Nets (Generalization of sequences)

In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, a sequence is a function with domain the natural numbers, and in the context of topology, the range of this function is usually any topological space. However, in the context of topology, sequences do not fully encode all information about a function between topological spaces. In particular, the following two conditions are not equivalent in general for a map ƒ between topological spaces X and Y:

1. The map ƒ is continuous
2. Given any point x in X, and any sequence in X converging to x, the composition of ƒ with this sequence converges to ƒ(x)

It is true however, that condition 1 implies condition 2, in the context of all spaces. The difficulty encountered when attempting to prove that condition 2 implies condition 1 lies in the fact that topological spaces are, in general, not first-countable. If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent. In particular, the two conditions are equivalent for metric spaces.

The purpose of the concept of a net, first introduced by E. H. Moore and H. L. Smith in 1922, is to generalize the notion of a sequence so as to confirm the equivalence of the conditions (with "sequence" being replaced by "net" in condition 2). In particular, rather than being defined on a countable linearly ordered set, a net is defined on an arbitrary directed set. In particular, this allows theorems similar to that asserting the equivalence of condition 1 and condition 2, to hold in the context of topological spaces which do not necessarily have a countable or linearly ordered neighbourhood basis around a point. Therefore, while sequences do not encode sufficient information about functions between topological spaces, nets do because of the fact that collections of open sets in topological spaces are much like directed sets in behaviour. The term "net" was coined by Kelley.

Nets are one of the many tools used in topology to generalize certain concepts that may only be general enough in the context of metric spaces. A related notion, that of the 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

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.

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 ƒ: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 ƒ(x_α) = ƒ(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 of 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.
• The set of cluster points of a net is equal to the set of limits of its convergent subnets.
• 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 = π_iX_i, then it converges to x if and only if  for each i. Armed with this observation and the above characterization of compactness in terms on nets, one can give a slick proof of Tychonoff's theorem.
• If ƒ:X→ Y and (x_α) is an ultranet on X, then (ƒ(x_α)) is an ultranet on Y.