Net (mathematics)
Encyclopedia
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
with domain the natural number
s, and in the context of topology, the range of this function is usually any topological space. However, in the context of topology, sequences do not fully encode all information about a function between topological spaces. In particular, the following two conditions are not equivalent in general for a map ƒ between topological spaces X and Y:
It is true however, that condition 1 implies condition 2, in the context of all spaces. The difficulty encountered when attempting to prove that condition 2 implies condition 1 lies in the fact that topological spaces are, in general, not first-countable. If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent. In particular, the two conditions are equivalent for metric space
s.
The purpose of the concept of a net, first introduced by E. H. Moore
and H. L. Smith in 1922, is to generalize the notion of a sequence so as to confirm the equivalence of the conditions (with "sequence" being replaced by "net" in condition 2). In particular, rather than being defined on a countable
linearly ordered set, a net is defined on an arbitrary directed set
. In particular, this allows theorems similar to that asserting the equivalence of condition 1 and condition 2, to hold in the context of topological spaces which 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 of the fact that collections of open sets in topological spaces are much like directed set
s in behaviour. The term "net" was coined by Kelley
.
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 space
s. A related notion, that of the filter
, was developed in 1937 by Henri Cartan
.
from some directed set
A to X.
If A is a directed set, we often write a net from A to X in the form (xα), which expresses the fact that the element α in A is mapped to the element xα in X.
s 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 x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if
S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.
If (xα) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write
if and only if
Intuitively, this means that the values xα come and stay as close as we want to x for large enough α.
Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.
Given a base for the topology
, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that (xα) is eventually in all members of the base containing this putative limit.
A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.
A net φ on set X is called universal, or an ultranet if for every subset A of X, either φ is eventually in A or φ is eventually in X-A.
A sequence (a1, a2, ...) in a topological space V can be considered a net in V defined on N.
The net is eventually in a subset Y of V if there exists an N in N such that for every n ≥ N, the point an is in Y.
We have limx → c an = L if and only if for every neighborhood Y of L, the net is eventually in Y.
The net is frequently in a subset Y of V if and only if for every N in N there exists some n ≥ N such that an is in Y, that is, if and only if infinitely many elements of the sequence are in Y. Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.
Function from a metric space to a topological space:
Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M\{c} reversely according to distance from c, that is, the relation is "has at least the same distance to c as", so that "large enough" with respect to the relation means "close enough to c". The function ƒ is a net in V defined on M\{c}.
The net ƒ is eventually in a subset Y of V if there exists an a in M\{c} such that for every x in M\{c} with d(x,c) ≤ d(a,c), the point f(x) is in Y.
We have limx → c ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.
The net ƒ is frequently in a subset Y of V if and only if for every a in M\{c} there exists some x in M\{c} with d(x,c) ≤ d(a,c) such that f(x) is in Y.
A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.
Function from a well-ordered set to a topological space:
Consider a well-ordered set
[0, c] with limit point c, and a function ƒ from [0, c) to a topological space V. This function is a net on [0, c).
It is eventually in a subset Y of V if there exists an a in [0, c) such that for every x ≥ a, the point f(x) is in Y.
We have limx → c ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.
The net ƒ is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y.
A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.
The first example is a special case of this with c = ω.
See also ordinal-indexed sequence.
. The following set of theorems and lemmas help cement that similarity:
or uniform space
, one can speak of Cauchy net
s in much the same way as Cauchy sequence
s.
The concept even generalises to Cauchy space
s.
The theory of filter
s also provides a definition of convergence in general topological spaces.
For a net
we put
Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g.
where equality holds whenever one of the nets is convergent.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, more specifically in general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
. In essence, a sequence is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
with domain the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, and in the context of topology, the range of this function is usually any topological space. However, in the context of topology, sequences do not fully encode all information about a function between topological spaces. In particular, the following two conditions are not equivalent in general for a map ƒ between topological spaces X and Y:
- The map ƒ is continuousContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
- Given any point x in X, and any sequence in X converging to x, the composition of ƒ with this sequence converges to ƒ(x)
It is true however, that condition 1 implies condition 2, in the context of all spaces. The difficulty encountered when attempting to prove that condition 2 implies condition 1 lies in the fact that topological spaces are, in general, not first-countable. If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent. In particular, the two conditions are equivalent for metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s.
