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{{Short description|Cluster point in a topological space}}
{{redirect|Limit point|uses where the word "point" is optional|Limit (mathematics)|and|Limit (disambiguation)#Mathematics}}
In mathematics, a '''limit point''', '''accumulation point''', or '''cluster point''' of a [[Set (mathematics)|set]] <math>S</math> in a [[topological space]] <math>X</math> is a point <math>x</math> that can be "approximated" by points of <math>S</math> in the sense that every [[neighbourhood (mathematics)|neighbourhood]] of <math>x</math> contains a point of <math>S</math> other than <math>x</math> itself. A limit point of a set <math>S</math> does not itself have to be an element of <math>S.</math> 
There is also a closely related concept for [[sequence]]s. A '''cluster point''' or '''accumulation point''' of a [[sequence]] <math>(x_n)_{n \in \N}</math> in a [[topological space]] <math>X</math> is a point <math>x</math> such that, for every neighbourhood <math>V</math> of <math>x,</math> there are infinitely many natural numbers <math>n</math> such that <math>x_n \in V.</math> This definition of a cluster or accumulation point of a sequence generalizes to [[Net (mathematics)|nets]] and [[Filter (set theory)|filters]]. 

The similarly named notion of a {{em|[[limit point of a sequence]]}}{{sfn|Dugundji|1966|pp=209-210}} (respectively, a [[limit point of a filter]],{{sfn|Bourbaki|1989|pp=68-83}} a [[limit point of a net]]) by definition refers to a point that the [[Convergent sequence|sequence converges to]] (respectively, the [[Convergent filter|filter converges to]], the [[Convergent net|net converges to]]). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is {{em|not}} synonymous with "cluster/accumulation point of a sequence". 

The limit points of a set should not be confused with [[adherent point]]s (also called {{em|points of [[Closure (topology)|closure]]}}) for which every neighbourhood of <math>x</math> contains ''some'' point of <math>S</math>. Unlike for limit points, an adherent point <math>x</math> of <math>S</math> may have a neighbourhood not containing points other than <math>x</math> itself. A limit point can be characterized as an adherent point that is not an [[isolated point]].

Limit points of a set should also not be confused with [[boundary point]]s. For example, <math>0</math> is a boundary point (but not a limit point) of the set <math>\{0\}</math> in <math>\R</math> with [[standard topology]]. However, <math>0.5</math> is a limit point (though not a boundary point) of interval <math>[0, 1]</math> in <math>\R</math> with standard topology (for a less trivial example of a limit point, see the first caption).<ref>{{Cite web|date=2021-01-13|title=Difference between boundary point & limit point.|url=https://math.stackexchange.com/a/1290541}}</ref><ref>{{Cite web|date=2021-01-13|title=What is a limit point|url=https://math.stackexchange.com/a/663768}}</ref><ref>{{Cite web|date=2021-01-13|title=Examples of Accumulation Points|url=https://www.bookofproofs.org/branches/examples-of-accumulation-points/|access-date=2021-01-14|archive-date=2021-04-21|archive-url=https://web.archive.org/web/20210421215655/https://www.bookofproofs.org/branches/examples-of-accumulation-points/|url-status=dead}}</ref>

This concept profitably generalizes the notion of a [[Limit (mathematics)|limit]] and is the underpinning of concepts such as [[closed set]] and [[topological closure]]. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.

[[File:Rational sequence with 2 accumulation points.svg|thumb|400px|With respect to the usual [[Topological space#Examples of topological spaces|Euclidean topology]], the sequence of rational numbers <math>x_n=(-1)^n \frac{n}{n+1}</math> has no {{em|[[Limit of a sequence#Topological spaces|limit]]}} (i.e. does not converge), but has two accumulation points (which are considered {{em|limit points}} here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set <math>S = \{x_n\}.</math>]]

==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]].

==Relation between accumulation point of a sequence and accumulation point of a set==

Every sequence <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> in <math>X</math> is by definition just a map <math>x_{\bull} : \N \to X</math> so that its [[Image (mathematics)|image]] <math>\operatorname{Im} x_{\bull} := \left\{ x_n : n \in \N \right\}</math> can be defined in the usual way. 

* If there exists an element <math>x \in X</math> that occurs infinitely many times in the sequence, <math>x</math> is an accumulation point of the sequence. But <math>x</math> need not be an accumulation point of the corresponding set <math>\operatorname{Im} x_{\bull}.</math> For example, if the sequence is the constant sequence with value <math>x,</math> we have <math>\operatorname{Im} x_{\bull} = \{ x \}</math> and <math>x</math> is an isolated point of <math>\operatorname{Im} x_{\bull}</math> and not an accumulation point of <math>\operatorname{Im} x_{\bull}.</math>

* If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an <math>\omega</math>-accumulation point of the associated set <math>\operatorname{Im} x_{\bull}.</math>

Conversely, given a countable infinite set <math>A \subseteq X</math> in <math>X,</math> we can enumerate all the elements of <math>A</math> in many ways, even with repeats, and thus associate with it many sequences <math>x_{\bull}</math> that will satisfy <math>A = \operatorname{Im} x_{\bull}.</math>

* Any <math>\omega</math>-accumulation point of <math>A</math> is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of <math>A</math> and hence also infinitely many terms in any associated sequence).

* A point <math>x \in X</math> that is {{em|not}} an <math>\omega</math>-accumulation point of <math>A</math> cannot be an accumulation point of any of the associated sequences without infinite repeats (because <math>x</math> has a neighborhood that contains only finitely many (possibly even none) points of <math>A</math> and that neighborhood can only contain finitely many terms of such sequences).

==Properties==

Every [[Limit of a sequence#Topological spaces|limit]] of a non-constant sequence is an accumulation point of the sequence.
And by definition, every limit point is an [[adherent point]].

