In mathematics, alimit point,accumulation point, orcluster point of aset${\displaystyle S}$ in atopological space${\displaystyle X}$ is a point${\displaystyle x}$ that can be "approximated" by points of${\displaystyle S}$ in the sense that everyneighbourhood of${\displaystyle x}$ contains a point of${\displaystyle S}$ other than${\displaystyle x}$ itself. A limit point of a set${\displaystyle S}$ does not itself have to be an element of${\displaystyle S.}$ There is also a closely related concept forsequences. Acluster point oraccumulation point of asequence${\displaystyle (x_n{)}_{n\in \mathbb{N}}}$ in atopological space${\displaystyle X}$ is a point${\displaystyle x}$ such that, for every neighbourhood${\displaystyle V}$ of${\displaystyle x,}$ there are infinitely many natural numbers${\displaystyle n}$ such that${\displaystyle x_n\in V.}$ This definition of a cluster or accumulation point of a sequence generalizes tonets andfilters.

The similarly named notion of alimit point of a sequence^[1] (respectively, alimit point of a filter,^[2] alimit point of a net) by definition refers to a point that thesequence converges to (respectively, thefilter converges to, thenet 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" isnot synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused withadherent points (also calledpoints ofclosure) for which every neighbourhood of${\displaystyle x}$ containssome point of${\displaystyle S}$. Unlike for limit points, an adherent point${\displaystyle x}$ of${\displaystyle S}$ may have a neighbourhood not containing points other than${\displaystyle x}$ itself. A limit point can be characterized as an adherent point that is not anisolated point.

Limit points of a set should also not be confused withboundary points. For example,${\displaystyle 0}$ is a boundary point (but not a limit point) of the set${\displaystyle \{0\}}$ in${\displaystyle \mathbb{R}}$ withstandard topology. However,${\displaystyle 0.5}$ is a limit point (though not a boundary point) of interval${\displaystyle [0,1]}$ in${\displaystyle \mathbb{R}}$ with standard topology (for a less trivial example of a limit point, see the first caption).^[3][4][5]

This concept profitably generalizes the notion of alimit and is the underpinning of concepts such asclosed set andtopological 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.

Let${\displaystyle S}$  be a subset of atopological space${\displaystyle X.}$  A point${\displaystyle x}$  in${\displaystyle X}$  is alimit point orcluster point oraccumulation point of the set${\displaystyle S}$  if everyneighbourhood 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 ametric 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 aFréchet–Urysohn space (which allmetric spaces andfirst-countable spaces are), then${\displaystyle x\in X}$  is a limit point of${\displaystyle S}$  if and only if there is asequence of points in${\displaystyle S\setminus \{x\}}$  whoselimit is${\displaystyle x.}$  In fact, Fréchet–Urysohn spaces are characterized by this property.

The set of limit points of${\displaystyle S}$  is called thederived 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 anω-accumulation point of${\displaystyle S.}$ 

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

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

In a topological space${\displaystyle X,}$  a point${\displaystyle x\in X}$  is said to be acluster point oraccumulation point of a sequence${\displaystyle x_{\bullet }={\left(x_n\right)}_{n=1}^{\mathrm{\infty }}}$  if, for everyneighbourhood${\displaystyle V}$  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${\displaystyle n\ge n_0}$  such that${\displaystyle x_n\in V.}$  If${\displaystyle X}$  is ametric space or afirst-countable space (or, more generally, aFréchet–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 thelimit set.

Note that there is already the notion oflimit 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 termlimit point of a sequence as a synonym for accumulation point of the sequence.

The concept of anet generalizes the idea of asequence. A net is a function${\displaystyle f:(P,\le )\to X,}$  where${\displaystyle (P,\le )}$  is adirected set and${\displaystyle X}$  is a topological space. A point${\displaystyle x\in X}$  is said to be acluster point oraccumulation point of a net${\displaystyle f}$  if, for everyneighbourhood${\displaystyle V}$  of${\displaystyle x}$  and every${\displaystyle p_0\in P,}$  there is some${\displaystyle p\ge p_0}$  such that${\displaystyle f(p)\in V,}$  equivalently, if${\displaystyle f}$  has asubnet which converges to${\displaystyle x.}$  Cluster points in nets encompass the idea of both condensation points and ω-accumulation points.Clustering andlimit points are also defined forfilters.

Every sequence${\displaystyle x_{\bullet }={\left(x_n\right)}_{n=1}^{\mathrm{\infty }}}$  in${\displaystyle X}$  is by definition just a map${\displaystyle x_{\bullet }:\mathbb{N}\to X}$  so that itsimage${\displaystyle \mathrm{Im}{x}_{\bullet }:=\left\{x_n:n\in \mathbb{N}\right\}}$  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${\displaystyle \mathrm{Im}{x}_{\bullet }.}$  For example, if the sequence is the constant sequence with value${\displaystyle x,}$  we have${\displaystyle \mathrm{Im}{x}_{\bullet }=\{x\}}$  and${\displaystyle x}$  is an isolated point of${\displaystyle \mathrm{Im}{x}_{\bullet }}$  and not an accumulation point of${\displaystyle \mathrm{Im}{x}_{\bullet }.}$ 

• 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${\displaystyle \mathrm{Im}{x}_{\bullet }.}$ 

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${\displaystyle A=\mathrm{Im}{x}_{\bullet }.}$ 

• 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 isnot 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).

Everylimit of a non-constant sequence is an accumulation point of the sequence.And by definition, every limit point is anadherent point.

The closure${\displaystyle \mathrm{cl}(S)}$  of a set${\displaystyle S}$  is adisjoint union of its limit points${\displaystyle L(S)}$  and isolated points${\displaystyle I(S)}$ ; that is,$${\displaystyle \mathrm{cl}(S)=L(S)\cup I(S)\text{and}L(S)\cap I(S)=\mathrm{\varnothing }.}$$ 

A point${\displaystyle x\in X}$  is a limit point of${\displaystyle S\subseteq X}$  if and only if it is in theclosure 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 thedefinition ofclosure.

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.

Noisolated point is a limit point of any set.

A space${\displaystyle X}$  isdiscrete if and only if no subset of${\displaystyle X}$  has a limit point.

If a space${\displaystyle X}$  has thetrivial 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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