• Source: Accumulation point

In mathematics, a limit point, accumulation point, or cluster point of a set



S


{\displaystyle S}

in a topological space



X


{\displaystyle X}

is a point



x


{\displaystyle x}

that can be "approximated" by points of



S


{\displaystyle S}

in the sense that every neighbourhood of



x


{\displaystyle x}

contains a point of



S


{\displaystyle S}

other than



x


{\displaystyle x}

itself. A limit point of a set



S


{\displaystyle S}

does not itself have to be an element of



S
.


{\displaystyle S.}


There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence



(

x

n



)

n
∈

N





{\displaystyle (x_{n})_{n\in \mathbb {N} }}

in a topological space



X


{\displaystyle X}

is a point



x


{\displaystyle x}

such that, for every neighbourhood



V


{\displaystyle V}

of



x
,


{\displaystyle x,}

there are infinitely many natural numbers



n


{\displaystyle n}

such that




x

n


∈
V
.


{\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 (respectively, a limit point of a filter, 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



x


{\displaystyle x}

contains some point of



S


{\displaystyle S}

. Unlike for limit points, an adherent point



x


{\displaystyle x}

of



S


{\displaystyle S}

may have a neighbourhood not containing points other than



x


{\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,



0


{\displaystyle 0}

is a boundary point (but not a limit point) of the set



{
0
}


{\displaystyle \{0\}}

in




R



{\displaystyle \mathbb {R} }

with standard topology. However,



0.5


{\displaystyle 0.5}

is a limit point (though not a boundary point) of interval



[
0
,
1
]


{\displaystyle [0,1]}

in




R



{\displaystyle \mathbb {R} }

with standard topology (for a less trivial example of a limit point, see the first caption).
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.




Definition






= Accumulation points of a set

=

Let



S


{\displaystyle S}

be a subset of a topological space



X
.


{\displaystyle X.}


A point



x


{\displaystyle x}

in



X


{\displaystyle X}

is a limit point or cluster point or accumulation point of the set



S


{\displaystyle S}

if every neighbourhood of



x


{\displaystyle x}

contains at least one point of



S


{\displaystyle S}

different from



x


{\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



X


{\displaystyle X}

is a




T

1




{\displaystyle T_{1}}

space (such as a metric space), then



x
∈
X


{\displaystyle x\in X}

is a limit point of



S


{\displaystyle S}

if and only if every neighbourhood of



x


{\displaystyle x}

contains infinitely many points of



S
.


{\displaystyle S.}

In fact,




T

1




{\displaystyle T_{1}}

spaces are characterized by this property.
If



X


{\displaystyle X}

is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then



x
∈
X


{\displaystyle x\in X}

is a limit point of



S


{\displaystyle S}

if and only if there is a sequence of points in



S
∖
{
x
}


{\displaystyle S\setminus \{x\}}

whose limit is



x
.


{\displaystyle x.}

In fact, Fréchet–Urysohn spaces are characterized by this property.
The set of limit points of



S


{\displaystyle S}

is called the derived set of



S
.


{\displaystyle S.}




Special types of accumulation point of a set

If every neighbourhood of



x


{\displaystyle x}

contains infinitely many points of



S
,


{\displaystyle S,}

then



x


{\displaystyle x}

is a specific type of limit point called an ω-accumulation point of



S
.


{\displaystyle S.}


If every neighbourhood of



x


{\displaystyle x}

contains uncountably many points of



S
,


{\displaystyle S,}

then



x


{\displaystyle x}

is a specific type of limit point called a condensation point of



S
.


{\displaystyle S.}


If every neighbourhood



U


{\displaystyle U}

of



x


{\displaystyle x}

is such that the cardinality of




U
∩
S


{\displaystyle U\cap S}

equals the cardinality of



S
,


{\displaystyle S,}

then



x


{\displaystyle x}

is a specific type of limit point called a complete accumulation point of



S
.


{\displaystyle S.}





= Accumulation points of sequences and nets

=

In a topological space



X
,


{\displaystyle X,}

a point



x
∈
X


{\displaystyle x\in X}

is said to be a cluster point or accumulation point of a sequence




x

∙


=


(

x

n


)


n
=
1


∞




{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}

if, for every neighbourhood



V


{\displaystyle V}

of



x
,


{\displaystyle x,}

there are infinitely many



n
∈

N



{\displaystyle n\in \mathbb {N} }

such that




x

n


∈
V
.


{\displaystyle x_{n}\in V.}


It is equivalent to say that for every neighbourhood



V


{\displaystyle V}

of



x


{\displaystyle x}

and every




n

0


∈

N

,


{\displaystyle n_{0}\in \mathbb {N} ,}

there is some



n
≥

n

0




{\displaystyle n\geq n_{0}}

such that




x

n


∈
V
.


