Editing Net (mathematics) (section)

==As generalization of sequences==

Every non-empty [[Total order|totally ordered set]] is directed. Therefore, every function on such a set is a net. In particular, the [[natural number]]s <math>\N</math> together with the usual integer comparison <math>\,\leq\,</math> preorder form the [[wiktionary:archetypical|archetypical]] example of a directed set. A sequence is a function on the natural numbers, so every sequence <math>a_1, a_2, \ldots</math> in a topological space <math>X</math> can be considered a net in <math>X</math> defined on <math>\N.</math> Conversely, any net whose domain is the natural numbers is a [[sequence]] because by definition, a sequence in <math>X</math> is just a function from <math>\N = \{1, 2, \ldots\}</math> into <math>X.</math> It is in this way that nets are generalizations of sequences: rather than being defined on a [[countable set|countable]] [[Total order|linearly ordered]] set (<math>\N</math>), a net is defined on an arbitrary [[directed set]]. Nets are frequently denoted using notation that is similar to (and inspired by) that used with sequences. For example, the subscript notation <math>x_a</math> is taken from sequences.

Similarly, every [[limit of a sequence]] and [[limit of a function]] can be interpreted as a limit of a net. Specifically, the net is eventually in a subset <math>S</math> of <math>X</math> if there exists an <math>N \in \N</math> such that for every integer <math>n \geq N,</math> the point <math>a_n</math> is in <math>S.</math> So <math>\lim {}_n a_n \to L</math> if and only if for every neighborhood <math>V</math> of <math>L,</math> the net is eventually in <math>V.</math> The net is frequently in a subset <math>S</math> of <math>X</math> if and only if for every <math>N \in \N</math> there exists some integer <math>n \geq N</math> such that <math>a_n \in S,</math> that is, if and only if infinitely many elements of the sequence are in <math>S.</math> Thus a point <math>y \in X</math> is a cluster point of the net if and only if every neighborhood <math>V</math> of <math>y</math> contains infinitely many elements of the sequence.

In the context of topology, sequences do not fully encode all information about functions between topological spaces. In particular, the following two conditions are, in general, not equivalent for a map <math>f</math> between topological spaces <math>X</math> and <math>Y</math>:

#The map <math>f</math> is [[Continuous function#Continuous functions between topological spaces|continuous in the topological sense]];
#Given any point <math>x</math> in <math>X,</math> and any sequence in <math>X</math> converging to <math>x,</math> the composition of <math>f</math> with this sequence converges to <math>f(x)</math> [[Continuous function#Definition in terms of limits of sequences|(continuous in the sequential sense)]].

While condition 1 always guarantees condition 2, the converse is not necessarily true. The spaces for which the two conditions are equivalent are called [[sequential space]]s. All [[first-countable space]]s, including [[metric space]]s, are sequential spaces, but not all topological spaces are sequential. Nets generalize the notion of a sequence so that condition 2 reads as follows:

# <li value="2">Given any point <math>x</math> in <math>X,</math> and any net in <math>X</math> converging to <math>x,</math> the composition of <math>f</math> with this net converges to <math>f(x)</math> (continuous in the net sense).</li>

With this change, the conditions become equivalent for all maps of topological spaces, including 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 set]]s in behavior.