The closure <math>\operatorname{cl}(S)</math> of a set <math>S</math> is a [[disjoint union]] of its limit points <math>L(S)</math> and isolated points <math>I(S)</math>; that is,
<math display="block">\operatorname{cl} (S) = L(S) \cup I(S)\quad\text{and}\quad L(S) \cap I(S) = \emptyset.</math>

A point <math>x \in X</math> is a limit point of <math>S \subseteq X</math> if and only if it is in the [[Closure (topology)|closure]] of <math>S \setminus \{ x \}.</math>
{{math proof | proof =
We use the fact that a point is in the closure of a set if and only if every neighborhood of the point meets the set. Now, <math>x</math> is a limit point of <math>S,</math> if and only if every neighborhood of <math>x</math> contains a point of <math>S</math> other than <math>x,</math> if and only if every neighborhood of <math>x</math> contains a point of <math>S \setminus \{x\},</math> if and only if <math>x</math> is in the closure of <math>S \setminus \{x\}.</math>
}}

If we use <math>L(S)</math> to denote the set of limit points of <math>S,</math> then we have the following characterization of the closure of <math>S</math>: The closure of <math>S</math> is equal to the union of <math>S</math> and <math>L(S).</math> This fact is sometimes taken as the {{em|definition}} of [[Closure (topology)|closure]].
{{math proof | proof =
("Left subset") Suppose <math>x</math> is in the closure of <math>S.</math> If <math>x</math> is in <math>S,</math> we are done. If <math>x</math> is not in <math>S,</math> then every neighbourhood of <math>x</math> contains a point of <math>S,</math> and this point cannot be <math>x.</math> In other words, <math>x</math> is a limit point of <math>S</math> and <math>x</math> is in <math>L(S).</math>

("Right subset") If <math>x</math> is in <math>S,</math> then every neighbourhood of <math>x</math> clearly meets <math>S,</math> so <math>x</math> is in the closure of <math>S.</math> If <math>x</math> is in <math>L(S),</math> then every neighbourhood of <math>x</math> contains a point of <math>S</math> (other than <math>x</math>), so <math>x</math> is again in the closure of <math>S.</math> This completes the proof.
}} 

A corollary of this result gives us a characterisation of closed sets: A set <math>S</math> is closed if and only if it contains all of its limit points.
{{math proof | proof =
''Proof'' 1: <math>S</math> is closed if and only if <math>S</math> is equal to its closure if and only if <math>S=S\cup L(S)</math> if and only if <math>L(S)</math> is contained in <math>S.</math>

''Proof'' 2: Let <math>S</math> be a closed set and <math>x</math> a limit point of <math>S.</math> If <math>x</math> is not in <math>S,</math> then the complement to <math>S</math> comprises an open neighbourhood of <math>x.</math> Since <math>x</math> is a limit point of <math>S,</math> any open neighbourhood of <math>x</math> should have a non-trivial intersection with <math>S.</math> However, a set can not have a non-trivial intersection with its complement. Conversely, assume <math>S</math> contains all its limit points. We shall show that the complement of <math>S</math> is an open set. Let <math>x</math> be a point in the complement of <math>S.</math> By assumption, <math>x</math> is not a limit point, and hence there exists an open neighbourhood <math>U</math> of <math>x</math> that does not intersect <math>S,</math> and so <math>U</math> lies entirely in the complement of <math>S.</math> Since this argument holds for arbitrary <math>x</math> in the complement of <math>S,</math> the complement of <math>S</math> can be expressed as a union of open neighbourhoods of the points in the complement of <math>S.</math> Hence the complement of <math>S</math> is open.
}} 

No [[isolated point]] is a limit point of any set.
{{math proof | proof =
If <math>x</math> is an isolated point, then <math>\{x\}</math> is a neighbourhood of <math>x</math> that contains no points other than <math>x.</math>
}} 

A space <math>X</math> is [[discrete space|discrete]] if and only if no subset of <math>X</math> has a limit point.
{{math proof | proof =
If <math>X</math> is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if <math>X</math> is not discrete, then there is a singleton <math>\{ x \}</math> that is not open. Hence, every open neighbourhood of <math>\{ x \}</math> contains a point <math>y \neq x,</math> and so <math>x</math> is a limit point of <math>X.</math>
}} 

If a space <math>X</math> has the [[trivial topology]] and <math>S</math> is a subset of <math>X</math> with more than one element, then all elements of <math>X</math> are limit points of <math>S.</math> If <math>S</math> is a singleton, then every point of <math>X \setminus S</math> is a limit point of <math>S.</math>
{{math proof | proof =
As long as <math>S \setminus \{ x \}</math> is nonempty, its closure will be <math>X.</math> It is only empty when <math>S</math> is empty or <math>x</math> is the unique element of <math>S.</math>
}}

==See also==

* {{annotated link|Adherent point}}
* {{annotated link|Condensation point}}
* {{annotated link|Convergent filter}}
* {{annotated link|Derived set (mathematics)}}
* {{annotated link|Filters in topology}}
* {{annotated link|Isolated point}}
* {{annotated link|Limit of a function}}
* {{annotated link|Limit of a sequence}}
* {{annotated link|Subsequential limit}}

==Citations==

{{reflist|group=note}}
{{reflist}}

==References==

* {{Bourbaki General Topology Part I Chapters 1-4}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Dugundji Topology}} <!--{{sfn|Dugundji|1966|p=}}-->
* {{Munkres Topology|edition=2}} <!--{{sfn|Munkres|2000|p=}}-->
* {{springer|title=Limit point of a set|id=p/l058880}}

{{Topology}}

[[Category:Limit sets| ]]
[[Category:Topology]]
[[Category:General topology]]