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 1922¹. 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 (x_α), which expresses the fact that the element α in A is mapped to the element x_α 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 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.

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 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 there exists some point x, such that (x_α) is eventually in all members of the base containing this putative limit.

Examples of limits of nets

• Limit of a sequence and limit of a function: see below.

• 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.

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://en.wikipedia.org/wiki/Limit_point#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.

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 a_n is in Y.

We have lim_{x → c} a_n = 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 a_n 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 lim_{x → 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 lim_{x → 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://en.wikipedia.org/wiki/Order_topology#Ordinal-indexed_sequences.

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 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 , then it converges to x if and only if for each i.

• If f:X→ Y and (x_α) is an ultranet on X, then (f(x_α)) 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

1. ^ 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