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 (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are not required to be total orders or even partial orders. A directed set may have greatest elements and/or maximal elements. In this case, the conditions
and
cannot be replaced by the strict inequalities
and
, since the strict inequalities cannot be satisfied if a or b is maximal.
A net in , denoted
, is a function of the form
whose domain
is some directed set, and whose values are
. Elements of a net's domain are called its indices. When the set
is clear from context it is simply called a net, and one assumes
is a directed set with preorder
Notation for nets varies, for example using angled brackets
. As is common in algebraic topology notation, the filled disk or "bullet" stands in place of the input variable or index
.
Limits of nets
A net is said to be eventually or residually in a set
if there exists some
such that for every
with
the point
A point
is called a limit point or limit of the net
in
whenever:
- for every open neighborhood
of
the net
is eventually in
,
expressed equivalently as: the net converges to/towards or has
as a limit; and variously denoted as:
If
is clear from context, it may be omitted from the notation.
If and this limit is unique (i.e.
only for
) then one writes:
using the equal sign in place of the arrow
[4] In a Hausdorff space, every net has at most one limit, and the limit of a convergent net is always unique.[4]Some authors do not distinguish between the notations
and
, but this can lead to ambiguities if the ambient space
is not Hausdorff.
Cluster points of nets
A net is said to be frequently or cofinally in
if for every
there exists some
such that
and
[5] A point
is said to be an accumulation point or cluster point of a net if for every neighborhood
of
the net is frequently/cofinally in
[5] In fact,
is a cluster point if and only if it has a subset that converges to
[6] The set
of all cluster points of
in
is equal to
for each
, where
.
Subnets
The analogue of "subsequence" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,[7] which is as follows: If and
are nets then
is called a subnet or Willard-subnet[7] of
if there exists an order-preserving map
such that
is a cofinal subset of
and
The map
is called order-preserving and an order homomorphism if whenever
then
The set
being cofinal in
means that for every
there exists some
such that
If is a cluster point of some subnet of
then
is also a cluster point of
[6]
Ultranets
A net in set
is called a universal net or an ultranet if for every subset
is eventually in
or
is eventually in the complement
[5]
Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet.[8] Assuming the axiom of choice, every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.[5] If is an ultranet in
and
is a function then
is an ultranet in
[5]
Given an ultranet clusters at
if and only it converges to
[5]
Cauchy nets
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 in
of a net in
necessarily lies in
.Explicitly, this means that if
is a net with
for all
, and
in
then
More generally, if is any subset, the closure of
is the set of points
with
for some net
in
.[6]
Open sets and characterizations of topologies
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 point
if and only if for every net
in the domain,
in
implies
in
[6] Briefly, a function
is continuous if and only if
in
implies
in
In general, this statement would not be true if the word "net" was replaced by "sequence"; that is, it is necessary to allow for directed sets other than just the natural numbers if
is not a first-countable space (or not a sequential space).
Proof |
---|
( ( We construct a net Now, for every open neighborhood |
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 |
---|
( Let ( |
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 Conversely, assume that |
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
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
:
- The map
is continuous in the topological sense;
- 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:
- Given any point
in
and any net in
converging to
the composition of
with this net converges to
(continuous in the net sense).
With this change, the conditions become equivalent for all maps of topological spaces, including topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behavior.
For an example where sequences do not suffice, interpret the set of all functions with prototype
as the Cartesian product
(by identifying a function
with the tuple
and conversely) and endow it with the product topology. This (product) topology on
is identical to the topology of pointwise convergence. Let
denote the set of all functions
that are equal to
everywhere except for at most finitely many points (that is, such that the set
is finite). Then the constant
function
belongs to the closure of
in
that is,
[8] This will be proven by constructing a net in
that converges to
However, there does not exist any sequence in
that converges to
[14] which makes this one instance where (non-sequence) nets must be used because sequences alone can not reach the desired conclusion. Compare elements of
pointwise in the usual way by declaring that
if and only if
for all
This pointwise comparison is a partial order that makes
a directed set since given any
their pointwise minimum
belongs to
and satisfies
and
This partial order turns the identity map
(defined by
) into an
-valued net. This net converges pointwise to
in
which implies that
belongs to the closure of
in
More generally, a subnet of a sequence is not necessarily a sequence.[5][a] Moreso, a subnet of a sequence may be a sequence, but not a subsequence.[b] But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences. Specifically, for a first-countable space, the net induces the sequence
where
is defined as the
smallest value in
– that is, let
and let
for every integer
.
Examples
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
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
[8] However, the converse does not hold in general.[8] For example, suppose
and let
denote the sequence
that alternates between
and
Then
and
are cluster points of both
and
in
but
is not a cluster point of
since the open ball of radius
centered at
does not contain even a single point
Tychonoff's theorem and relation to the axiom of choice
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
[8]
A net is eventually in a subset
of a topological space
if and only if there exists some
such that for every
satisfying
the point
is in
Such a net
converges in
to a given point
if and only if
in the usual sense (meaning that for every neighborhood
of
is eventually in
).[8]
The net is frequently in a subset
of
if and only if for every
there exists some
with
such that
is in
Consequently, a point
is a cluster point of the net
if and only if for every neighborhood
of
the net is frequently in
Function from a well-ordered set to a topological space
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
- Characterizations of the category of topological spaces
- Filter (set theory) – Family of sets representing "large" sets
- Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
- Preorder – Reflexive and transitive binary relation
- Sequential space – Topological space characterized by sequences
- Ultrafilter (set theory) – Maximal proper filter
Notes
Citations
References
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- Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii, 340. ISBN 0-7923-2531-1. MR 1269778.
- Howes, Norman R. (23 June 1995). Modern Analysis and Topology. Graduate Texts in Mathematics. New York: Springer-Verlag Science & Business Media. ISBN 978-0-387-97986-1. OCLC 31969970. OL 1272666M.
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