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In mathematics the term net has at least two
meanings. See the glossary of Riemannian and metric geometry for its meaning for metric spaces.
This article is about its meaning in topology, where the concept of a
net is a generalization of that 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. Nets were first introduced by Eliakim Hastings
Moore and H. L. Smith in 1922.
Definition and examples
If X is a topological space, a net in X is a function from some directed set A to
X.
Since the natural numbers with the normal order form a directed set,
this definition includes all sequences among the nets. Other examples arise from real functions: suppose x0 is a real number and f :
R − {x0} -> R is a function. The set A =
R − {x0} can be directed towards x0 (see directed set for an explanation), and the function then turns into a net.
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.
Supplementary definitions
If D and E are directed sets, and h is a function from D to E, then h is
called final if for every e in E there is a d in D so that if q in D is
greater than or equal to d then h(q) is greater than or equal to e.
If D and E are directed sets, h is a final function from D to E, and φ is a
net on set X based on E, then φ following h is called a subnet of φ.
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 greater than or equal to α so
that φ(β) is in A.
If φ is a net on X based on directed set D and A is a subset of X, then φ is
eventually in A if there exists an α in D so that for every β in D with β greater than
or equal to α, 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.
Limits of nets
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.
The three most frequently seen examples of limits of nets
These are:
- Limits of functions of a real variable: limx →
c f(x).
- 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.
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, which is widely used in the theory of
metric spaces.
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 that one limit, but if X is a Hausdorff space, the limit of a net, if it exists, is unique.
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
theorems of Bolzano-Weierstrass and
Heine-Borel.
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.
See also
The theory of filters also provides a definition of
convergence in general topological spaces.
Reference
E. H. Moore and H. L. Smith (1922). A General Theory of Limits. American Journal of Mathematics 44
(2), 102–121. edut
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