Editing Net (mathematics) (section)

=== 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>.