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 f between topological spaces X and Y:
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 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.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
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.
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 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 (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.
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.
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)
in which case, this net is then also said to converges 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 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 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 it not being true that
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.
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 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 is depends solely on this topological subspace consisting of and the image of (i.e. the points of) the net
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 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.
Let be a net in based on the directed set and let be a subset of then is said to be frequently in (or cofinally in) if for every there exists some such that and
A point is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood of the net is frequently in
A net in set is called universal or an ultranet if for every subset is eventually in or is eventually in Ultranets are closely related to ultrafilters.
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.
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
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.
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:
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.
|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
|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 above property, 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.
A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces.
A net is a Cauchy net 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 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.
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. 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. 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.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). For instance, any net
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 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.
In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be discontinuous. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it.
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M.
In mathematics, a filter is a special subset of a partially ordered set. Filters appear in order and lattice theory, but can also be found in topology, from where they originate. The dual notion of a filter is an order ideal.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. They can be thought of in a similar fashion for a function. For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant. Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
In mathematics, a limit point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. In contrast to sets, for a sequence, net, or filter, the term "limit point" is not synonymous with a "cluster/accumulation point". The similarly named notions of a limit point of a filter, a limit point of a sequence, or a limit point of a net; each of these respectively refers to a point that a filter, sequence, or net converges to.
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.
In topology and related areas of mathematics, a subnet is a generalization of the concept of subsequence to the case of nets. The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
In mathematics, a càdlàg, RCLL, or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space.
In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.
In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.
The strongest locally convex topological vector space (TVS) topology on the tensor product of two locally convex TVSs, making the canonical map continuous is called the projective topology or the π-topology. When is endowed with this topology then it is denoted by and called the projective tensor product of and
In functional analysis and related areas of mathematics, a metrizable topological vector space (TVS) is a TVS whose topology is induced by a metric. An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.
In mathematics, particularly in functional analysis and topology, the closed graph theorem is a fundamental result stating that a linear operator with a closed graph will, under certain conditions, be continuous. The original result has been generalized many times so there are now many theorems referred to as "closed graph theorems."
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain properties relating elements of X with the family of filters on X. Convergence spaces generalize the notions of convergence that are found in point-set topology, including metric convergence and uniform convergence. Every topological space gives rise to a canonical convergence but there are convergences, known as non-topological convergences, that do not arise from any topological space. Examples of convergences that are in general non-topological include convergence in measure and almost everywhere convergence. Many topological properties have generalizations to convergence spaces.