Editing Accumulation point (section)

{{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>]]