Accumulation point

"Limit point" redirects here. For uses where the word "point" is optional, see Limit (mathematics) and Limit (disambiguation) § Mathematics.

In mathematics, a limit point, accumulation point, or cluster point of a set ${\displaystyle S}$ in a topological space ${\displaystyle X}$ is a point ${\displaystyle x}$ that can be "approximated" by points of ${\displaystyle S}$ in the sense that every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself. A limit point of a set ${\displaystyle S}$ does not itself have to be an element of ${\displaystyle S.}$ There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ in a topological space ${\displaystyle X}$ is a point ${\displaystyle x}$ such that, for every neighbourhood ${\displaystyle V}$ of ${\displaystyle x,}$ there are infinitely many natural numbers ${\displaystyle n}$ such that ${\displaystyle x_{n}\in V.}$ This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.

The similarly named notion of a limit point of a sequence^[1] (respectively, a limit point of a filter,^[2] a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the 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 not synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called points of closure) for which every neighbourhood of ${\displaystyle x}$ contains some point of ${\displaystyle S}$. Unlike for limit points, an adherent point ${\displaystyle x}$ of ${\displaystyle S}$ may have a neighbourhood not containing points other than ${\displaystyle x}$ 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 points. For example, ${\displaystyle 0}$ is a boundary point (but not a limit point) of the set ${\displaystyle \{0\}}$ in ${\displaystyle \mathbb {R} }$ with standard topology. However, ${\displaystyle 0.5}$ is a limit point (though not a boundary point) of interval ${\displaystyle [0,1]}$ in ${\displaystyle \mathbb {R} }$ with standard topology (for a less trivial example of a limit point, see the first caption).^[3][4][5]

This concept profitably generalizes the notion of a 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.

With respect to the usual Euclidean topology, the sequence of rational numbers ${\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}}$ has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set ${\displaystyle S=\{x_{n}\}.}$

Definition

Accumulation points of a set

 

A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

Let ${\displaystyle S}$  be a subset of a topological space ${\displaystyle X.}$  A point ${\displaystyle x}$  in ${\displaystyle X}$  is a limit point or cluster point or accumulation point of the set ${\displaystyle S}$  if every neighbourhood of ${\displaystyle x}$  contains at least one point of ${\displaystyle S}$  different from ${\displaystyle x}$  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 ${\displaystyle X}$  is a ${\displaystyle T_{1}}$  space (such as a metric space), then ${\displaystyle x\in X}$  is a limit point of ${\displaystyle S}$  if and only if every neighbourhood of ${\displaystyle x}$  contains infinitely many points of ${\displaystyle S.}$ ^[6] In fact, ${\displaystyle T_{1}}$  spaces are characterized by this property.

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

The set of limit points of ${\displaystyle S}$  is called the derived set of ${\displaystyle S.}$ 

Special types of accumulation point of a set

If every neighbourhood of ${\displaystyle x}$  contains infinitely many points of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called an ω-accumulation point of ${\displaystyle S.}$ 

If every neighbourhood of ${\displaystyle x}$  contains uncountably many points of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called a condensation point of ${\displaystyle S.}$ 

If every neighbourhood ${\displaystyle U}$  of ${\displaystyle x}$  is such that the cardinality of ${\displaystyle U\cap S}$  equals the cardinality of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called a complete accumulation point of ${\displaystyle S.}$ 

Accumulation points of sequences and nets

See also: Net (mathematics) § Cluster point of a net, and Cluster point of a filter

In a topological space ${\displaystyle X,}$  a point ${\displaystyle x\in X}$  is said to be a cluster point or accumulation point of a sequence ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}$  if, for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x,}$  there are infinitely many ${\displaystyle n\in \mathbb {N} }$  such that ${\displaystyle x_{n}\in V.}$  It is equivalent to say that for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x}$  and every ${\displaystyle n_{0}\in \mathbb {N} ,}$  there is some ${\displaystyle n\geq n_{0}}$  such that ${\displaystyle x_{n}\in V.}$  If ${\displaystyle X}$  is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then ${\displaystyle x}$  is a cluster point of ${\displaystyle x_{\bullet }}$  if and only if ${\displaystyle x}$  is a limit of some subsequence of ${\displaystyle x_{\bullet }.}$  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 to mean a point ${\displaystyle x}$  to which the sequence converges (that is, every neighborhood of ${\displaystyle x}$  contains all but finitely many elements of the sequence). That is why we do not use the term limit point of a sequence as a synonym for accumulation point of the sequence.

