Editing Net (mathematics) (section)

==Definitions==

A [[directed set]] is a non-empty set <math>A</math> together with a [[preorder]], typically automatically assumed to be denoted by <math>\,\leq\,</math> (unless indicated otherwise), with the property that it is also ({{em|upward}}) {{em|directed}}, which means that for any <math>a, b \in A,</math> there exists some <math>c \in A</math> such that <math>a \leq c</math> and <math>b \leq c.</math> 
In words, this property means that given any two elements (of <math>A</math>), there is always some element that is "above" both of them (greater than or equal to each); in this way, directed sets generalize the notion of "a direction" in a mathematically rigorous way. Importantly though, directed sets are {{em|not}} required to be [[total order]]s or even [[partial order]]s. A directed set may have the [[greatest element]]. In this case, the conditions <math>a \leq c</math> and <math>b \leq c</math> cannot be replaced by the strict inequalities <math>a < c</math> and <math>b < c</math>, since the strict inequalities cannot be satisfied if ''a'' or ''b'' is the greatest element.

A '''net in''' <math>X</math>, denoted <math>x_\bull = \left(x_a\right)_{a \in A}</math>, is a [[Function (mathematics)|function]] of the form <math>x_\bull : A \to X</math> whose [[Domain of a function|domain]] <math>A</math> is some directed set, and whose values are <math>x_\bullet(a)= x_a </math>. Elements of a net's domain are called its {{em|indices}}. When the set <math>X</math> is clear from context it is simply called a '''net''', and one assumes <math>A</math> is a directed set with preorder <math>\,\leq.</math> Notation for nets varies, for example using angled brackets <math>\left\langle x_a \right\rangle_{a \in A}</math>. As is common in [[algebraic topology]] notation, the filled disk or "bullet" stands in place of the input variable or index <math>a \in A</math>.

=== Limits of nets ===

{{anchor|Limit of a net|Limit point of a net|Convergent net|Net convergence}}
A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is said to be {{em|eventually}} or {{em|residually}} {{em|in}} a set <math>S</math> if there exists some <math>a \in A</math> such that for every <math>b \in A</math> with <math>b \geq a,</math> the point <math>x_b \in S.</math> A point <math>x \in X</math> is called a {{em|{{visible anchor|limit point}}}} or {{em|{{visible anchor|limit|Limit of a net}}}} of the net <math>x_\bull</math> in <math>X</math> whenever:
:for every open [[Topological neighborhood|neighborhood]] <math>U</math> of <math>x,</math> the net <math>x_\bull</math> is eventually in <math>U</math>,
expressed equivalently as: the net {{em|{{visible anchor|converges|Convergent net}} to/towards <math>x</math>}} or {{em|has <math>x</math> as a limit}}; and variously denoted as:<math display="block">\begin{alignat}{4}
                  & x_\bull && \to\; && x && \;\;\text{ in } X \\
                  & x_a     && \to\; && x && \;\;\text{ in } X \\
\lim        \; & x_\bull && \to\; && x && \;\;\text{ in } X \\
\lim_{a \in A} \; & x_a     && \to\; && x && \;\;\text{ in } X \\
\lim_a   \; & x_a     && \to\; && x && \;\;\text{ in } X.
\end{alignat}</math>If <math>X</math> is clear from context, it may be omitted from the notation.

If <math>\lim x_\bull \to x</math> and this limit is unique (i.e. <math>\lim x_\bull \to y</math> only for <math>x = y</math>) then one writes:<math display=block>\lim x_\bull = x \;~~ \text{ or } ~~\; \lim x_a = x \;~~ \text{ or } ~~\; \lim_{a \in A} x_a = x</math>using the equal sign in place of the arrow <math>\to.</math>{{sfn|Kelley|1975|pp=65–72}} In a [[Hausdorff space]], every net has at most one limit, and the limit of a convergent net is always unique.{{sfn|Kelley|1975|pp=65–72}}
Some authors do not distinguish between the notations <math>\lim x_\bull = x</math> and <math>\lim x_\bull \to x</math>, but this can lead to ambiguities if the ambient space ''<math>X</math>'' is not Hausdorff.

=== Cluster points of nets ===

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is said to be {{em|{{visible anchor|frequently in|text=frequently}}}} or {{em|{{visible anchor|cofinally in}}}} <math>S</math> if for every <math>a \in A</math> there exists some <math>b \in A</math> such that <math>b \geq a</math> and <math>x_b \in S.</math>{{sfn|Willard|2004|pp=73–77}} A point <math>x \in X</math> is said to be an {{em|{{visible anchor|accumulation point}}}} or ''cluster point'' of a net if for every neighborhood <math>U</math> of <math>x,</math> the net is frequently/cofinally in <math>U.</math>{{sfn|Willard|2004|pp=73–77}} In fact, <math>x \in X</math> is a cluster point if and only if it has a subnet that converges to <math>x.</math>{{sfn|Willard|2004|p=75}} The set <math display="inline">\operatorname{cl}_X \left( x_{\bullet} \right)   </math> of all cluster points of <math>x_\bull</math> in <math>X</math> is equal to <math display="inline">\operatorname{cl}_X \left(x_{\geq a} \right) </math> for each <math>a\in A </math>, where <math>x_{\geq a} := \left\{x_b : b \geq a, b \in A\right\}</math>.

