Net (mathematics)

 This article is about nets in topological spaces and not about εnets in metric spaces
In topology and related areas of mathematics a net or MooreSmith 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 firstcountable 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.
Contents 
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 (x_{α}), which expresses the fact that the element α in A is mapped to the element x_{α} in X. We usually use ≥ to denote the binary relation given on A.
Examples
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 N_{x} denote the set of all neighbourhoods containing x. Then N_{x} 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 N_{x}, let x_{S} be a point in S. Then x_{S} is a net. As S increases with respect to ≥, the points x_{S} in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are lead to the idea that x_{S} must tend towards x in some sense. We can make this limiting concept precise.
Limits of nets
If (x_{α}) is a net from a directed set A into X, and if Y is a subset of X, then we say that (x_{α}) is eventually in Y if there exists an α in A so that for every β in A with β ≥ α, the point x_{β} lies in Y.
If (x_{α}) 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 x_{α} = x
if and only if
 for every neighborhood U of x, (x_{α}) is eventually in U.
Intuitively, this means that the values x_{α} 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.
Examples of limits of nets
 Limits of sequences.
 Limits of functions of a real variable: lim_{x → 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 RiemannStieltjes integral.
Supplementary definitions
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.
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 φoh 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 XA.
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 (x_{α}) with
 lim x_{α} = x
we have
 lim f(x_{α}) = 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 firstcountable.
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.
If U is a subset of X, then x is in the closure of U if and only if there exists a net (x_{α}) with limit x and such that x_{α} is in U for all α. In particular, U is closed if and only if, whenever (x_{α}) 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 (x_{α}) in X has a subnet with a limit in X. This can be seen as a generalization of the BolzanoWeierstrass theorem and HeineBorel 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.
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.de:Netz (Mathematik)