The concept of a net generalizes the idea of a sequence. A net is a function ${\displaystyle f:(P,\leq )\to X,}$  where ${\displaystyle (P,\leq )}$  is a directed set and ${\displaystyle X}$  is a topological space. A point ${\displaystyle x\in X}$  is said to be a cluster point or accumulation point of a net ${\displaystyle f}$  if, for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x}$  and every ${\displaystyle p_{0}\in P,}$  there is some ${\displaystyle p\geq p_{0}}$  such that ${\displaystyle f(p)\in V,}$  equivalently, if ${\displaystyle f}$  has a subnet which converges to ${\displaystyle x.}$  Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.

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

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).

Properties

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 ${\displaystyle S\setminus \{x\}.}$ 

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, ${\displaystyle x}$  is a limit point of ${\displaystyle S,}$  if and only if every neighborhood of ${\displaystyle x}$  contains a point of ${\displaystyle S}$  other than ${\displaystyle x,}$  if and only if every neighborhood of ${\displaystyle x}$  contains a point of ${\displaystyle S\setminus \{x\},}$  if and only if ${\displaystyle x}$  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.

Proof

("Left subset") Suppose ${\displaystyle x}$  is in the closure of ${\displaystyle S.}$  If ${\displaystyle x}$  is in ${\displaystyle S,}$  we are done. If ${\displaystyle x}$  is not in ${\displaystyle S,}$  then every neighbourhood of ${\displaystyle x}$  contains a point of ${\displaystyle S,}$  and this point cannot be ${\displaystyle x.}$  In other words, ${\displaystyle x}$  is a limit point of ${\displaystyle S}$  and ${\displaystyle x}$  is in ${\displaystyle L(S).}$ 

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

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.

Proof

Proof 1: ${\displaystyle S}$  is closed if and only if ${\displaystyle S}$  is equal to its closure if and only if ${\displaystyle S=S\cup L(S)}$  if and only if ${\displaystyle L(S)}$  is contained in ${\displaystyle S.}$ 

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

No isolated point is a limit point of any set.

Proof

If ${\displaystyle x}$  is an isolated point, then ${\displaystyle \{x\}}$  is a neighbourhood of ${\displaystyle x}$  that contains no points other than ${\displaystyle x.}$ 

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

Proof

If ${\displaystyle X}$  is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if ${\displaystyle X}$  is not discrete, then there is a singleton ${\displaystyle \{x\}}$  that is not open. Hence, every open neighbourhood of ${\displaystyle \{x\}}$  contains a point ${\displaystyle y\neq x,}$  and so ${\displaystyle x}$  is a limit point of ${\displaystyle X.}$ 

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 ${\displaystyle S.}$ 

Proof

As long as ${\displaystyle S\setminus \{x\}}$  is nonempty, its closure will be ${\displaystyle X.}$  It is only empty when ${\displaystyle S}$  is empty or ${\displaystyle x}$  is the unique element of ${\displaystyle S.}$ 

See also

• 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

Citations

1. Dugundji 1966, pp. 209–210.
2. Bourbaki 1989, pp. 68–83.
3. "Difference between boundary point & limit point". 2021-01-13.
4. "What is a limit point". 2021-01-13.
5. . 2021-01-13. Archived from the original on 2021-04-21. Retrieved 2021-01-14.
6. Munkres 2000, pp. 97–102.

References

• 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]