In mathematics, a limit point (or cluster point or accumulation point) of a set $S$ in a

sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. A net is a function $f\; :\; (P,\backslash leq)\; \backslash to\; X,$ where $(P,\backslash leq)$ is a neighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

$V$ of $x$ and every $p\_0\; \backslash in\; P,$ there is some $p\; \backslash geq\; p\_0$ such that $f(p)\; \backslash in\; V,$ equivalently, if $f$ has a

topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

$X$ is a point $x$ that can be "approximated" by points of $S$ in the sense that every neighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

of $x$ with respect to the topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

on $X$ also contains a point of $S$ other than $x$ itself. A limit point of a set $S$ does not itself have to be an element of $S.$
There is also a closely related concept for sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s. A cluster point or accumulation point of a sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

$(x\_n)\_$ in a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

$X$ is a point $x$ such that, for every neighbourhood $V$ of $x,$ there are infinitely many natural numbers $n$ such that $x\_n\; \backslash in\; V.$ This definition of a cluster or accumulation point of a sequence generalizes to nets and filters
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

.
In contrast to sets, for a sequence, net, or filter, the term "limit point" is synonymous with a "cluster/accumulation point"; by definition, the similarly named notion of a limit point of a filter (respectively, a limit point of a sequence, a limit point of a net) refers to a point that the filter converges to (respectively, the sequence converges to, the net converges to).
The limit points of a set should not be confused with adherent point
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s for which every neighbourhood of $x$ contains a point of $S$. Unlike for limit points, this point of $S$ may be $x$ itself. A limit point can be characterized as an adherent point that is not an isolated point
]
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
Limit points of a set should also not be confused with boundary point
In topology and mathematics in general, the boundary of a subset ''S'' of a topological space ''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More precisely, it is the set of points in the closure ...

s. For example, $0$ is a boundary point (but not a limit point) of set $\backslash $ in $\backslash R$ with standard topology
of ordered pairs . Blue lines denote coordinate axes, horizontal green lines are integer , vertical cyan lines are integer , brown-orange lines show half-integer or , magenta and its tint show multiples of one tenth (best seen under magnification ...

. However, $0.5$ is a limit point (though not a boundary point) of interval $$, 1
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

/math> in $\backslash R$ 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
Limit or Limits may refer to:
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and is the underpinning of concepts such as closed set
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

and topological closure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. 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 $S$ be a subset of atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

$X.$
A point $x$ in $X$ is a limit point or cluster point or $S$ if every neighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

of $x$ contains at least one point of $S$ different from $x$ 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 $X$ is a $T\_1$ space (such as a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

), then $x\; \backslash in\; X$ is a limit point of $S$ if and only if every neighbourhood of $x$ contains infinitely many points of $S.$ In fact, $T\_1$ spaces are characterized by this property.
If $X$ is a Fréchet–Urysohn space
In the field of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, g ...

(which all metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s and first-countable space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

s are), then $x\; \backslash in\; X$ is a limit point of $S$ if and only if there is a sequence
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of points in $S\; \backslash setminus\; \backslash $ whose limit
Limit or Limits may refer to:
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* Limit (song), "Limit" (song), a 2016 single by Luna Sea
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is $x.$ In fact, Fréchet–Urysohn spaces are characterized by this property.
The set of limit points of $S$ is called the derived set of $S.$
Types of accumulation points

If every neighbourhood of $x$ contains infinitely many points of $S,$ then $x$ is a specific type of limit point called an of $S.$ If every neighbourhood of $x$ contains uncountably many points of $S,$ then $x$ is a specific type of limit point called acondensation point
In mathematics, a condensation point ''p'' of a subset ''S'' of a topological space, is any point ''p'' such that every open neighborhood of ''p'' contains uncountably many points of ''S''. Thus "condensation point" is synonymous with "\aleph_1-accu ...

of $S.$
If every neighbourhood $U$ of $x$ satisfies $\backslash left,\; U\; \backslash cap\; S\backslash \; =\; \backslash left,\; S\; \backslash ,$ then $x$ is a specific type of limit point called a of $S.$
Accumulation points of sequences and nets

In a topological space $X,$ a point $x\; \backslash in\; X$ is said to be a cluster point or $x\_\; =\; \backslash left(x\_n\backslash right)\_^$ if, for everyneighbourhood
A neighbourhood (British English, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...

$V$ of $x,$ there are infinitely many $n\; \backslash in\; \backslash mathbb$ such that $x\_n\; \backslash in\; V.$
It is equivalent to say that for every neighbourhood $V$ of $x$ and every $n\_0\; \backslash in\; \backslash mathbb,$ there is some $n\; \backslash geq\; n\_0$ such that $x\_n\; \backslash in\; V.$
If $X$ is a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

or a first-countable space
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...

(or, more generally, a Fréchet–Urysohn space
In the field of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, g ...

