Net (mathematics)
A generalization of a sequence of points
In mathematics , more specifically in general topology and related branches, a net or Moore – Smith sequence is a generalization of the notion of a sequence . In essence, a sequence is a function whose domain is the natural numbers . The codomain of this function is usually some topological space .
The motivation for generalizing the notion of a sequence is that, in the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map
between topological spaces
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 if the topological spaces are not both first-countable . In particular, the two conditions are equivalent for metric spaces . The spaces for which the converse holds are the sequential spaces .
The concept of a net, first introduced by E. H. Moore and Herman L. Smith in 1922, [1] is to generalize the notion of a sequence so that the above conditions (with "sequence" being replaced by "net" in condition 2) are in fact equivalent for all maps of topological spaces. In particular, rather than being defined on a countable linearly ordered set, a net is defined on an arbitrary directed set . This allows for theorems similar to the assertion that the conditions 1 and 2 above are equivalent to hold in the context of topological spaces that do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do, because collections of open sets in topological spaces are much like directed sets in behavior. The term "net" was coined by John L. Kelley . [2] [3]
Nets are one of the many tools used in topology to generalize certain concepts that may not be general enough in the context of metric spaces . A related notion, that of the filter , was developed in 1937 by Henri Cartan .
Definitions
Any
function
whose
domain
is a
directed set
is called a
net
. If this function takes values in some set
then it may also be referred to as a
net in
Explicitly, a
net in
is a function of the form
where
is some
directed set
. Elements of a net's domain are called its
indices
.
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.
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 (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.
Nets are primarily used in the fields of Analysis and Topology , where they are used to characterize many important topological properties that (in general), sequences are unable to characterize (this shortcoming of sequences motivated the study of sequential spaces and Fréchet–Urysohn spaces ). Nets are intimately related to filters , which are also often used in topology . Every net may be associated with a filter and every filter may be associated with a net, where the properties of these associated objects are closely tied together (see the article about Filters in topology for more details). Nets directly generalize sequences and they may often be used very similarly to sequences. Consequently, the learning curve for using nets is typically much less steep than that for filters, which is why many mathematicians, especially analysts , prefer them over filters. However, filters, and especially ultrafilters , have some important technical advantages over nets that ultimately result in nets being encountered much less often than filters outside of the fields of Analysis and Topology.
A
subnet
is not merely the restriction of a net
to a directed subset of
see the linked page for a definition.
Examples of nets
Every non-empty totally ordered set is directed. Therefore, every function on such a set is a net. In particular, the natural numbers with the usual order form such a set, and a sequence is a function on the natural numbers, so every sequence is a net.
Another important example is as follows. Given a point
in a topological space, let
denote the set of all
neighbourhoods
containing
Then
is a directed set, where the direction is given by reverse inclusion, so that
if and only if
is contained in
For
let
be a point in
Then
is a net. As
increases with respect to
the points
in the net are constrained to lie in decreasing neighbourhoods of
so intuitively speaking, we are led to the idea that
must tend towards
in some sense. We can make this limiting concept precise.
A subnet of a sequence is
not
necessarily a sequence.
[4]
For an example, let
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
).
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
And it is said to be
frequently
or
cofinally in
if for every
there exists some
such that
and
[4]
A point is called a
limit point
(respectively,
cluster point
) of a net if that net is eventually (respectively, cofinally) in every neighborhood of that point.
Explicitly, a point
is said to be an
accumulation point
or
cluster point
of a net if for every neighborhood
of
the net is frequently in
[4]
A point
is called a
limit point
or
limit
of the net
in
if (and only if)
-
for every open
neighborhood
of
the net
is eventually in
in which case, this net is then also said to
converge
to/towards
and to
have
as a limit
.
Intuitively, convergence of a net
means that the values
come and stay as close as we want to
for large enough
The example net given above on the
neighborhood system
of a point
does indeed converge to
according to this definition.
