reversely according to distance from by declaring that if and only if In other words, the relation is "has at least the same distance to as", so that "large enough" with respect to this relation means "close enough to ". Given any function with domain its restriction to can be canonically interpreted as a net directed by ^{[7]}
A net is eventually in a subset of a topological space if and only if there exists some such that for every satisfying the point is in Such a net converges in to a given point if and only if in the usual sense (meaning that for every neighborhood of is eventually in ).^{[7]}
The net is frequently in a subset of if and only if for every there exists some with such that is in Consequently, a point is a cluster point of the net if and only if for every neighborhood of the net is frequently in
Consider a wellordered set with limit point and a function from to a topological space This function is a net on
It is eventually in a subset of if there exists an such that for every 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 such that
A point is a cluster point of the net if and only if for every neighborhood of the net is frequently in
The first example is a special case of this with
See also ordinalindexed sequence.
Main article: Subnet (mathematics) 
See also: Filters in topology § Subnets 
The analogue of "subsequence" for nets is the notion of a "subnet". There are several different nonequivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,^{[10]} which is as follows: If and are nets then is called a subnet or Willardsubnet^{[10]} of if there exists an orderpreserving map such that is a cofinal subset of and
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:
Closed sets and closure
A subset is closed in if and only if every limit point of every convergent net in necessarily belongs to Explicitly, a subset is closed if and only if whenever and is a net valued in (meaning that for all ) such that in then necessarily
More generally, if is any subset then a point is in the closure of if and only if there exists a net in with limit and such that for every index ^{[8]}
Open sets and characterizations of topologies
See also: Characterizations of the category of topological spaces § Convergent net characterization 
A subset is open if and only if no net in converges to a point of ^{[11]} Also, subset is open if and only if every net converging to an element of is eventually contained in 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 between topological spaces is continuous at a given point if and only if for every net in its domain, if in then in ^{[8]} Said more succinctly, a function is continuous if and only if whenever in then in In general, this 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 is not a firstcountable space (or not a sequential space).
Proof


() Let be continuous at point and let be a net such that Then for every open neighborhood of its preimage under is a neighborhood of (by the continuity of at ). Thus the interior of which is denoted by is an open neighborhood of and consequently is eventually in Therefore is eventually in and thus also eventually in which is a subset of Thus and this direction is proven. () Let be a point such that for every net such that Now suppose that is not continuous at Then there is a neighborhood of whose preimage under is not a neighborhood of Because necessarily Now the set of open neighborhoods of with the containment preorder is a directed set (since the intersection of every two such neighborhoods is an open neighborhood of as well). We construct a net such that for every open neighborhood of whose index is is a point in this neighborhood that is not in ; that there is always such a point follows from the fact that no open neighborhood of is included in (because by assumption, is not a neighborhood of ). It follows that is not in Now, for every open neighborhood of this neighborhood is a member of the directed set whose index we denote For every the member of the directed set whose index is is contained within ; therefore Thus and by our assumption But is an open neighborhood of and thus is eventually in and therefore also in in contradiction to not being in for every This is a contradiction so must be continuous at This completes the proof. 
Compactness
A space is compact if and only if every net in has a subnet with a limit in This can be seen as a generalization of the Bolzano–Weierstrass theorem and Heine–Borel theorem.
Proof


() First, suppose that is compact. We will need the following observation (see finite intersection property). Let be any nonempty set and be a collection of closed subsets of such that for each finite Then as well. Otherwise, would be an open cover for with no finite subcover contrary to the compactness of Let be a net in directed by For every define
The collection has the property that every finite subcollection has nonempty intersection. Thus, by the remark above, we have that
and this is precisely the set of cluster points of By the proof given in the next section, it is equal to the set of limits of convergent subnets of Thus has a convergent subnet.
() Conversely, suppose that every net in has a convergent subnet. For the sake of contradiction, let be an open cover of with no finite subcover. Consider Observe that is a directed set under inclusion and for each there exists an such that for all Consider the net This net cannot have a convergent subnet, because for each there exists such that is a neighbourhood of ; however, for all we have that This is a contradiction and completes the proof. 
The set of cluster points of a net is equal to the set of limits of its convergent subnets.
Proof


Let be a net in a topological space (where as usual automatically assumed to be a directed set) and also let If is a limit of a subnet of then is a cluster point of Conversely, assume that is a cluster point of Let be the set of pairs where is an open neighborhood of in and is such that The map mapping to is then cofinal. Moreover, giving the product order (the neighborhoods of are ordered by inclusion) makes it a directed set, and the net defined by converges to 
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.
In general, a net in a space can have more than one limit, but if is a Hausdorff space, the limit of a net, if it exists, is unique. Conversely, if is not Hausdorff, then there exists a net on with two distinct limits. Thus the uniqueness of the limit is equivalent 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 or partial order may have distinct limit points even in a Hausdorff space.
A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.^{[12]}
A net is a Cauchy net if for every entourage there exists such that for all is a member of ^{[12]}^{[13]} More generally, in a Cauchy space, a net is Cauchy if the filter generated by the net is a Cauchy filter.
A topological vector space (TVS) is called complete 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 sequential completeness). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly nonnormable) topological vector spaces.
See also: Filters in topology § Filters and nets 
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.^{[14]} More specifically, for every filter base an associated net 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).^{[15]} For instance, any net in induces a filter base of tails where the filter in generated by this filter base is called the net's eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other.^{[15]} 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.^{[15]} 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 in algebraic topology. In any case, he shows how the two can be used in combination to prove various theorems in general topology.
Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences.^{[16]}^{[17]}^{[18]} Some authors work even with more general structures than the real line, like complete lattices.^{[19]}
For a net put
Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,