Net (mathematics)
This article is about nets in topological spaces and not about ε-nets in metric spaces.In
topology and related areas of
mathematics a
net or
Moore-Smith sequence is a generalization of a
sequence, intended to unify the various notions of
limit and generalize them to arbitrary
topological spaces. Limits of nets accomplish for all topological spaces what limits of sequences accomplish for
first-countable spaces such as
metric spaces.
A sequence is usually indexed by the
natural numbers which are a
totally ordered set. Nets generalize this concept by weakening the
order relation on the index set to that of a
directed set.
Nets were first introduced by
E. H. Moore and
H. L. Smith in
1922. An equivalent notion, called
filter, was developed in
1937 by
Henri Cartan.
If
X is a topological space, a
net in
X is a
function 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. We usually use ≥ to denote the binary relation given on
A.
Since the
natural numbers with the usual order form a directed set and a sequence is a function on the natural numbers, 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 from a directed set
A into
X, and if
Y is a subset of
X, then we say that (
xα) is
eventually 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 =
xif 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
neighbourhood system of a point
x does indeed converge to
x according to this definition.
*
Limit of a sequence.
*
Limit of a function of a
real variable: lim
x â†' c f(
x). Here we direct the set
R\{
c} according to distance from
c.
* Limits of nets of
Riemann sums, in the definition of the
Riemann integral. In this example, the directed set is the set of
partitions of the interval of integration, partially ordered by inclusion. A similar thing is done in the definition of the
Riemann-Stieltjes integral.
If
D and
E are directed sets, and
h is a function from
D to
E, then
h is called
cofinal if for every
e in
E there is a
d in
D so that if
q is in
D and
q ≥
d then
h(
q) ≥
e. In other words, the
image h(
D) is
cofinal in
E.
If
D and
E are directed sets,
h is a cofinal function from
D to
E, and φ is a net on set
X based on
E, then φo
h is called a
subnet of φ. All subnets are of this form, by definition.
If φ is a net on
X based on directed set
D and
A is a subset of
X, then φ is
frequently in A if for every α in
D there is a β in
D, β ≥ α so that φ(β) is in
A.
A net φ on set
X is called
universal if for every subset
A of
X, either φ is eventually in
A or φ is eventually in
X-
A.
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, which is widely used in the theory of
metric spaces.
A function
f :
X â†'
Y between topological spaces is
continuous at the point
x if and only if for every net (
xα) with:lim
xα =
xwe have:lim
f(
xα) =
f(
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-countable.
In general, a net in a space
X can have more than one limit, but if
X is a
Hausdorff space, 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 preorder 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
closure of
U if and only if there exists a net (
xα) with limit
x and such that
xα is in
U for all α.In particular,
U is closed if and only if, whenever (
xα) is a net with elements in
U and limit
x, then
x is in
U.
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
compact 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 theorem and
Heine-Borel theorem.
In a
metric space or
uniform space, one can speak of
Cauchy nets in much the same way as
Cauchy sequences.The concept even generalises to
Cauchy spaces.
The theory of
filters also provides a definition of
convergence in general topological spaces.
E. H. Moore and H. L. Smith (1922). A General Theory of Limits.
American Journal of Mathematics 44 (2), 102–121.
