Editing Net (mathematics) (section)

===Metric spaces===

Suppose <math>(M, d)</math> is a [[metric space]] (or a [[pseudometric space]]) and <math>M</math> is endowed with the [[metric topology]]. If <math>m \in M</math> is a point and <math>m_\bull = \left(m_i\right)_{a \in A}</math> is a net, then <math>m_\bull \to m</math> in <math>(M, d)</math> if and only if <math>d\left(m, m_\bull\right) \to 0</math> in <math>\R,</math> where <math>d\left(m, m_\bull\right) := \left(d\left(m, m_a\right)\right)_{a \in A}</math> is a net of [[real number]]s. 
In [[plain English]], this characterization says that a net converges to a point in a metric space if and only if the distance between the net and the point converges to zero. 
If <math>(M, \|\cdot\|)</math> is a [[normed space]] (or a [[seminormed space]]) then <math>m_\bull \to m</math> in <math>(M, \|\cdot\|)</math> if and only if <math>\left\|m - m_\bull\right\| \to 0</math> in <math>\Reals,</math> where <math>\left\|m - m_\bull\right\| := \left(\left\|m - m_a\right\|\right)_{a \in A}.</math>

If <math>(M, d)</math> has at least two points, then we can fix a point <math>c \in M</math> (such as <math>M := \R^n</math> with the [[Euclidean metric]] with <math>c := 0</math> being the origin, for example) and direct the set <math>I := M \setminus \{c\}</math> reversely according to distance from <math>c</math> by declaring that <math>i \leq j</math> if and only if <math>d(j, c) \leq d(i, c).</math> In other words, the relation is "has at least the same distance to <math>c</math> as", so that "large enough" with respect to this relation means "close enough to <math>c</math>". 
Given any function with domain <math>M,</math> its restriction to <math>I := M \setminus \{c\}</math> can be canonically interpreted as a net directed by <math>(I, \leq).</math>{{sfn|Willard|2004|p=77}} 

A net <math>f : M \setminus \{c\} \to X</math> is eventually in a subset <math>S</math> of a topological space <math>X</math> if and only if there exists some <math>n \in M \setminus \{c\}</math> such that for every <math>m \in M \setminus \{c\}</math> satisfying <math>d(m, c) \leq d(n, c),</math> the point <math>f(m)</math> is in <math>S.</math> 
Such a net <math>f</math> converges in <math>X</math> to a given point <math>L \in X</math> if and only if <math>\lim_{m \to c} f(m) \to L</math> in the usual sense (meaning that for every neighborhood <math>V</math> of <math>L,</math> <math>f</math> is eventually in <math>V</math>).{{sfn|Willard|2004|p=77}} 

The net <math>f : M \setminus \{c\} \to X</math> is frequently in a subset <math>S</math> of <math>X</math> if and only if for every <math>n \in M \setminus \{c\}</math> there exists some <math>m \in M \setminus \{c\}</math> with <math>d(m, c) \leq d(n, c)</math> such that <math>f(m)</math> is in <math>S.</math>
Consequently, a point <math>L \in X</math> is a cluster point of the net <math>f</math> if and only if for every neighborhood <math>V</math> of <math>L,</math> the net is frequently in <math>V.</math>