Notation for limits
If the net
converges in
to a point
then this fact may be expressed by writing any of the following:
where if the topological space
is clear from context then the words "in
" may be omitted.
If
in
and if this limit in
is unique (uniqueness in
means that if
is such that
then necessarily
) then this fact may be indicated by writing
where an equals sign is used in place of the arrow
[5]
In a
Hausdorff space
, every net has at most one limit so the limit of a convergent net in a Hausdorff space is always unique.
[5]
Some authors instead use the notation "
" to mean
with
out
also requiring that the limit be unique; however, if this notation is defined in this way then the
equals sign
is no longer guaranteed to denote a
transitive relationship
and so no longer denotes
equality
. Specifically, without the uniqueness requirement, if
are distinct and if each is also a limit of
in
then
and
could be written (using the equals sign
) despite
being false.
Bases and subbases
Given a
subbase
for the topology on
(where note that every
base
for a topology is also a subbase) and given a point
a net
in
converges to
if and only if it is eventually in every neighborhood
of
This characterization extends to
neighborhood subbases
(and so also
neighborhood bases
) of the given point
Convergence in metric spaces
Suppose
is a
metric space
(or a
pseudometric space
) and
is endowed with the
metric topology
.
If
is a point and
is a net, then
in
if and only if
in
where
is a net of
real numbers
.
In
plain English
, this characterization says that a net converges to a point in a metric space if and only if the distance between the net and the point converges to zero.
If
is a
normed space
(or a
seminormed space
) then
in
if and only if
in
where
Convergence in topological subspaces
If the set
is endowed with the
subspace topology
induced on it by
then
in
if and only if
in
In this way, the question of whether or not the net
converges to the given point
depends
solely
on this topological subspace
consisting of
and the
image
of (that is, the points of) the net
Limits in a Cartesian product
A net in the product space has a limit if and only if each projection has a limit.
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
[6]
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 .
Cluster points of a net
A point
is a cluster point of a given net if and only if it has a subset that converges to
[8]
If
is a net in
then the set of all cluster points of
in
is equal to
[7]
where
for each
If
is a cluster point of some subnet of
then
is also a cluster point of
[8]
Ultranets
A net
in set
is called a
universal net
or an
ultranet
if for every subset
is eventually in
or
is eventually in the complement
[4]
Ultranets
are closely related to
ultrafilters
.
Every constant net is an ultranet. Every subnet of an ultranet is an ultranet.
[7]
Every net has some subnet that is an ultranet.
[4]
If
is an ultranet in
and
is a function then
is an ultranet in
[4]
Given
an ultranet clusters at
if and only it converges to
[4]
Examples of limits of nets
Every limit of a sequence and limit of a function can be interpreted as a limit of a net (as described below).
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.
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
[9]
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
Examples
Sequence in a topological space
A sequence
in a topological space
can be considered a net in
defined on
The net is eventually in a subset
of
if there exists an
such that for every integer
the point
is in
So
if and only if for every neighborhood
of
the net is eventually in
The net is frequently in a subset
of
if and only if for every
there exists some integer
such that
that is, if and only if infinitely many elements of the sequence are in
Thus a point
is a cluster point of the net if and only if every neighborhood
of
contains infinitely many elements of the sequence.
Function from a metric space to a topological space
Fix a point
in a metric space
that has at least two 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 .
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
,
[10]
which is as follows:
If
and
are nets then
is called a
subnet
or
Willard-subnet
[10]
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
Properties
Virtually all concepts of topology can be rephrased in the language of nets and limits. This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence . The following set of theorems and lemmas help cement that similarity:
Characterizations of topological properties
Closed sets and closure
A subset
is closed in
if and only if every limit point of every convergent net in
necessarily belongs to
Explicitly, a subset
is closed if and only if whenever
and
is a net valued in
(meaning that
for all
) such that
in
then necessarily
More generally, if
is any subset then a point
is in the
closure
of
if and only if there exists a net
in
with limit
and such that
for every index
[8]
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 given point
if and only if for every net
in its domain, if
in
then
in
[8]
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
|
---|
(
(
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
The collection
and this is precisely the set of cluster points of
(
|
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.
