Net (mathematics)

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 codomain 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 f between topological spaces X and Y:

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

It is true, however, that condition 1 implies condition 2. 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,^[1] 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 that 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 collections of open sets in topological spaces are much like directed sets in behaviour. The term "net" was coined by Kelley.^[2][3]

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 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.

Supplementary definitions

Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be 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_n 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) = 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.

Proof

One direction: Let f be continuous at point x, and let (xα) be a net such that lim (xα) = x. Then for every open neighborhood U of f(x), its preimage by f, V, is a neighborhood of x (by the continuity of f at x). Thus the interior of V, int(V), is an open neighborhood of x, and thus (xα) is eventually in int(V) . Therefore f(xα) is eventually in f(int(V)) and thus also eventually in f(V) = U. Thus lim f(xα) = f(x), and this direction is proven. The other direction: Let x be a point such that for every net (xα) such that lim (xα) = x, lim f(xα) = f(x). Now suppose that f is not continuous at x. Then there is a neighborhood U of f(x) whose preimage under f, V, is not a neighborhood of x. Note however that since f(x) is in U, x is in V. Now the set of open neighborhoods of x with the containment preoreder is a directed set (since the intersection of every two such neighborhoods is an open neighborhood of x as well). We construct a net (xα) such that for every open neighborhood of x whose index is α, xα is a point in this neighborhood that is not in V; that there is always such a point follows from the fact that no open neighborhood of x is included in V (since by our assumption V is not a neighborhood of x). Note that it follows that f(xα) is not in U. Now, for every open neighborhood W of x, this neighborhood is a member of the directed set whose index we denote α0. For every β ≥ α0, the member of the directed set whose index is β is contained within W; therefore xβ is in W. Thus lim (xα) = x and by our assumption lim f(xα) = f(x). But int(U) is an open neighborhood of f(x) and thus f(xα) is eventually in int(U) and therefore also in U, in contradiction to f(xα) not being in U for every α. Thus we have arrived at a contradiction, and we are forced to conclude that f is continuous in x. So the other direction is proven as well.

• 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.

Proof

Let X be a topological space, A a directed set, $\langle x_{\alpha} \rangle_{\alpha \in A}$ be a net in X, and $y \in X$. It is easily seen that if y is a limit of a subnet of $\langle x_{\alpha} \rangle_{\alpha \in A}$, then y is a cluster point of $\langle x_{\alpha} \rangle_{\alpha \in A}$. Conversely, assume that y is a cluster point of $\langle x_{\alpha} \rangle_{\alpha \in A}$. Let B be the set of pairs $(U,\alpha)$ where U is an open neighborhood of y in X and $\alpha \in A$ is such that $x_{\alpha} \in U$. The map $h : B \rightarrow A$ mapping $(U,\alpha)$ to $\alpha$ is then cofinal. Moreover, giving B the product order (the neighborhoods of y are ordered by inclusion) makes it a directed set, and the net $\langle y_{\beta} \rangle_{\beta \in B}$ defined by $y_{\beta}=x_{h(\beta)}$ converges to y.

• 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 contrary to the compactness of X. 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 $\bigcap_{\alpha\in A} \operatorname{cl}(E_{\alpha}) \neq \emptyset$ and this is precisely the set of cluster points of $\langle x_{\alpha} \rangle_{\alpha \in A}$. By the above property, it is equal to the set of limits of convergent subnets of $\langle x_{\alpha} \rangle_{\alpha \in A}$. Thus $\langle x_{\alpha} \rangle_{\alpha \in A}$ has a convergent subnet. 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 $D \triangleq \{J\subset I : |J|<\infty \}$. Observe that D is a directed set under inclusion and for each $C\in D$, there exists an $x_C\in X$ such that $x_C \notin U_a$ for all $a \in C$. Consider the net $\langle x_{C} \rangle_{C\in D}$. This net cannot have a convergent subnet, because for each $x \in X$ there exists $c \in I$ such that $U_c$ is a neighbourhood of x; however, for all $B \supseteq \{c\}$, we have that $x_B \notin U_c$. 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. 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.

Cauchy nets

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.

Relation to filters

A filter is another idea in topology that allows for a general definition for convergence in general topological spaces. The two ideas are equivalent in the sense that they give the same concept of convergence.^[4] More specifically, for every filter base an associated net can be constructed, and convergence of the filter base implies convergence of the associated net—and the other way around (for every net there is a filter base, and convergence of the net implies convergence of the filter base).^[5] For instance, any net $(x_\alpha)_{\alpha \in A}$ in $X$ induces a filter base of tails $\{ \{ x_\alpha : \alpha \in A, \alpha_0 \leq \alpha \} : \alpha_0 \in A \}\,$ where the filter in $X$ generated by this filter base is called the net's eventuality filter. This correspondence allows for any theorem that can be proven with one concept to be proven with the other.^[5] For instance, continuity of a function from one topological space to the other can be characterized either by the convergence of a net in the domain implying the convergence of the corresponding net in the codomain, or by the same statement with filter bases.

Robert G. Bartle argues that despite their equivalence, it is useful to have both concepts.^[5] He argues that nets are enough like sequences to make natural proofs and definitions in analogy to sequences, especially ones using sequential elements, such as is common in analysis, while filters are most useful in algebraic topology. In any case, he shows how the two can be used in combination to prove various theorems in general topology.

Limit superior

Limit superior and limit inferior of a net of real numbers can be defined in a similar manner as for sequences.^[6][7][8] Some authors work even with more general structures than the real line, like complete lattices.^[9]

For a net $(x_\alpha)_{\alpha\in I}$ we put

$\limsup x_\alpha = \lim_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta=\inf_{\alpha\in I} \sup_{\beta\succeq\alpha} x_\beta.$

Limit superior of a net of real numbers has many properties analogous to the case of sequences, e.g.

$\limsup (x_\alpha+y_\alpha) \le \limsup x_\alpha+\limsup y_\alpha,$

where equality holds whenever one of the nets is convergent.

References

1. Moore, E. H.; Smith, H. L. (1922). "A General Theory of Limits". American Journal of Mathematics 44 (2): 102–121. doi:10.2307/2370388. JSTOR 2370388
2. (Sundström 2010, p. 16n)
3. Megginson, p. 143
4. http://www.math.wichita.edu/~pparker/classes/handout/netfilt.pdf
5. R. G. Bartle, American Mathematical Monthly, Vol. 62, No. 8 (1955), pp. 551–557.
6. Aliprantis-Border, p. 32
7. Megginson, p. 217, p. 221, Exercises 2.53–2.55
8. Beer, p. 2
9. Schechter, Sections 7.43–7.47

• Kelley, John L. (1991). General Topology. Springer. ISBN 3-540-90125-6.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
• Sundström, Manya Raman (2010). "A pedagogical history of compactness". arXiv:1006.4131v1 [math.HO].
• Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite dimensional analysis: A hitchhiker's guide (3rd ed.). Berlin: Springer. pp. xxii,703. ISBN 978-3-540-32696-0. MR 2378491.
• Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii,340. ISBN 0-7923-2531-1. MR 1269778.
• Megginson, Robert E. (1998). An Introduction to Banach Space Theory. Graduate Texts in Mathematics 193. New York: Springer. ISBN 0-387-98431-3.
• Schechter, Eric (1997). Handbook of Analysis and Its Foundations. San Diego: Academic Press. ISBN 9780080532998. Retrieved 22 June 2013.
• net at PlanetMath.org.