Net (mathematics)


In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize the concept of a sequence in a metric space. 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 in one-to-one correspondence with filters.

History edit

The concept of a net was first introduced by E. H. Moore and Herman L. Smith in 1922.[1] The term "net" was coined by John L. Kelley.[2][3]

The related concept of a filter was developed in 1937 by Henri Cartan.

Definitions edit

A directed set 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 (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 (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are not required to be total orders or even partial orders. A directed set may have greatest elements and/or maximal elements. In this case, the conditions   and   cannot be replaced by the strict inequalities   and  , since the strict inequalities cannot be satisfied if a or b is maximal.

A net in  , denoted  , is a function of the form   whose domain   is some directed set, and whose values are  . Elements of a net's domain are called its indices. When the set   is clear from context it is simply called a net, and one assumes   is a directed set with preorder   Notation for nets varies, for example using angled brackets  . As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index  .

Limits of nets edit

A net   is said to be eventually or residually in a set   if there exists some   such that for every   with   the point   A point   is called a limit point or limit of the net   in   whenever:

for every open neighborhood   of   the net   is eventually in  ,

expressed equivalently as: the net converges to/towards   or has   as a limit; and variously denoted as:

If   is clear from context, it may be omitted from the notation.

If   and this limit is unique (i.e.   only for  ) then one writes:

using the equal sign in place of the arrow  [4] In a Hausdorff space, every net has at most one limit, and the limit of a convergent net is always unique.[4] Some authors do not distinguish between the notations   and  , but this can lead to ambiguities if the ambient space   is not Hausdorff.

Cluster points of nets edit

A net   is said to be frequently or cofinally in   if for every   there exists some   such that   and  [5] A point   is said to be an accumulation point or cluster point of a net if for every neighborhood   of   the net is frequently/cofinally in  [5] In fact,   is a cluster point if and only if it has a subset that converges to  [6] The set   of all cluster points of   in   is equal to   for each  , where  .

Subnets edit

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,[7] which is as follows: If   and   are nets then   is called a subnet or Willard-subnet[7] of   if there exists an order-preserving map   such that   is a cofinal subset of   and

The map   is called order-preserving and an order homomorphism if whenever   then   The set   being cofinal in   means that for every   there exists some   such that  

If   is a cluster point of some subnet of   then   is also a cluster point of  [6]

Ultranets edit

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  [5]

Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet.[8] Assuming the axiom of choice, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.[5] If   is an ultranet in   and   is a function then   is an ultranet in  [5]

Given   an ultranet clusters at   if and only it converges to  [5]

Cauchy nets edit

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.[9]

A net   is a Cauchy net if for every entourage   there exists   such that for all     is a member of  [9][10] 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 non-normable) topological vector spaces.

Characterizations of topological properties edit

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 edit

A subset   is closed in   if and only if every limit point in   of a net in   necessarily lies in  . Explicitly, this means that if   is a net with   for all  , and   in   then  

More generally, if   is any subset, the closure of   is the set of points   with   for some net   in  .[6]

Open sets and characterizations of topologies edit

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 edit

A function   between topological spaces is continuous at a point   if and only if for every net   in the domain,   in   implies   in  [6] Briefly, a function   is continuous if and only if   in   implies   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 edit

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 edit

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 edit

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.

Relation to filters edit

A filter is a related 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.[12] More specifically, every filter base induces an associated net using the filter's pointed sets, and convergence of the filter base implies convergence of the associated net. Similarly, 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. Convergence of the net implies convergence of the eventuality filter.[13] This correspondence allows for any theorem that can be proven with one concept to be proven with the other.[13] 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.[13] 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.

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.

As generalization of sequences edit

Every non-empty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers   together with the usual integer comparison   preorder form the archetypical example of a directed set. A sequence is a function on the natural numbers, so every sequence   in a topological space   can be considered a net in   defined on   Conversely, any 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: rather than being defined on a countable linearly ordered set ( ), a net is defined on an arbitrary directed set. Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences. For example, the subscript notation   is taken from sequences.

Similarly, every limit of a sequence and limit of a function can be interpreted as a limit of a net. Specifically, 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.

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  :

  1. The map   is continuous in the topological sense;
  2. 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. The spaces for which the two conditions are equivalent are called sequential spaces. All first-countable spaces, including metric spaces, are sequential spaces, but not all topological spaces are sequential. Nets generalize the notion of a sequence so that condition 2 reads as follows:

  1. Given any point   in   and any net in   converging to   the composition of   with this net converges to   (continuous in the net sense).

With this change, the conditions become equivalent for all maps of topological spaces, including 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.

For an example where sequences do not suffice, 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,  [8] This will be proven by constructing a net in   that converges to   However, there does not exist any sequence in   that converges to  [14] 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  

More generally, a subnet of a sequence is not necessarily a sequence.[5][a] Moreso, a subnet of a sequence may be a sequence, but not a subsequence.[b] But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences. Specifically, for a first-countable space, the net   induces the sequence   where   is defined as the   smallest value in   – that is, let   and let   for every integer  .

Examples edit

Subspace topology edit

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  

Neighborhood systems edit

Intuitively, convergence of a net   means that the values   come and stay as close as we want to   for large enough   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  . Therefore, in this neighborhood system of a point  ,   does indeed converge to   according to the definition of net convergence.

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  

Limits in a Cartesian product edit

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  [15] And whenever the net   clusters at   in   then   clusters at   for every index  [8] However, the converse does not hold in general.[8] 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 edit

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.

Limit superior/inferior edit

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,

where equality holds whenever one of the nets is convergent.

Riemann integral edit

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.

Metric spaces edit

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  

If   has at least two points, then we can fix a 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  [8]

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  ).[8]

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 edit

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.

See also edit

Notes edit

  1. ^ 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).
  2. ^ The sequence   is not a subsequence of  , although it is a subnet, because the map   defined by   is an order-preserving map whose image is   and satisfies   for all   Indeed, this is because   and   for every   in other words, when considered as functions on   the sequence   is just the identity map on   while  

Citations edit

  1. ^ Moore, E. H.; Smith, H. L. (1922). "A General Theory of Limits". American Journal of Mathematics. 44 (2): 102–121. doi:10.2307/2370388. JSTOR 2370388.
  2. ^ (Sundström 2010, p. 16n)
  3. ^ Megginson, p. 143
  4. ^ a b Kelley 1975, pp. 65–72.
  5. ^ a b c d e f g Willard 2004, pp. 73–77.
  6. ^ a b c d Willard 2004, p. 75.
  7. ^ a b Schechter 1996, pp. 157–168.
  8. ^ a b c d e f Willard 2004, p. 77.
  9. ^ a b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, p. 260, ISBN 9780486131788.
  10. ^ Joshi, K. D. (1983), Introduction to General Topology, New Age International, p. 356, ISBN 9780852264447.
  11. ^ Howes 1995, pp. 83–92.
  12. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-04-24. Retrieved 2013-01-15.{{cite web}}: CS1 maint: archived copy as title (link)
  13. ^ a b c R. G. Bartle, Nets and Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
  14. ^ Willard 2004, pp. 71–72.
  15. ^ Willard 2004, p. 76.
  16. ^ Aliprantis-Border, p. 32
  17. ^ Megginson, p. 217, p. 221, Exercises 2.53–2.55
  18. ^ Beer, p. 2
  19. ^ Schechter, Sections 7.43–7.47

References edit