===Subnets===
{{Main|Subnet (mathematics)}}
{{See also|Filters in topology#Subnets}}

The analogue of "[[subsequence]]" for nets is the notion of a "subnet". There are several different non-equivalent definitions of "subnet" and this article will use the definition introduced in 1970 by Stephen Willard,{{sfn|Schechter|1996|pp=157–168}} which is as follows: 
If <math>x_\bull = \left(x_a\right)_{a \in A}</math> and <math>s_\bull = \left(s_i\right)_{i \in I}</math> are nets then <math>s_\bull</math> is called a {{em|subnet}} or {{em|{{visible anchor|Willard-subnet}}}}{{sfn|Schechter|1996|pp=157–168}} of <math>x_\bull</math> if there exists an order-preserving map <math>h : I \to A</math> such that <math>h(I)</math> is a [[Cofinal (mathematics)|cofinal]] subset of <math>A</math> and 
<math display=block>s_i = x_{h(i)} \quad \text{ for all } i \in I.</math> 
The map <math>h : I \to A</math> is called {{em|[[order-preserving]]}} and an {{em|order homomorphism}} if whenever <math>i \leq j</math> then <math>h(i) \leq h(j).</math> 
The set <math>h(I)</math> being {{em|[[Cofinal (mathematics)|cofinal]]}} in <math>A</math> means that for every <math>a \in A,</math> there exists some <math>b \in h(I)</math> such that <math>b \geq a.</math>

If <math>x \in X</math> is a cluster point of some subnet of <math>x_\bull</math> then <math>x</math> is also a cluster point of <math>x_\bull.</math>{{sfn|Willard|2004|p=75}}

===Ultranets===

A net <math>x_\bull</math> in set <math>X</math> is called a {{em|{{visible anchor|universal net}}}} or an {{em|{{visible anchor|ultranet}}}} if for every subset <math>S \subseteq X,</math> <math>x_\bull</math> is eventually in <math>S</math> or <math>x_\bull</math> is eventually in the complement <math>X \setminus S.</math>{{sfn|Willard|2004|pp=73–77}} 

Every constant net is a (trivial) ultranet. Every subnet of an ultranet is an ultranet.{{sfn|Willard|2004|p=77}} Assuming the [[axiom of choice]], every net has some subnet that is an ultranet, but no nontrivial ultranets have ever been constructed explicitly.{{sfn|Willard|2004|pp=73–77}} 
If <math>x_\bull = \left(x_a\right)_{a \in A}</math> is an ultranet in <math>X</math> and <math>f : X \to Y</math> is a function then <math>f \circ x_\bull = \left(f\left(x_a\right)\right)_{a \in A}</math> is an ultranet in <math>Y.</math>{{sfn|Willard|2004|pp=73–77}} 

Given <math>x \in X,</math> an ultranet clusters at <math>x</math> if and only it converges to <math>x.</math>{{sfn|Willard|2004|pp=73–77}}

===Cauchy nets===

A Cauchy net generalizes the notion of [[Cauchy sequence]] to nets defined on [[uniform space]]s.<ref name="willard">{{citation|title=General Topology|series=Dover Books on Mathematics|first=Stephen|last=Willard|publisher=Courier Dover Publications|year=2012|isbn=9780486131788|page=260|url=https://books.google.com/books?id=UrsHbOjiR8QC&pg=PA26}}.</ref>

A net <math>x_\bull = \left(x_a\right)_{a \in A}</math> is a {{em|{{visible anchor|Cauchy net}}}} if for every [[Entourage (mathematics)|entourage]] <math>V</math> there exists <math>c \in A</math> such that for all <math>a, b \geq c,</math> <math>\left(x_a, x_b\right)</math> is a member of <math>V.</math><ref name="willard"/><ref>{{citation|title=Introduction to General Topology|first=K. D.|last=Joshi|publisher=New Age International|year=1983|isbn=9780852264447|page=356|url=https://books.google.com/books?id=fvCpXrube5wC&pg=PA356}}.</ref> More generally, in a [[Cauchy space]], a net <math>x_\bull</math> is Cauchy if the filter generated by the net is a [[Cauchy filter]].

A [[topological vector space]] (TVS) is called {{em|[[Complete topological vector space|complete]]}} if every Cauchy net converges to some point. A [[normed space]], which is a special type of topological vector space, is a complete TVS (equivalently, a [[Banach space]]) if and only if every Cauchy sequence converges to some point (a property that is called {{em|sequential completeness}}). Although Cauchy nets are not needed to describe completeness of normed spaces, they are needed to describe completeness of more general (possibly non-[[Normable space|normable]]) topological vector spaces.