limit point

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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 Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
$X$ is a point $x$ that can be "approximated" by points of $S$ in the sense that every
neighbourhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
of $x$ with respect to the
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
s. A cluster point or accumulation point of a
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
$\left(x_n\right)_$ in a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
$X$ 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. 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 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. 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 $\$ in $\R$ with standard topology. However, $0.5$ is a limit point (though not a boundary point) of interval
closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
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 $S$ be a subset of a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
$X.$ A point $x$ in $X$ is a limit point or cluster point or $S$ if every
neighbourhood A neighbourhood (British English British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
), 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 (which all
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
s and first-countable spaces 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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
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 British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
$V$ 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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...
or a first-countable space (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. 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 mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
. 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 (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
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 British English (BrE, en-GB, or BE) is, according to Oxford Dictionaries, " English as used in Great Britain, as distinct from that used elsewhere". More narrowly, it can refer specifically to the En ...
$V$ 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 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.

# 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. 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 is a limit point of any set. A space $X$ is discrete if and only if no subset of $X$ has a limit point. If a space $X$ has the trivial topology 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 $S.$