- 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 using more general index sets: directed sets. This allows a weaker order relation on the index set
and also, even without weakening the order, a larger index set.
Nets were first introduced by E. H. Moore and H. L.
Smith in 1922[1]. A related notion, called
filter, was developed in 1937 by Henri
Cartan.
Definition
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.
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 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.
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, in order to prove convergence of a net it is necessary and sufficient to prove that
(xα) is eventually in all members of the base.
Examples of limits of nets
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
pointhttp://leroy.atomant.net/mediawiki/index.php/Limit_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.
One can also define the concept of a subnet of a net.
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
f is a net in V defined on M\{c}.
The net f 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 f(x) = L if and only if for every neighborhood Y of
L, f is eventually in Y.
The net f 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 f 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
f 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 f(x) = L if and only if for every neighborhood Y of
L, f is eventually in Y.
The net f 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 f 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 sequencehttp://leroy.atomant.net/mediawiki/index.php/Order_topology#.
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:
- A function f : 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 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.
- A subset
is closed
if and only if, whenever (xα) is a net with elements in A and limit x, then x is in
A.
- A net has a cluster point x if and only if it has a subnet which
converges to x.
- 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 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
, then it converges to x if and only
if πi(xα)→πi(x) for each i.
- If f:X→ Y and (xα) is an ultranet on X, then (f(xα))
is an ultranet on Y.
Related ideas
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.
References
- ^ E. H. Moore and H. L. Smith. "A General Theory of Limits". American
Journal of Mathematics (1922) 44 (2), 102–121.
- Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.
- net on
PlanetMath
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
