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

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

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 (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. Importantly though, 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 ).

A net in is a function of the form whose domain is some directed set. Elements of a net's domain are called its indices. When the set is clear from context it is simply called a net. 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 Notation for nets varies, with some authors using, for instance, angled brackets instead of parentheses. The function's value at an element in its domain is denoted by instead of the usual parentheses notation that is typically used with functions. 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.

Limits of nets

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 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. 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 [4] 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.[4] 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.

Cluster points of nets

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] A point is a cluster point of a given net if and only if it has a subset that converges to [6] If is a net in then the set of all cluster points of in is equal to[7]

where for each

Subnets

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 non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,[8] which is as follows: If and are nets then is called a subnet or Willard-subnet[8] 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

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.[7] 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

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

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 [6]

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 [6] 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).

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 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.

Relation to filters

See also: Filters in topology § Filters and nets

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

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, [7] 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

Subspace topology

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

Main article: Neighborhood system

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

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 [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.

Limit superior/inferior

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

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

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 [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

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.

See also

Notes

  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

  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 c d e f g Willard 2004, p. 77.
  8. ^ a b Schechter 1996, pp. 157–168.
  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