Editing Net (mathematics) (section)

===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}}