Editing Net (mathematics) (section)

===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}}