Editing Accumulation point (section)

===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>