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In mathematics, a limit point, accumulation point, or cluster point of a set $S$ 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 po ...
$X$ 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 ...
$\left(x_n\right)_$ 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 po ...
$X$ is a point $x$ such that, for every neighbourhood $V$ of $x,$ there are infinitely many natural numbers $n$ such that $x_n \in V.$ 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 Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component th ...
. The similarly named notion of a (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 synonymous with "cluster/accumulation point of a sequence". The limit points of a set should not be confused with
adherent point In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (or equivalentl ...
s (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 point In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bounda ...
s. For example, $0$ is a boundary point (but not a limit point) of the set $\$ in $\R$ 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 o ...
/math> in $\R$ 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 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 po ...
$X.$ 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 set ...
), then $x \in X$ 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, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a speci ...
(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 set ...
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 $x \in X$ 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 \setminus \$ 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 $\left, U \cap S\ = \left, S \,$ 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 $x \in X$ is said to be a or $x_ = \left\left(x_n\right\right)_^$ if, for 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, ...
$V$ of $x,$ there are infinitely many $n \in \N$ such that $x_n \in V.$ It is equivalent to say that for every neighbourhood $V$ of $x$ and every $n_0 \in \N,$ there is some $n \geq n_0$ such that $x_n \in V.$ 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 set ...
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 In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a speci ...
), 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
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 $f : \left(P,\leq\right) \to X,$ where $\left(P,\leq\right)$ is 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 $x \in X$ is said to be a or $f$ if, for 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, ...
$V$ of $x$ and every $p_0 \in P,$ there is some $p \geq p_0$ such that $f\left(p\right) \in V,$ equivalently, if $f$ has a
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 identica ...
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 Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component th ...
.

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

Every sequence $x_ = \left\left(x_n\right\right)_^$ in $X$ is by definition just a map $x_ : \N \to X$ so that its
image 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-dimensiona ...
$\operatorname x_ := \left\$ can be defined in the usual way. * If there exists an element $x \in X$ 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 $\operatorname x_.$ For example, if the sequence is the constant sequence with value $x,$ we have $\operatorname x_ = \$ and $x$ is an isolated point of $\operatorname x_$ and not an accumulation point of $\operatorname x_.$ * 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 $\omega$-accumulation point of the associated set $\operatorname x_.$ Conversely, given a countable infinite set $A \subseteq X$ 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 $A = \operatorname x_.$ * Any $\omega$-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 $x \in X$ that is an $\omega$-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 In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (or equivalentl ...
. The closure $\operatorname\left(S\right)$ of a set $S$ is a disjoint union of its limit points $L\left(S\right)$ and isolated points $I\left(S\right)$: $\operatorname (S) = L(S) \cup I(S), L(S) \cap I(S) = \varnothing.$ A point $x \in X$ is a limit point of $S \subseteq X$ if and only if it is in the closure of $S \setminus \.$ If we use $L\left(S\right)$ 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 $L\left(S\right).$ 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. No
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 ...
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 $X \setminus S$ is a limit point of $S.$