Cauchy nets
A Cauchy net generalizes the notion of Cauchy sequence to nets defined on uniform spaces . [12]
A net
is a
Cauchy net
if for every
entourage
there exists
such that for all
is a member of
[12]
[13]
More generally, in a
Cauchy space
, a net
is Cauchy if the filter generated by the net is a
Cauchy filter
.
A topological vector space (TVS) is called complete if every Cauchy net converges to some point. A normed space , which is a special type of topological vector space, is a complete TVS (equivalently, a Banach space ) if and only if every Cauchy sequence converges to some point (a property that is called sequential completeness ). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non- normable ) topological vector spaces.
Relation to filters
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.
[14]
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).
[15]
For instance, any net
in
induces a filter base of tails
where the filter in
generated by this filter base is called the net's
eventuality filter
. This correspondence allows for any theorem that can be proven with one concept to be proven with the other.
[15]
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. [15] He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis , while filters are most useful in algebraic topology . In any case, he shows how the two can be used in combination to prove various theorems in general topology .
Limit superior
Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences. [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.
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 Pages displaying short descriptions of redirect targets
Citations
- ↑ 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 .
- ↑ ( Sundström 2010 , p. 16n)
- ↑ Megginson, p. 143
- 1 2 3 4 5 6 7 Willard 2004 , pp. 73–77.
- 1 2 Kelley 1975 , pp. 65–72.
- ↑ Willard 2004 , p. 76.
- 1 2 3 4 5 6 7 Willard 2004 , p. 77.
- 1 2 3 4 Willard 2004 , p. 75.
- ↑ Willard 2004 , pp. 71–72.
- 1 2 Schechter 1996 , pp. 157–168.
- ↑ Howes 1995 , pp. 83–92.
- 1 2 Willard, Stephen (2012), General Topology , Dover Books on Mathematics, Courier Dover Publications, p. 260, ISBN 9780486131788 .
- ↑ Joshi, K. D. (1983), Introduction to General Topology , New Age International, p. 356, ISBN 9780852264447 .
-
↑
"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 ) - 1 2 3 R. G. Bartle, Nets and Filters In Topology, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
- ↑ Aliprantis-Border, p. 32
- ↑ Megginson, p. 217, p. 221, Exercises 2.53–2.55
- ↑ Beer, p. 2
- ↑ Schechter, Sections 7.43–7.47
References
- Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv : 1006.4131v1 [ math.HO ].
- Aliprantis, Charalambos D. ; Border, Kim C. (2006). Infinite dimensional analysis: A hitchhiker's guide (3rd ed.). Berlin: Springer. pp. xxii, 703. ISBN 978-3-540-32696-0 . MR 2378491 .
- 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 .
- Kelley, John L. (1975). General Topology . Graduate Texts in Mathematics . Vol. 27. New York: Springer Science & Business Media. ISBN 978-0-387-90125-1 . OCLC 338047 .
- Kelley, John L. (1991). General Topology . Springer. ISBN 3-540-90125-6 .
- Megginson, Robert E. (1998). An Introduction to Banach Space Theory . Graduate Texts in Mathematics . Vol. 193. New York: Springer. ISBN 0-387-98431-3 .
- Schechter, Eric (1997). Handbook of Analysis and Its Foundations . San Diego: Academic Press. ISBN 9780080532998 . Retrieved 22 June 2013 .
- Schechter, Eric (1996). Handbook of Analysis and Its Foundations . San Diego, CA: Academic Press. ISBN 978-0-12-622760-4 . OCLC 175294365 .
- Willard, Stephen (2004) [1970]. General Topology . Mineola, N.Y. : Dover Publications . ISBN 978-0-486-43479-7 . OCLC 115240 .