Net (mathematics)

This article is about nets in topological spaces and not about ε-nets in metric spaces.

In mathematics, more specifically in point-set 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 neighbourhood basis about a point that is countable, or linearly ordered by inclusion. 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.

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 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 T is contained in S. 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 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.

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 ƒ is a net in V defined on M\{c}.

The net ƒ 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} ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.

The net ƒ 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 ƒ 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 ƒ 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} ƒ(x) = L if and only if for every neighborhood Y of L, ƒ is eventually in Y.

The net ƒ 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 ƒ 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 sequence.

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) = (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.

• 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.

Proof

First, suppose that X is compact.We will need the following observation (see Finite intersection property). Let I be any set and $\{C_i\}_{i \in I}$ be a collection of closed subsets of X such that $\bigcap_{i\in J} C_i \neq \emptyset$ for each finite $J\subseteq I$. Then $\bigcap_{i\in I} C_i \neq \emptyset$ as well. Otherwise, $\{C_i^c \}_{i \in I}$ would be an open cover for X with no finite subcover. Let A be a directed set and $\langle x_{\alpha} \rangle_{\alpha \in A}$ be a net in X. For every $\alpha \in A$ define $E_{\alpha}\triangleq\{x_{\beta} : \beta \geq \alpha \}.$ The collection $\{\operatorname{cl}(E_{\alpha}) : \alpha\in A \}$ has the property that every finite subcollection has non-empty intersection. Thus, by the remark above, we have that $\bigcap_{\alpha\in A} \operatorname{cl}(E_{\alpha}) \neq \emptyset$ Let x be an element of this intersection. Then, each open neighborhood U of x meets $\operatorname{cl}(E_{\alpha})$ for each $\alpha \in A$. Thus, for each $\alpha \in A$ there exists $\beta \geq \alpha$ with $x_{\beta}\in U$. Hence, the net $\langle x_{\alpha} \rangle_{\alpha \in A}$ meets U infinitely often and so has a subnet that converges to x. Conversely, suppose that every net in X has a convergent subnet. For the sake of contradiction, let $\{U_i : i\in I\}$ be an open cover of X with no finite subcover. Consider $B \triangleq \{J\subset I : |J|<\infty \}$. Observe that B is a directed set under inclusion and for each $b\in B$, there exists an $x_b\in X$ such that $x_b\notin a$ for all $a\leq b$. Consider the net $\langle x_{b} \rangle_{b\in B}$. This net cannot have a convergent subnet because, for each x there exists $b$ such that $x\in b$, but for all $c\geq b$, we have that $x_c \notin b$ and thus is not in some open neighborhood of x. This is a contradiction and completes the proof.

• 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 $\pi_i(x_\alpha)\to \pi_i(x)$ for each i.

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

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.