In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a subnet is a generalization of the concept of
subsequence to the case of
nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition is not completely straightforward, but is designed to allow as many theorems about subsequences to generalize to nets as possible.
There are three non-equivalent definitions of "subnet".
The first definition of a subnet was introduced by
John L. Kelley in 1955 and later,
Stephen Willard
Stephen Willard (born 27 August 1958 in Swindon) is a former professional English darts player. Who played in Professional Darts Corporation events.
He won a PDC Tour Card in 2015, which was the year he also won the Saints Open defeating G ...
introduced his own (non-equivalent) variant of Kelley's definition in 1970.
Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet" but they are each equivalent to the concept of "subordinate filter", which is the analog of "subsequence" for
filters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component that ...
(they are not equivalent in the sense that there exist subordinate filters on
whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship).
A third definition of "subnet" (not equivalent to those given by Kelley or Willard) that equivalent to the concept of "subordinate filter" was introduced independently by Smiley (1957), Aarnes and Andenaes (1972), Murdeshwar (1983), and possibly others, although it is not often used.
This article discusses the definition due to
Stephen Willard
Stephen Willard (born 27 August 1958 in Swindon) is a former professional English darts player. Who played in Professional Darts Corporation events.
He won a PDC Tour Card in 2015, which was the year he also won the Saints Open defeating G ...
(the other definitions are described in the article
Filters in topology#Subnets).
Definitions
There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in
1970 by
Stephen Willard
Stephen Willard (born 27 August 1958 in Swindon) is a former professional English darts player. Who played in Professional Darts Corporation events.
He won a PDC Tour Card in 2015, which was the year he also won the Saints Open defeating G ...
, which is as follows:
If
and
are nets in a set
from
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
s
and
respectively, then
is said to be a of
( or a ) if there exists a
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
final function
such that
A function
is , , and an if whenever
then
and it is called if its
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
is
cofinal in
The set
being in
means that for every
there exists some
such that
that is, for every
there exists an
such that
[Some authors use a more general definition of a subnet. In this definition, the map is required to satisfy the condition: For every there exists a such that whenever Such a map is final but not necessarily monotone.]
Since the net
is the function
and the net
is the function
the defining condition
may be written more succinctly and cleanly as either
or
where
denotes
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
and
is just notation for the function
Subnets versus subsequences
Importantly, a subnet is not merely the restriction of a net
to a directed subset of its domain
In contrast, by definition, a of a given sequence
is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. Explicitly, a sequence
is said to be a of
if there exists a strictly increasing sequence of positive integers
such that
for every
(that is to say, such that
). The sequence
can be canonically identified with the function
defined by
Thus a sequence
is a subsequence of
if and only if there exists a strictly increasing function
such that
Subsequences are subnets
Every
subsequence is a subnet because if
is a subsequence of
then the map
defined by
is an order-preserving map whose image is cofinal in its codomain and satisfies
for all
Sequence and subnet but not a subsequence
The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
is not a
subsequence of
although it is a subnet because the map
defined by
is an order-preserving map whose image is
and satisfies
for all
[Indeed, this is because and for every in other words, when considered as functions on the sequence is just the identity map on while ]
While a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
is a net, a sequence has subnets that are not subsequences. The key difference is that subnets can use the same point in the net multiple times and the indexing set of the subnet can have much larger
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
. Using the more general definition where we do not require monotonicity, a sequence is a subnet of a given sequence, if and only if it can be obtained from some subsequence by repeating its terms and reordering them.
[, Satz 2.8.3, p. 81]
Subnet of a sequence that is not a sequence
A subnet of a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
is necessarily a sequence.
For an example, let
be directed by the usual order
and define
by letting
be the
ceiling
A ceiling is an overhead interior surface that covers the upper limits of a room. It is not generally considered a structural element, but a finished surface concealing the underside of the roof structure or the floor of a story above. Ceilings ...
of
Then
is an order-preserving map (because it is a non-decreasing function) whose image
is a
cofinal subset of its codomain. Let
be any sequence (such as a constant sequence, for instance) and let
for every
(in other words, let
). This net
is not a sequence since its domain
is an
uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal nu ...
. However,
is a subnet of the sequence
since (by definition)
holds for every
Thus
is a subnet of
that is not a sequence.
Furthermore, the sequence
is also a subnet of
since the inclusion map
(that sends
) is an order-preserving map whose image
is a cofinal subset of its codomain and
holds for all
Thus
and
are (simultaneously) subnets of each another.
Subnets induced by subsets
Suppose
is an infinite set and
is a sequence. Then
is a net on
that is also a subnet of
(take
to be the inclusion map
). This subnet
in turn induces a subsequence
by defining
as the
smallest value in
(that is, let
and let
for every integer
). In this way, every infinite subset of
induces a canonical subnet that may be written as a subsequence. However, as demonstrated below, not every subnet of a sequence is a subsequence.
Applications
The definition generalizes some key theorems about subsequences:
* A net
converges to
if and only if every subnet of
converges to
* A net
has a
cluster point if and only if it has a subnet
that converges to
* A topological space
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
if and only if every net in
has a convergent subnet (see
net for a proof).
Taking
be the identity map in the definition of "subnet" and requiring
to be a
cofinal subset of
leads to the concept of a , which turns out to be inadequate since, for example, the second theorem above fails for the
Tychonoff plank if we restrict ourselves to cofinal subnets.
Clustering and closure
If
is a net in a subset
and if
is a cluster point of
then
In other words, every cluster point of a net in a subset belongs to the
closure of that set.
If
is a net in
then the set of all cluster points of
in
is equal to
where
for each
Convergence versus clustering
If a net converges to a point
then
is necessarily a cluster point of that net. The converse is not guaranteed in general. That is, it is possible for
to be a cluster point of a net
but for
to converge to
However, if
clusters at
then there exists a subnet of
that converges to
This subnet can be explicitly constructed from
and the
neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbou ...
at
as follows: make
into a directed set by declaring that
then
and
is a subnet of
since the map
is a
monotone function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
whose image
is a cofinal subset of
and
Thus, a point
is a cluster point of a given net if and only if it has a subnet that converges to
See also
*
*
Notes
Citations
References
*
*
*
*
*
{{Order theory
Topology