Net (mathematics)

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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^[1]. A related notion, called filter, was developed in 1937 by Henri Cartan.

1 Definition
2 Examples
3 Limits of nets
4 Examples of limits of nets
5 Supplementary definitions
6 Properties
7 Related ideas
8 References

[edit] 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.

[edit] 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 led to the idea that x_S must tend towards x in some sense. We can make this limiting concept precise.

[edit] 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 (or residually 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.

[edit] Examples of limits of nets

Limit of a sequence.

Limit of a function 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 Riemann-Stieltjes integral.

[edit] 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 whenever 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 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 (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 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.

[edit] 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 (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 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 (x_α) with limit x and such that x_α is in U for all α.

A subset $A\subset X$ is closed if and only if, whenever (x_α) 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 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 Bolzano-Weierstrass theorem and Heine-Borel theorem.

A net in the product space has a limit if and only if each projection has a limit. Symbolically, if (x_α) is a net in the product $X=\prod X_i$ , then it converges to x if and only if $\pi_i(x_\alpha)\to \pi_i(x)$ for each i.