Editing Accumulation point (section)

===Accumulation points of sequences and nets===
{{anchor|sequence accumulation point|Cluster points of sequences and nets}}
{{See also|Net (mathematics)#Cluster point of a net|Cluster point of a filter}}

In a topological space <math>X,</math> a point <math>x \in X</math> is said to be a '''{{visible anchor|cluster point of a sequence|text=cluster point}}''' or '''{{visible anchor|accumulation point of a sequence|Accumulation point of a sequence}}''' <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> if, for every [[Neighbourhood (mathematics)|neighbourhood]] <math>V</math> of <math>x,</math> there are infinitely many <math>n \in \N</math> such that <math>x_n \in V.</math> 
It is equivalent to say that for every neighbourhood <math>V</math> of <math>x</math> and every <math>n_0 \in \N,</math> there is some <math>n \geq n_0</math> such that <math>x_n \in V.</math> 
If <math>X</math> is a [[metric space]] or a [[first-countable space]] (or, more generally, a [[Fréchet–Urysohn space]]), then <math>x</math> is a cluster point of <math>x_{\bull}</math> if and only if <math>x</math> is a limit of some subsequence of <math>x_{\bull}.</math> 
The set of all cluster points of a sequence is sometimes called the [[limit set]]. 

Note that there is already the notion of [[Limit of a sequence#Topological spaces|limit of a sequence]] to mean a point <math>x</math> to which the sequence converges (that is, every neighborhood of <math>x</math> contains all but finitely many elements of the sequence). That is why we do not use the term {{em|limit point}} of a sequence as a synonym for accumulation point of the sequence.

The concept of a [[Net (mathematics)|net]] generalizes the idea of a [[sequence]]. A net is a function <math>f : (P,\leq) \to X,</math> where <math>(P,\leq)</math> is a [[directed set]] and <math>X</math> is a topological space. A point <math>x \in X</math> is said to be a [[Cluster point of a net|'''{{visible anchor|cluster point of a net|text=cluster point}}''']] or [[Accumulation point of a net|'''{{visible anchor|accumulation point of a net|Accumulation point of a net}}''']] <math>f</math> if, for every [[Neighbourhood (mathematics)|neighbourhood]] <math>V</math> of <math>x</math> and every <math>p_0 \in P,</math> there is some <math>p \geq p_0</math> such that <math>f(p) \in V,</math> equivalently, if <math>f</math> has a [[Subnet (mathematics)|subnet]] which converges to <math>x.</math> Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. [[Cluster point of a filter|Clustering]] and [[Limit point of a filter|limit points]] are also defined for [[Filter (set theory)|filters]].