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 X and Y:

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

While it is necessarily true that condition 1 implies condition 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 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 behaviour. 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 only 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 net where if this function takes values in some set then it may also be referred to as a net in . Elements of a net's domain are called its indices. Explicitly, a net in is a function of the form where is some directed set. 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 (i.e. that is greater than or equal to each of them); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. The natural 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 (non-strict) 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 (this subscript notation being taken from sequences) instead of the usual parentheses notation that is used with functions more generally (e.g. in set theory, not denotes a function's value). As in the field of algebraic topology, the filled disk or "bullet" denotes the location where arguments to the net (i.e. 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 not necessarily a sequence.[4] For an example, let be directed by the usual order and define by letting be the ceiling of so that is an order morphism (that is to say, a non-decreasing function) whose image is a cofinal subset of its codomain. Let be any sequence (say a constant sequence, for instance) and let for every Then holds for every which 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). Furthermore, the sequence is also a subnet of since the inclusion map (that sends ) is an order morphism whose image is a cofinal subset of its codomain and holds for all Thus and are (simultaneously) subnets of each another.

Limits of nets

If is a net from a directed set into and if is a subset of then is said to be eventually in (or residually in ) if there exists some such that for every with the point A point is called a limit point or limit 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 converge to/towards and to have as a limit.

Intuitively, convergence of this 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

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 [5] 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.[5] Some authors instead use the notation "" to mean without 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 solely 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.

Symbolically, suppose that the Cartesian product

of the spaces is endowed with the product topology and that for every index the canonical projection to is denoted 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,

If given then

  if and only if   for every

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.

Cluster points of a net

A net in is said to be frequently in or cofinally in a given subset if for every there exists some such that and [4] 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]

If is a net in a subset and if is a cluster point of then

If is a net in then the set of all cluster points of in is equal to[4]

where for each If is a cluster point of some subnet of then is also a cluster point of [4]

If a net converges to a point then is necessarily a cluster point of that net.[4] The converse is not guaranteed in general. That is, it is possible for to be a cluster point of a net but for to not converge to However, if clusters at then there exists a subnet of that converges to This subnet can be explicitly constructed from and the neighborhood filter at as follows: make

into a directed set by declaring that
then and is a subnet of since the map
is a monotone function whose image is a cofinal subset of and

Thus, a point is a cluster point of a given net if and only if it has a subnet that converges to [4]

Ultranets

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. 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]

Examples of limits of nets

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

Consider a function from a metric space to a topological space and a point We direct the set reversely according to distance from that is, the relation is "has at least the same distance to as", so that "large enough" with respect to the relation means "close enough to ". The function is a net in defined on

The net is eventually in a subset of if there exists some such that for every with 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 with such that is in

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.

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

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 [6] 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 the point if and only if for every net in the domain [4]

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 is not a first-countable 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 if then 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

It follows that a function is continuous if and only if whenever in then in

In addition, a function is continuous if and only if whenever a net clusters at a point in then clusters at in

Proof

() Assume is continuous at and is a net that clusters at Then there exists a subnet of that converges to Continuity implies that Because is a subnet of this implies that is a cluster point of

() Assume that whenever a net clusters at a point then clusters at in Let By the closure characterization of continuity, it remains to show that so let Then there exists a net in such that in which implies that clusters at The assumption guarantees that clusters at in Thus has a subset (specifically, ) that clusters at in which implies that

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 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.

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.

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 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.

If and is an ultranet on then is an ultranet on

Cauchy nets

A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.[7]

A net is a Cauchy net if for every entourage there exists such that for all is a member of [7][8] 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.

Relation to filters

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.[9] 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).[10] 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.[10] 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.[10] 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.[11][12][13] Some authors work even with more general structures than the real line, like complete lattices.[14]

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

  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 c d e f g h i j k l Willard 2004, pp. 73–77.
  5. ^ a b Kelley 1975, pp. 65–72.
  6. ^ Howes 1995, pp. 83–92.
  7. ^ a b Willard, Stephen (2012), General Topology, Dover Books on Mathematics, Courier Dover Publications, p. 260, ISBN 9780486131788.
  8. ^ Joshi, K. D. (1983), Introduction to General Topology, New Age International, p. 356, ISBN 9780852264447.
  9. ^ "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)
  10. ^ a b c R. G. Bartle, Nets and Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
  11. ^ Aliprantis-Border, p. 32
  12. ^ Megginson, p. 217, p. 221, Exercises 2.53–2.55
  13. ^ Beer, p. 2
  14. ^ Schechter, Sections 7.43–7.47

References