Net (mathematics)

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This article is about nets in topological spaces. For unfoldings of polyhedra, see Net (polyhedron).

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 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 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,[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[edit]

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

Examples of nets[edit]

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 Nx denote the set of all neighbourhoods containing x. Then Nx is a directed set, where the direction is given by reverse inclusion, so that ST if and only if S is contained in T. For S in Nx, let xS be a point in S. Then (xS) is a net. As S increases with respect to ≥, the points xS in the net are constrained to lie in decreasing neighbourhoods of x, so intuitively speaking, we are led to the idea that xS must tend towards x in some sense. We can make this limiting concept precise.

Limits of nets[edit]

If (xα) is a net from a directed set A into X, and if Y is a subset of X, then we say that (xα) 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 xβ lies in Y.

If (xα) 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 xα = x

if and only if

for every neighborhood U of x, (xα) is eventually in U.

Intuitively, this means that the values xα 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 (xα) is eventually in all members of the base containing this putative limit.

Examples of limits of nets[edit]

Supplementary definitions[edit]

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[edit]

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 nN, the point an is in Y.

We have limn an = 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 nN such that an 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 limxc 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 xa, the point f(x) is in Y.

We have limxc 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[edit]

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:

lim xα = x
we have
lim ƒ(xα) = 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.

Cauchy nets[edit]

Main article: Cauchy net

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[edit]

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] Therefore, any theorems that can be proven with one concept can be proven in 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[edit]

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[edit]

  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. ^ a b c 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