), then $x$ is cluster point of $x\_$ if and only if $x$ is a limit of some subsequence of $x\_.$
The set of all cluster points of a sequence is sometimes called the limit set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

.
Note that there is already the notion of limit of a sequence
As the positive integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. ...

to mean a point $x$ to which the sequence converges (that is, every neighborhood of $x$ contains all but finitely many elements of the sequence). That is why we do not use the term of a sequence as a synonym for accumulation point of the sequence.
The concept of a net
Net or net may refer to:
Mathematics and physics
* Net (mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...

generalizes the idea of a directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

and $X$ is a topological space. A point $x\; \backslash in\; X$ is said to be a cluster point or $f$ if, for every subnet
A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting.
Computers that belong to the same subnet are addressed with an identical ...

which converges to $x.$ Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

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

Every sequence $x\_\; =\; \backslash left(x\_n\backslash right)\_^$ in $X$ is by definition just a map $x\_\; :\; \backslash mathbb\; \backslash to\; X$ so that itsimage
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

$\backslash operatorname\; x\_\; :=\; \backslash left\backslash $ can be defined in the usual way.
* If there exists an element $x\; \backslash in\; X$ that occurs infinitely many times in the sequence, $x$ is an accumulation point of the sequence. But $x$ need not be an accumulation point of the corresponding set $\backslash operatorname\; x\_.$ For example, if the sequence is the constant sequence with value $x,$ we have $\backslash operatorname\; x\_\; =\; \backslash $ and $x$ is an isolated point of $\backslash operatorname\; x\_$ and not an accumulation point of $\backslash operatorname\; x\_.$
* 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 $\backslash omega$-accumulation point of the associated set $\backslash operatorname\; x\_.$
Conversely, given a countable infinite set $A\; \backslash subseteq\; X$ in $X,$ we can enumerate all the elements of $A$ in many ways, even with repeats, and thus associate with it many sequences $x\_$ that will satisfy $A\; =\; \backslash operatorname\; x\_.$
* Any $\backslash omega$-accumulation point of $A$ is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of $A$ and hence also infinitely many terms in any associated sequence).
* A point $x\; \backslash in\; X$ that is an $\backslash omega$-accumulation point of $A$ cannot be an accumulation point of any of the associated sequences without infinite repeats (because $x$ has a neighborhood that contains only finitely many (possibly even none) points of $A$ and that neighborhood can only contain finitely many terms of such sequences).
Properties

Everylimit
Limit or Limits may refer to:
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of a non-constant sequence is an accumulation point of the sequence.
And by definition, every limit point is an adherent point
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

.
The closure $\backslash operatorname(S)$ of a set $S$ is a disjoint union of its limit points $L(S)$ and isolated points $I(S)$:
:$\backslash operatorname\; (S)\; =\; L(S)\; \backslash cup\; I(S),\; L(S)\; \backslash cap\; I(S)\; =\; \backslash varnothing.$
A point $x\; \backslash in\; X$ is a limit point of $S\; \backslash subseteq\; X$ if and only if it is in the closure of $S\; \backslash setminus\; \backslash .$
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, $x$ is a limit point of $S,$ if and only if every neighborhood of $x$ contains a point of $S$ other than $x,$ if and only if every neighborhood of $x$ contains a point of $S\; \backslash setminus\; \backslash ,$ if and only if $x$ is in the closure of $S\; \backslash setminus\; \backslash .$
If we use $L(S)$ to denote the set of limit points of $S,$ then we have the following characterization of the closure of $S$: The closure of $S$ is equal to the union of $S$ and $L(S).$ This fact is sometimes taken as the of closure.
("Left subset") Suppose $x$ is in the closure of $S.$ If $x$ is in $S,$ we are done. If $x$ is not in $S,$ then every neighbourhood of $x$ contains a point of $S,$ and this point cannot be $x.$ In other words, $x$ is a limit point of $S$ and $x$ is in $L(S).$
("Right subset") If $x$ is in $S,$ then every neighbourhood of $x$ clearly meets $S,$ so $x$ is in the closure of $S.$ If $x$ is in $L(S),$ then every neighbourhood of $x$ contains a point of $S$ (other than $x$), so $x$ is again in the closure of $S.$ This completes the proof.
A corollary of this result gives us a characterisation of closed sets: A set $S$ is closed if and only if it contains all of its limit points.
''Proof'' 1: $S$ is closed if and only if $S$ is equal to its closure if and only if $S=S\backslash cup\; L(S)$ if and only if $L(S)$ is contained in $S.$
''Proof'' 2: Let $S$ be a closed set and $x$ a limit point of $S.$ If $x$ is not in $S,$ then the complement to $S$ comprises an open neighbourhood of $x.$ Since $x$ is a limit point of $S,$ any open neighbourhood of $x$ should have a non-trivial intersection with $S.$ However, a set can not have a non-trivial intersection with its complement. Conversely, assume $S$ contains all its limit points. We shall show that the complement of $S$ is an open set. Let $x$ be a point in the complement of $S.$ By assumption, $x$ is not a limit point, and hence there exists an open neighbourhood $U$ of $x$ that does not intersect $S,$ and so $U$ lies entirely in the complement of $S.$ Since this argument holds for arbitrary $x$ in the complement of $S,$ the complement of $S$ can be expressed as a union of open neighbourhoods of the points in the complement of $S.$ Hence the complement of $S$ is open.
No isolated point
]
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

is a limit point of any set.
If $x$ is an isolated point, then $\backslash $ is a neighbourhood of $x$ that contains no points other than $x.$
A space $X$ is discrete space, discrete if and only if no subset of $X$ has a limit point.
If $X$ is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if $X$ is not discrete, then there is a singleton $\backslash $ that is not open. Hence, every open neighbourhood of $\backslash $ contains a point $y\; \backslash neq\; x,$ and so $x$ is a limit point of $X.$
If a space $X$ has the trivial topologyIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

and $S$ is a subset of $X$ with more than one element, then all elements of $X$ are limit points of $S.$ If $S$ is a singleton, then every point of $X\; \backslash setminus\; S$ is a limit point of $S.$
As long as $S\; \backslash setminus\; \backslash $ is nonempty, its closure will be $X.$ It is only empty when $S$ is empty or $x$ is the unique element of $S.$
See also

* * * * * * * * *Citations

References

* * * * {{Topology Topology General topology