For an example where sequences do not suffice, interpret the set <math>\Reals^\Reals</math> of all functions with prototype <math>f : \Reals \to \Reals</math> as the Cartesian product <math>{\textstyle\prod\limits_{x \in \Reals}} \Reals</math> (by identifying a function <math>f</math> with the tuple <math>(f(x))_{x \in \Reals},</math> and conversely) and endow it with the [[product topology]]. This (product) topology on <math>\Reals^\Reals</math> is identical to the [[topology of pointwise convergence]]. Let <math>E</math> denote the set of all functions <math>f : \Reals \to \{0, 1\}</math> that are equal to <math>1</math> everywhere except for at most finitely many points (that is, such that the set <math>\{x : f(x) = 0\}</math> is finite). Then the constant <math>0</math> function <math>\mathbf{0} : \Reals \to \{0\}</math> belongs to the closure of <math>E</math> in <math>\Reals^\Reals;</math> that is, <math>\mathbf{0} \in \operatorname{cl}_{\Reals^\Reals} E.</math>{{sfn|Willard|2004|p=77}} This will be proven by constructing a net in <math>E</math> that converges to <math>\mathbf{0}.</math> However, there does not exist any {{em|sequence}} in <math>E</math> that converges to <math>\mathbf{0},</math>{{sfn|Willard|2004|pp=71–72}} which makes this one instance where (non-sequence) nets must be used because sequences alone can not reach the desired conclusion. Compare elements of <math>\Reals^\Reals</math> pointwise in the usual way by declaring that <math>f \geq g</math> if and only if <math>f(x) \geq g(x)</math> for all <math>x.</math> This pointwise comparison is a partial order that makes <math>(E, \geq)</math> a directed set since given any <math>f, g \in E,</math> their pointwise minimum <math>m := \min \{f, g\}</math> belongs to <math>E</math> and satisfies <math>f \geq m</math> and <math>g \geq m.</math> This partial order turns the [[identity map]] <math>\operatorname{Id} : (E, \geq) \to E</math> (defined by <math>f \mapsto f</math>) into an <math>E</math>-valued net. This net converges pointwise to <math>\mathbf{0}</math> in <math>\Reals^\Reals,</math> which implies that <math>\mathbf{0}</math> belongs to the closure of <math>E</math> in <math>\Reals^\Reals.</math>

More generally, a subnet of a sequence is {{em|not}} necessarily a sequence.{{sfn|Willard|2004|pp=73–77}}{{efn|For an example, let <math>X = \Reals^n</math> and let <math>x_i = 0</math> for every <math>i \in \N,</math> so that <math>x_\bull = (0)_{i \in \N} : \N \to X</math> is the constant zero sequence. 
Let <math>I = \{r \in \Reals : r > 0\}</math> be directed by the usual order <math>\,\leq\,</math> and let <math>s_r = 0</math> for each <math>r \in R.</math> 
Define <math>\varphi : I \to \N</math> by letting <math>\varphi(r) = \lceil r \rceil</math> be the [[Ceiling function|ceiling]] of <math>r.</math> 
The map <math>\varphi : I \to \N</math> is an order morphism whose image is cofinal in its codomain and <math>\left(x_\bull \circ \varphi\right)(r) = x_{\varphi(r)} = 0 = s_r</math> holds for every <math>r \in R.</math> This shows that <math>\left(s_{r}\right)_{r \in R} = x_\bull \circ \varphi</math> is a subnet of the sequence <math>x_\bull</math> (where this subnet is not a subsequence of <math>x_\bull</math> because it is not even a sequence since its domain is an [[uncountable set]]).}} Moreso, a subnet of a sequence may be a sequence, but not a subsequence.{{efn|The sequence <math>\left(s_i\right)_{i \in \N} := (1, 1, 2, 2, 3, 3, \ldots)</math> is not a subsequence of <math>\left(x_i\right)_{i \in \N} := (1, 2, 3, \ldots)</math>, although it is a subnet, because the map <math>h : \N \to \N</math> defined by <math>h(i) := \left\lfloor \tfrac{i + 1}{2} \right\rfloor</math> is an order-preserving map whose image is <math>h(\N) = \N</math> and satisfies <math>s_i = x_{h(i)}</math> for all <math>i \in \N.</math> Indeed, this is because <math>x_i = i</math> and <math>s_i = h(i)</math> for every <math>i \in \N;</math> in other words, when considered as functions on <math>\N,</math> the sequence <math>x_{\bull}</math> is just the identity map on <math>\N</math> while <math>s_{\bull} = h.</math>}} But, in the specific case of a sequential space, every net induces a corresponding sequence, and this relationship maps subnets to subsequences. Specifically, for a first-countable space, the net <math>\left(x_a\right)_{a \in A}</math> induces the sequence <math>\left(x_{h_n}\right)_{n \in \N}</math> where <math>h_n</math> is defined as the <math>n^{\text{th}}</math> smallest value in <math>A</math>{{spaced ndash}}that is, let <math>h_1 := \inf A</math> and let <math>h_n := \inf \{a \in A : a > h_{n-1}\}</math> for every integer <math>n > 1</math>.