{\displaystyle x_{n}\in V.}


If



X


{\displaystyle X}

is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then



x


{\displaystyle x}

is a cluster point of




x

∙




{\displaystyle x_{\bullet }}

if and only if



x


{\displaystyle x}

is a limit of some subsequence of




x

∙


.


{\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



x


{\displaystyle x}

to which the sequence converges (that is, every neighborhood of



x


{\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



f
:
(
P
,
≤
)
→
X
,


{\displaystyle f:(P,\leq )\to X,}

where



(
P
,
≤
)


{\displaystyle (P,\leq )}

is a directed set and



X


{\displaystyle X}

is a topological space. A point



x
∈
X


{\displaystyle x\in X}

is said to be a cluster point or accumulation point of a net



f


{\displaystyle f}

if, for every neighbourhood



V


{\displaystyle V}

of



x


{\displaystyle x}

and every




p

0


∈
P
,


{\displaystyle p_{0}\in P,}

there is some



p
≥

p

0




{\displaystyle p\geq p_{0}}

such that



f
(
p
)
∈
V
,


{\displaystyle f(p)\in V,}

equivalently, if



f


{\displaystyle f}

has a subnet which converges to



x
.


{\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




x

∙


=


(

x

n


)


n
=
1


∞




{\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}

in



X


{\displaystyle X}

is by definition just a map




x

∙


:

N

→
X


{\displaystyle x_{\bullet }:\mathbb {N} \to X}

so that its image



Im
⁡

x

∙


:=

{


x

n


:
n
∈

N


}



{\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



x
∈
X


{\displaystyle x\in X}

that occurs infinitely many times in the sequence,



x


{\displaystyle x}

is an accumulation point of the sequence. But



x


{\displaystyle x}

need not be an accumulation point of the corresponding set



Im
⁡

x

∙


.


{\displaystyle \operatorname {Im} x_{\bullet }.}

For example, if the sequence is the constant sequence with value



x
,


{\displaystyle x,}

we have



Im
⁡

x

∙


=
{
x
}


{\displaystyle \operatorname {Im} x_{\bullet }=\{x\}}

and



x


{\displaystyle x}

is an isolated point of



Im
⁡

x

∙




{\displaystyle \operatorname {Im} x_{\bullet }}

and not an accumulation point of



Im
⁡

x

∙


.


{\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



ω


{\displaystyle \omega }

-accumulation point of the associated set



Im
⁡

x

∙


.


{\displaystyle \operatorname {Im} x_{\bullet }.}


Conversely, given a countable infinite set



A
⊆
X


{\displaystyle A\subseteq X}

in



X
,


{\displaystyle X,}

we can enumerate all the elements of



A


{\displaystyle A}

in many ways, even with repeats, and thus associate with it many sequences




x

∙




{\displaystyle x_{\bullet }}

that will satisfy



A
=
Im
⁡

x

∙


.


{\displaystyle A=\operatorname {Im} x_{\bullet }.}


Any



ω


{\displaystyle \omega }

-accumulation point of



A


{\displaystyle A}

is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of



A


{\displaystyle A}

and hence also infinitely many terms in any associated sequence).
A point



x
∈
X


{\displaystyle x\in X}

that is not an



ω


{\displaystyle \omega }

-accumulation point of



A


{\displaystyle A}

cannot be an accumulation point of any of the associated sequences without infinite repeats (because



x


{\displaystyle x}

has a neighborhood that contains only finitely many (possibly even none) points of



A


{\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



cl
⁡
(
S
)


{\displaystyle \operatorname {cl} (S)}

of a set



S


{\displaystyle S}

is a disjoint union of its limit points



L
(
S
)


{\displaystyle L(S)}

and isolated points



I
(
S
)


{\displaystyle I(S)}

; that is,




cl
⁡
(
S
)
=
L
(
S
)
∪
I
(
S
)


and


L
(
S
)
∩
I
(
S
)
=
∅
.


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


A point



x
∈
X


{\displaystyle x\in X}

is a limit point of



S
⊆
X


{\displaystyle S\subseteq X}

if and only if it is in the closure of



S
∖
{
x
}
.


{\displaystyle S\setminus \{x\}.}


If we use



L
(
S
)


{\displaystyle L(S)}

to denote the set of limit points of



S
,


{\displaystyle S,}

then we have the following characterization of the closure of



S


{\displaystyle S}

: The closure of



S


{\displaystyle S}

is equal to the union of



S


{\displaystyle S}

and



L
(
S
)
.


{\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



S


{\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



X


{\displaystyle X}

is discrete if and only if no subset of



X


{\displaystyle X}

has a limit point.


If a space



X


{\displaystyle X}

has the trivial topology and



S


{\displaystyle S}

is a subset of



X


{\displaystyle X}

with more than one element, then all elements of



X


{\displaystyle X}

are limit points of



S
.


{\displaystyle S.}

If



S


{\displaystyle S}

is a singleton, then every point of



X
∖
S


{\displaystyle X\setminus S}

is a limit point of



S
.


{\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 set
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






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]

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