The purpose of the concept of a net, first introduced by E. H. Moore
E. H. Moore
Eliakim Hastings Moore was an American mathematician.-Life:Moore, the son of a Methodist minister and grandson of US Congressman Eliakim H. Moore, discovered mathematics through a summer job at the Cincinnati Observatory while in high school. He learned mathematics at Yale University, where he was...
and H. L. Smith in 1922, is to generalize the notion of a sequence so as to confirm the equivalence of the conditions (with "sequence" being replaced by "net" in condition 2). In particular, rather than being defined on a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
linearly ordered set, a net is defined on an arbitrary directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
. In particular, this allows theorems similar to that asserting the equivalence of condition 1 and condition 2, to hold in the context of topological spaces which 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 of the fact that collections of open sets in topological spaces are much like directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
s in behaviour. The term "net" was coined by Kelley
John L. Kelley
John Leroy Kelley was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis....
.
Nets are one of the many tools used in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
to generalize certain concepts that may only be general enough in the context of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s. A related notion, that of the filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
, was developed in 1937 by Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
.
Definition
If X is a topological space, a net in X is a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from some directed set
Directed set
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ , with the additional property that every pair of elements has an upper bound: In other words, for any a and b in A there must exist a c in A with a ≤ c and b ≤...
A to X.
If A is a directed set, we often write a net from A to X in the form (xα), which expresses the fact that the element α in A is mapped to the element xα in X.
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 numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s 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 x in a topological space, let Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.
Limits of nets
If (xα) is a net from a directed set A into X, and if Y is a subset of X, then we say that (xα) is eventually in Y (or residually in Y) if there exists an α in A so that for every β in A with β ≥ α, the point xβ lies in Y.If (xα) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write
- lim xα = x
if and only if
- for every neighborhood U of x, (xα) is eventually in U.
Intuitively, this means that the values xα come and stay as close as we want to x for large enough α.
Note that the example net given above on the neighborhood system of a point x does indeed converge to x according to this definition.
Given a base for the topology
Base (topology)
In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
, in order to prove convergence of a net it is necessary and sufficient to prove that there exists some point x, such that (xα) is eventually in all members of the base containing this putative limit.
Examples of limits of nets
- Limit of a sequenceLimit of a sequenceThe limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
and limit of a functionLimit of a functionIn mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
: see below.
- Limits of nets of Riemann sumRiemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
s, in the definition of the Riemann integralRiemann integralIn the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The Riemann integral is unsuitable for many theoretical purposes...
. In this example, the directed set is the set of partitions of the intervalPartition of an intervalIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the formIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form...
of integration, partially ordered by inclusion.
Supplementary definitions
If φ is a net on X based on directed set D and A is a subset of X, then φ is frequently in (or cofinally in) A if for every α in D there exists some β ≥ α, β in D, so that φ(β) is in A.A point x in X is said to be an accumulation point or cluster point of a net if (and only if) for every neighborhood U of x, the net is frequently in U.
A net φ on set X is called universal, or an ultranet if for every subset A of X, either φ is eventually in A or φ is eventually in X-A.
Examples
Sequence in a topological space:A sequence (a1, a2, ...) in a topological space V can be considered a net in V defined on N.
The net is eventually in a subset Y of V if there exists an N in N such that for every n ≥ N, the point an is in Y.
We have limx → c an = L if and only if for every neighborhood Y of L, the net is eventually in Y.
The net is frequently in a subset Y of V if and only if for every N in N there exists some n ≥ N such that an is in Y, that is, if and only if infinitely many elements of the sequence are in Y. Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.
Function from a metric space to a topological space:
Consider a function from a metric space M to a topological space V, and a point c of M. We direct the set M\{c} reversely according to distance from c, that is, the relation is "has at least the same distance to c as", so that "large enough" with respect to the relation means "close enough to c". The function ƒ is a net in V defined on M\{c}.
The net ƒ is eventually in a subset Y of V if there exists an a in M\{c} such that for every x in M\{c} with d(x,c) ≤ d(a,c), the point f(x) is in Y.
We have limx → c ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.
The net ƒ is frequently in a subset Y of V if and only if for every a in M\{c} there exists some x in M\{c} with d(x,c) ≤ d(a,c) such that f(x) is in Y.
A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.
Function from a well-ordered set to a topological space:
Consider a well-ordered set
Well-order
In mathematics, a well-order relation on a set S is a strict total order on S with the property that every non-empty subset of S has a least element in this ordering. Equivalently, a well-ordering is a well-founded strict total order...
[0, c] with limit point c, and a function ƒ from [0, c) to a topological space V. This function is a net on [0, c).
