Accumulation point

Let ${\displaystyle S}$  be a subset of a topological space ${\displaystyle X.}$  A point ${\displaystyle x}$  in ${\displaystyle X}$  is a limit point or cluster point or accumulation point of the set ${\displaystyle S}$  if every neighbourhood of ${\displaystyle x}$  contains at least one point of ${\displaystyle S}$  different from ${\displaystyle x}$  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 ${\displaystyle X}$  is a ${\displaystyle T_{1}}$  space (such as a metric space), then ${\displaystyle x\in X}$  is a limit point of ${\displaystyle S}$  if and only if every neighbourhood of ${\displaystyle x}$  contains infinitely many points of ${\displaystyle S.}$ ^[6] In fact, ${\displaystyle T_{1}}$  spaces are characterized by this property.

If ${\displaystyle X}$  is a Fréchet–Urysohn space (which all metric spaces and first-countable spaces are), then ${\displaystyle x\in X}$  is a limit point of ${\displaystyle S}$  if and only if there is a sequence of points in ${\displaystyle S\setminus \{x\}}$  whose limit is ${\displaystyle x.}$  In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of ${\displaystyle S}$  is called the derived set of ${\displaystyle S.}$ 

If every neighbourhood of ${\displaystyle x}$  contains infinitely many points of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called an ω-accumulation point of ${\displaystyle S.}$ 

If every neighbourhood of ${\displaystyle x}$  contains uncountably many points of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called a condensation point of ${\displaystyle S.}$ 

If every neighbourhood ${\displaystyle U}$  of ${\displaystyle x}$  is such that the cardinality of ${\displaystyle U\cap S}$  equals the cardinality of ${\displaystyle S,}$  then ${\displaystyle x}$  is a specific type of limit point called a complete accumulation point of ${\displaystyle S.}$ 

In a topological space ${\displaystyle X,}$  a point ${\displaystyle x\in X}$  is said to be a cluster point or accumulation point of a sequence ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}$  if, for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x,}$  there are infinitely many ${\displaystyle n\in \mathbb {N} }$  such that ${\displaystyle x_{n}\in V.}$  It is equivalent to say that for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x}$  and every ${\displaystyle n_{0}\in \mathbb {N} ,}$  there is some ${\displaystyle n\geq n_{0}}$  such that ${\displaystyle x_{n}\in V.}$  If ${\displaystyle X}$  is a metric space or a first-countable space (or, more generally, a Fréchet–Urysohn space), then ${\displaystyle x}$  is a cluster point of ${\displaystyle x_{\bullet }}$  if and only if ${\displaystyle x}$  is a limit of some subsequence of ${\displaystyle x_{\bullet }.}$  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 ${\displaystyle x}$  to which the sequence converges (that is, every neighborhood of ${\displaystyle x}$  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 ${\displaystyle f:(P,\leq )\to X,}$  where ${\displaystyle (P,\leq )}$  is a directed set and ${\displaystyle X}$  is a topological space. A point ${\displaystyle x\in X}$  is said to be a cluster point or accumulation point of a net ${\displaystyle f}$  if, for every neighbourhood ${\displaystyle V}$  of ${\displaystyle x}$  and every ${\displaystyle p_{0}\in P,}$  there is some ${\displaystyle p\geq p_{0}}$  such that ${\displaystyle f(p)\in V,}$  equivalently, if ${\displaystyle f}$  has a subnet which converges to ${\displaystyle x.}$  Cluster points in nets encompass the idea of both condensation points and ω-accumulation points. Clustering and limit points are also defined for filters.

Every sequence ${\displaystyle x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty }}$  in ${\displaystyle X}$  is by definition just a map ${\displaystyle x_{\bullet }:\mathbb {N} \to X}$  so that its image ${\displaystyle \operatorname {Im} x_{\bullet }:=\left\{x_{n}:n\in \mathbb {N} \right\}}$  can be defined in the usual way.

• If there exists an element ${\displaystyle x\in X}$  that occurs infinitely many times in the sequence, ${\displaystyle x}$  is an accumulation point of the sequence. But ${\displaystyle x}$  need not be an accumulation point of the corresponding set ${\displaystyle \operatorname {Im} x_{\bullet }.}$  For example, if the sequence is the constant sequence with value ${\displaystyle x,}$  we have ${\displaystyle \operatorname {Im} x_{\bullet }=\{x\}}$  and ${\displaystyle x}$  is an isolated point of ${\displaystyle \operatorname {Im} x_{\bullet }}$  and not an accumulation point of ${\displaystyle \operatorname {Im} x_{\bullet }.}$ 

• 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 ${\displaystyle \omega }$ -accumulation point of the associated set ${\displaystyle \operatorname {Im} x_{\bullet }.}$ 

Conversely, given a countable infinite set ${\displaystyle A\subseteq X}$  in ${\displaystyle X,}$  we can enumerate all the elements of ${\displaystyle A}$  in many ways, even with repeats, and thus associate with it many sequences ${\displaystyle x_{\bullet }}$  that will satisfy ${\displaystyle A=\operatorname {Im} x_{\bullet }.}$ 

• Any ${\displaystyle \omega }$ -accumulation point of ${\displaystyle A}$  is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of ${\displaystyle A}$  and hence also infinitely many terms in any associated sequence).

• A point ${\displaystyle x\in X}$  that is not an ${\displaystyle \omega }$ -accumulation point of ${\displaystyle A}$  cannot be an accumulation point of any of the associated sequences without infinite repeats (because ${\displaystyle x}$  has a neighborhood that contains only finitely many (possibly even none) points of ${\displaystyle A}$  and that neighborhood can only contain finitely many terms of such sequences).

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 ${\displaystyle \operatorname {cl} (S)}$  of a set ${\displaystyle S}$  is a disjoint union of its limit points ${\displaystyle L(S)}$  and isolated points ${\displaystyle I(S)}$ ; that is, $${\displaystyle \operatorname {cl} (S)=L(S)\cup I(S)\quad {\text{and}}\quad L(S)\cap I(S)=\emptyset .}$$ 

A point ${\displaystyle x\in X}$  is a limit point of ${\displaystyle S\subseteq X}$  if and only if it is in the closure of ${\displaystyle S\setminus \{x\}.}$ 

If we use ${\displaystyle L(S)}$  to denote the set of limit points of ${\displaystyle S,}$  then we have the following characterization of the closure of ${\displaystyle S}$ : The closure of ${\displaystyle S}$  is equal to the union of ${\displaystyle S}$  and ${\displaystyle L(S).}$  This fact is sometimes taken as the definition of closure.

A corollary of this result gives us a characterisation of closed sets: A set ${\displaystyle S}$  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 ${\displaystyle X}$  is discrete if and only if no subset of ${\displaystyle X}$  has a limit point.

If a space ${\displaystyle X}$  has the trivial topology and ${\displaystyle S}$  is a subset of ${\displaystyle X}$  with more than one element, then all elements of ${\displaystyle X}$  are limit points of ${\displaystyle S.}$  If ${\displaystyle S}$  is a singleton, then every point of ${\displaystyle X\setminus S}$  is a limit point of ${\displaystyle S.}$ 

• 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

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• 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]