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

sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

. A net is a function $f\; :\; (P,\backslash leq)\; \backslash to\; X,$ where $(P,\backslash leq)$ is a neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

$V$ 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 subnet 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.

topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

$X$ is a point $x$ that can be "approximated" by points of $S$ in the sense that every neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

of $x$ with respect to the topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

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, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

s. A cluster point or accumulation point of a sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

$(x\_n)\_$ in a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

$X$ 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.
The similarly named notion of a (respectively, a limit point of a filter
Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters, which are special families of subsets of some gi ...

, 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 synonymous with "cluster/accumulation point of a sequence".
The limit points of a set should not be confused with adherent points (also called ) for which every neighbourhood of $x$ contains a point of $S$ (that is, any point belonging to closure of the set). Unlike for limit points, an adherent 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, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...

.
Limit points of a set should also not be confused with boundary points. For example, $0$ is a boundary point (but not a limit point) of the set $\backslash $ in $\backslash R$ with standard topology
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...

. However, $0.5$ is a limit point (though not a boundary point) of interval $$, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

/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 and is the underpinning of concepts such as closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...

and topological closure
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection ...

. 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, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...

$X.$
A point $x$ in $X$ is a limit point or cluster point or $S$ if every neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

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, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

), 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 (which all metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

s and first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...

s are), then $x\; \backslash in\; X$ is a limit point of $S$ if and only if there is a sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

of points in $S\; \backslash setminus\; \backslash $ whose limit 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 a condensation point 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 or $x\_\; =\; \backslash left(x\_n\backslash right)\_^$ if, for everyneighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

$V$ of $x,$ there are infinitely many $n\; \backslash in\; \backslash N$ such that $x\_n\; \backslash in\; V.$
It is equivalent to say that for every neighbourhood $V$ of $x$ and every $n\_0\; \backslash in\; \backslash N,$ there is some $n\; \backslash geq\; n\_0$ such that $x\_n\; \backslash in\; V.$
If $X$ is a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

or a first-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...

(or, more generally, a Fréchet–Urysohn space), then $x$ is a 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, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they ca ...

.
Note that there is already the notion of limit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limit ...

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 generalizes the idea of a directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...

and $X$ is a topological space. A point $x\; \backslash in\; X$ is said to be a or $f$ if, for every 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 $x\_\; :\; \backslash N\; \backslash to\; X$ so that itsimage
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

$\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

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 $\backslash operatorname(S)$ 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 $S\; \backslash setminus\; \backslash .$ 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. 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. Noisolated point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...

is a limit point of any set.
A space $X$ is discrete space, discrete if and only if no subset of $X$ has a limit point.
If a space $X$ has the trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...

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 $S.$
See also

* * * * * * * * *Citations

References

* * * * {{Topology Topology General topology