It is eventually in a subset Y of V if there exists an a in [0, c) such that for every x ≥ a, the point f(x) is in Y.
We have limx → c ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.
The net ƒ is frequently in a subset Y of V if and only if for every a in [0, c) there exists some x in [a, c) such that f(x) is in Y.
A point y in V is a cluster point of the net ƒ if and only if for every neighborhood Y of y, the net is frequently in Y.
The first example is a special case of this with c = ω.
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 sequenceLimit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
. The following set of theorems and lemmas help cement that similarity:
- A function ƒ:X→ Y between topological spaces is continuous at the point x if and only if for every net (xα) with
-
- lim xα = x
- we have
- lim ƒ(xα) = ƒ(x).
- Note that this theorem is in general not true if we replace "net" by "sequence". We have to allow for more directed sets than just the natural numbers if X is not first-countableFirst-countable spaceIn topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis...
.
- In general, a net in a space X can have more than one limit, but if X is a Hausdorff spaceHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
, the limit of a net, if it exists, is unique. Conversely, if X is not Hausdorff, then there exists a net on X 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. Note that this result depends on the directedness condition; a set indexed by a general preorderPreorderIn mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
or partial order may have distinct limit points even in a Hausdorff space.
- If U is a subset of X, then x is in the closureClosure (topology)In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of U if and only if there exists a net (xα) with limit x and such that xα is in U for all α.
- A subset A of X is closed if and only if, whenever (xα) is a net with elements in A and limit x, then x is in A.
- The set of cluster points of a net is equal to the set of limits of its convergent subnetSubnet (mathematics)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.If and are nets from...
s.
| Proof |
|---|
Let X be a topological space, A a directed set,  be a net in X, and  .It is easily seen that if y is a limit of a subnet of  , then y is a cluster point of  .Conversely, assume that y is a cluster point of  .Let B be the set of pairs  where U is an open neighborhood of y in X and  is such that  .The map  mapping  to  is then cofinal.Moreover, giving B the product order (the neighborhoods of y are ordered by inclusion) makes it a directed set, and the net  defined by  converges to y. |
- 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.
- A space X is compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
if and only if every net (xα) in X has a subnet with a limit in X. This can be seen as a generalization of the Bolzano–Weierstrass theoremBolzano–Weierstrass theoremIn real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...
and Heine–Borel theoremHeine–Borel theoremIn the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:For a subset S of Euclidean space Rn, the following two statements are equivalent:*S is closed and bounded...
.
| Proof |
|---|
| Finite intersection property Finite intersection property In general topology, a branch of mathematics, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty.... ). Let I be any set and  be a collection of closed subsets of X such that  for each finite  . Then  as well. Otherwise,  would be an open cover for X with no finite subcover contrary to the compactness of X.Let A be a directed set and  be a net in X. 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 X has a convergent subnet. For the sake of contradiction, let  be an open cover of X with no finite subcover. Consider  . Observe that D 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 x; however, for all  , we have that  . This is a contradiction and completes the proof. |
- A net in the product space has a limit if and only if each projection has a limit. Symbolically, if (xα) is a net in the product X = πiXi, then it converges to x if and only if
for each i. Armed with this observation and the above characterization of compactness in terms on nets, one can give a slick proof of Tychonoff's theoremTychonoff's theoremIn mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
.
- If ƒ:X→ Y and (xα) is an ultranet on X, then (ƒ(xα)) is an ultranet on Y.
Related ideas
In a metric spaceMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
or uniform space
Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence.The conceptual difference between...
, one can speak of Cauchy net
Cauchy net
In mathematics, 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 V there exists γ such that for all α, β ≥ γ, is a member of V. More generally, in a Cauchy space, a net is Cauchy if the filter generated by the...
s in much the same way as Cauchy sequence
Cauchy sequence
In mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
s.
The concept even generalises to Cauchy space
Cauchy space
In general topology and analysis, a Cauchy space is a generalization of metric spaces and uniform spaces for which the notion of Cauchy convergence still makes sense. Cauchy spaces were introduced by H. H. Keller in 1968, as an axiomatic tool derived from the idea of a Cauchy filter, in order to...
s.
The theory of filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
s also provides a definition of convergence in general topological spaces.
Limit superior
Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences. Some authors work even with more general structures, like complete lattices.For a net
we putLimit superior of a net of real numbers has many properties analogous to the case of sequences, e.g.
where equality holds whenever one of the nets is convergent.