Editing Accumulation point (section)

==Definition==

===Accumulation points of a set===
[[File:Diagonal argument.svg|thumb|A sequence enumerating all positive [[rational number]]s. Each positive [[real number]] is a cluster point.]]

Let <math>S</math> be a subset of a [[topological space]] <math>X.</math> 
A point <math>x</math> in <math>X</math> is a '''limit point''' or '''cluster point''' or '''{{visible anchor|accumulation point of the set}}''' <math>S</math> if every [[Neighbourhood (mathematics)|neighbourhood]] of <math>x</math> contains at least one point of <math>S</math> different from <math>x</math> itself. 

It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point. 

If <math>X</math> is a [[T1 space|<math>T_1</math> space]] (such as a [[metric space]]), then <math>x \in X</math> is a limit point of <math>S</math> if and only if every neighbourhood of <math>x</math> contains infinitely many points of <math>S.</math>{{sfn|Munkres|2000|pp=97-102}} In fact, <math>T_1</math> spaces are characterized by this property. 

If <math>X</math> is a [[Fréchet–Urysohn space]] (which all [[metric space]]s and [[first-countable space]]s are), then <math>x \in X</math> is a limit point of <math>S</math> if and only if there is a [[sequence]] of points in <math>S \setminus \{x\}</math> whose [[Limit of a sequence|limit]] is <math>x.</math> In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of <math>S</math> is called the [[Derived set (mathematics)|derived set]] of <math>S.</math>

====Special types of accumulation point of a set====

If every neighbourhood of <math>x</math> contains infinitely many points of <math>S,</math> then <math>x</math> is a specific type of limit point called an '''{{visible anchor|ω-accumulation point}}''' of <math>S.</math> 

If every neighbourhood of <math>x</math> contains [[Uncountable set|uncountably many]] points of <math>S,</math> then <math>x</math> is a specific type of limit point called a '''[[condensation point]]''' of <math>S.</math> 

If every neighbourhood <math>U</math> of <math>x</math> is such that the [[cardinality]] of
<math>U \cap S</math> equals the cardinality of <math>S,</math> then <math>x</math> is a specific type of limit point called a '''{{visible anchor|complete accumulation point}}''' of <math>S.</math>

===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]].