Editing Net (mathematics) (section)

===Cluster and limit points===

The set of cluster points of a net is equal to the set of limits of its convergent [[Subnet (mathematics)|subnet]]s.

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Let <math>x_\bull = \left(x_a\right)_{a \in A}</math> be a net in a topological space <math>X</math> (where as usual <math>A</math> automatically assumed to be a directed set) and also let <math>y \in X.</math> If <math>y</math> is a limit of a subnet of <math>x_\bull</math> then <math>y</math> is a cluster point of <math>x_\bull.</math>

Conversely, assume that <math>y</math> is a cluster point of <math>x_\bull.</math>
Let <math>B</math> be the set of pairs <math>(U, a)</math> where <math>U</math> is an open neighborhood of <math>y</math> in <math>X</math> and <math>a \in A</math> is such that <math>x_a \in U.</math>
The map <math>h : B \to A</math> mapping <math>(U, a)</math> to <math>a</math> is then cofinal.
Moreover, giving <math>B</math> the [[product order]] (the neighborhoods of <math>y</math> are ordered by inclusion) makes it a directed set, and the net <math>\left(y_b\right)_{b \in B}</math> defined by <math>y_b = x_{h(b)}</math> converges to <math>y.</math>
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A net has a limit if and only if all of its subnets have limits. In that case, every limit of the net is also a limit of every subnet.