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 :

The map  is continuous in the topological sense;
Given any point  in  and any sequence in  converging to  the composition of  with this sequence converges to  (continuous in the sequential sense).

While condition 1 always guarantees condition 2, the converse is not necessarily true if the topological spaces are not both first-countable. 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, 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.

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.

Definitions

Any function whose domain is a directed set is called a . If this function takes values in some set  then it may also be referred to as a  Explicitly, a  is a function of the form  where  is some directed set. Elements of a net's domain are called its . 
A  is a non-empty set  together with a preorder, typically automatically assumed to be denoted by  (unless indicated otherwise), with the property that it is also () , which means that for any  there exists some  such that  and  
In words, this property means that given any two elements (of ), there is always some element that is "above" both of them (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  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 (non-strict) preorder ; in particular, if a directed set  has a greatest element  then there does  exist any  such that  (in contrast, there  exists some  such that ).

Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences. 
A net in  may be denoted by  where unless there is reason to think otherwise, it should automatically be assumed that the set  is directed and that its associated preorder is denoted by  
However, notation for nets varies with some authors using, for instance, angled brackets  instead of parentheses. 
A net in  may also be written as  which expresses the fact that this net  is a function  whose value at an element  in its domain is denoted by  instead of the usual parentheses notation  that is typically used with functions (this subscript notation being taken from sequences). As in the field of algebraic topology, 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.

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 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  necessarily a sequence. 
For an example, let  and let  for every  so that  is the constant zero sequence. 
Let  be directed by the usual order  and let  for each  
Define  by letting  be the ceiling 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).

Limits of nets

A net  is said to be  or   a set  if there exists some  such that for every  with  the point  
And it is said to be  or   if for every  there exists some  such that  and  
A point is called a  (respectively, ) of a net if that net is eventually (respectively, cofinally) in every neighborhood of that point.

Explicitly, a point  is said to be an  or  of a net if for every neighborhood  of  the net is frequently in  

A point  is called a  or  of the net  in  if (and only if)
for every open neighborhood  of  the net  is eventually in 
in which case, this net is then also said to  and to .

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:

where if the topological space  is clear from context then the words "in " may be omitted.

If  in  and if this limit in  is unique (uniqueness in  means that if  is such that  then necessarily ) then this fact may be indicated by writing

where an equals sign is used in place of the arrow  In a Hausdorff space, every net has at most one limit so the limit of a convergent net in a Hausdorff space is always unique. 
Some authors instead use the notation "" to mean  with also requiring that the limit be unique; however, if this notation is defined in this way then the equals sign  is no longer guaranteed to denote a transitive relationship and so no longer denotes equality. Specifically, without the uniqueness requirement, if  are distinct and if each is also a limit of  in  then  and  could be written (using the equals sign ) despite  being false.

Bases and subbases

Given a subbase  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  on this topological subspace  consisting of  and the image of (that is, the points of) the net

Limits in a Cartesian product

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 

with the product topology, and that for every index  denote the canonical projection to  by

Let  be a net in  directed by  and for every index  let 

denote the result of "plugging  into ", which results in the net  
It is sometimes useful to think of this definition in terms of function composition: the net  is equal to the composition of the net  with the projection  that is, 

For any given point  the net  converges to  in the product space  if and only if for every index   converges to  in  
And whenever the net  clusters at  in  then  clusters at  for every index  However, the converse does not hold in general. 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  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.

Cluster points of a net

A point  is a cluster point of a given net if and only if it has a subset that converges to 
If  is a net in  then the set of all cluster points of  in  is equal to

where  for each  
If  is a cluster point of some subnet of  then  is also a cluster point of

Ultranets

A net  in set  is called a  or an  if for every subset   is eventually in  or  is eventually in the complement  
Ultranets are closely related to 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  is an ultranet in  and  is a function then  is an ultranet in  

Given  an ultranet clusters at  if and only it converges to

Examples of limits of nets

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,  This will be proven by constructing a net in  that converges to  However, there does not exist any  in  that converges to  which makes this one instance where (non-sequence) 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

Examples

Sequence in a topological space

A sequence  in a topological space  can be considered a net in  defined on 

The net is eventually in a subset  of  if there exists an  such that for every integer  the point  is in 

So  if and only if for every neighborhood  of  the net is eventually in 

The net is frequently in a subset  of  if and only if for every  there exists some integer  such that  that is, if and only if infinitely many elements of the sequence are in  Thus a point  is a cluster point of the net if and only if every neighborhood  of  contains infinitely many elements of the sequence.

Function from a metric space to a topological space

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  

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 ). 

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

Function from a well-ordered set to a topological space

Consider a well-ordered 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 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  and  are nets then  is called a  or  of  if there exists an order-preserving map  such that  is a cofinal subset of  and 
 
The map  is called  and an  if whenever  then  
The set  being  in  means that for every  there exists some  such that

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  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  

Open sets and characterizations of topologies

A subset  is open if and only if no net in  converges to a point of  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  
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 first-countable space (or not a sequential space).

() 
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.

() 
First, suppose that  is compact. We will need the following observation (see finite intersection property). Let  be any non-empty 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 non-empty 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.

Cluster and limit points

The set of cluster points of a net is equal to the set of limits of its convergent subnets.

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.

Other properties

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  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.

Cauchy nets

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.

A net  is a  if for every entourage  there exists  such that for all   is a member of  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  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). 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 . 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 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

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.

For a net  put

Limit superior of a net of real numbers has many properties analogous to the case of sequences. For example,

where equality holds whenever one of the nets is convergent.

See also

Citations

References

 
 
 
  
  
 
 
 
  
  

Articles containing proofs
General topology