Accumulation point

Every sequence ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}$ in ${\displaystyle X}$ is by definition just a map ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$ so that its image ${\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{n}:n\in \mathbb {N} \right\}}$ can be defined in the usual way.

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

• 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 ${\displaystyle \omega }$-accumulation point of the associated set ${\displaystyle \operatorname {Im} x_{\bullet }.}$

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

• Any ${\displaystyle \omega }$-accumulation point of ${\displaystyle A}$ is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of ${\displaystyle A}$ and hence also infinitely many terms in any associated sequence).

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

Every 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 ${\displaystyle \operatorname {cl} (S)}$ of a set ${\displaystyle S}$ is a disjoint union of its limit points ${\displaystyle L(S)}$ and isolated points ${\displaystyle I(S)}$; that is,

$${\displaystyle \operatorname {cl} (S)=L(S)\cup I(S)\quad {\text{and}}\quad L(S)\cap I(S)=\emptyset .}$$

A point ${\displaystyle x\in X}$ is a limit point of ${\displaystyle S\subseteq X}$ if and only if it is in the closure of ${\displaystyle S\setminus \{x\}.}$

If we use ${\displaystyle L(S)}$ to denote the set of limit points of ${\displaystyle S,}$ then we have the following characterization of the closure of ${\displaystyle S}$: The closure of ${\displaystyle S}$ is equal to the union of ${\displaystyle S}$ and ${\displaystyle L(S).}$ This fact is sometimes taken as the definition of closure.

A corollary of this result gives us a characterisation of closed sets: A set ${\displaystyle S}$ is closed if and only if it contains all of its limit points.

No isolated point is a limit point of any set.

A space ${\displaystyle X}$ is discrete if and only if no subset of ${\displaystyle X}$ has a limit point.

If a space ${\displaystyle X}$ has the trivial topology and ${\displaystyle S}$ is a subset of ${\displaystyle X}$ with more than one element, then all elements of ${\displaystyle X}$ are limit points of ${\displaystyle S.}$ If ${\displaystyle S}$ is a singleton, then every point of ${\displaystyle X\setminus S}$ is a limit point of ${\displaystyle S.}$

• Adherent point – Point that belongs to the closure of some given subset of a topological space
• Condensation point – a stronger analog of limit pointPages displaying wikidata descriptions as a fallback
• Convergent filter – Use of filters to describe and characterize all basic topological notions and results.Pages displaying short descriptions of redirect targets
• Derived set (mathematics) – set of all limit points of a setPages displaying wikidata descriptions as a fallback
• Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
• Isolated point – Point of a subset S around which there are no other points of S
• Limit of a function – Point to which functions converge in analysis
• Limit of a sequence – Value to which tends an infinite sequence
• Subsequential limit – The limit of some subsequence

• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
• "Limit point of a set", Encyclopedia of Mathematics, EMS Press, 2001 [1994]