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Accumulation point

Cluster point in a topological space

Wikipedia contributorsAdapted for Fathomly · View authors
7 min read
Source updated September 10, 2026

In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x 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 sequences. A cluster point or accumulation point of a sequence (x_{n})_{n\in \mathbb {N} } in a topological space 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 limit point of a sequence (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 not synonymous with "cluster/accumulation point of a sequence".

The limit points of a set should not be confused with adherent points (also called points of closure) for which every neighbourhood of x contains some point of S. Unlike for limit points, an adherent point x of S may have a neighbourhood not containing points other than 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 \{0\} in \mathbb {R} with standard topology. However, 0.5 is a limit point (though not a boundary point) of interval [0,1] in \mathbb {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 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.

01Definition

Accumulation points of a set

Let S be a subset of a topological space X. A point x in X is a limit point or cluster point or accumulation point of the set S if every neighbourhood 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), 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 spaces and first-countable spaces are), then x\in X is a limit point of S if and only if there is a sequence of points in S\setminus \{x\} 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.

Special types of accumulation point of a set

If every neighbourhood of x contains infinitely many points of S, then x is a specific type of limit point called an ω-accumulation point 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 is such that the cardinality of U\cap S equals the cardinality of S, then x is a specific type of limit point called a complete accumulation point of S.

Accumulation points of sequences and nets

In a topological space X, a point x\in X is said to be a cluster point or accumulation point of a sequence x_{\bullet }=\left(x_{n}\right)_{n=1}^{\infty } if, for every neighbourhood V of x, there are infinitely many n\in \mathbb {N} such that x_{n}\in V. It is equivalent to say that for every neighbourhood V of x and every n_{0}\in \mathbb {N} , there is some n\geq n_{0} such that x_{n}\in V. If X is a metric space or a first-countable space (or, more generally, a Fréchet-Urysohn space), then x is a cluster point of x_{\bullet } if and only if x is a limit of some subsequence of 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 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 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 f:(P,\leq )\to X, where (P,\leq ) is a directed set and X is a topological space. A point x\in X is said to be a cluster point or accumulation point of a net f if, for every neighbourhood V of x and every p_{0}\in P, there is some p\geq p_{0} such that f(p)\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.

With respect to the usual Euclidean topology, the sequence of rational numbers has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set
With respect to the usual Euclidean topology, the sequence of rational numbers has no limit (i.e. does not converge), but has two accumulation points (which are considered limit points here), viz. -1 and +1. Thus, thinking of sets, these points are limit points of the set

02Relation between accumulation point of a sequence and accumulation point of a set

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

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_{\bullet } that will satisfy A=\operatorname {Im} x_{\bullet }.

  • 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 not 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).
A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.
A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

03Properties

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

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 \{x\}.

Proof

We use the fact that a point is in the closure of a set if and only if every neighborhood of the point meets the set. Now, x is a limit point of S, if and only if every neighborhood of x contains a point of S other than x, if and only if every neighborhood of x contains a point of S\setminus \{x\}, if and only if x is in the closure of S\setminus \{x\}.

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 definition of closure.

Proof

("Left subset") Suppose x is in the closure of S. If x is in S, we are done. If x is not in S, then every neighbourhood of x contains a point of S, and this point cannot be x. In other words, x is a limit point of S and x is in L(S).

("Right subset") If x is in S, then every neighbourhood of x clearly meets S, so x is in the closure of S. If x is in L(S), then every neighbourhood of x contains a point of S (other than x), so x is again in the closure of S. This completes the proof.

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.

Proof

Proof 1: S is closed if and only if S is equal to its closure if and only if S=S\cup L(S) if and only if L(S) is contained in S.

Proof 2: Let S be a closed set and x a limit point of S. If x is not in S, then the complement to S comprises an open neighbourhood of x. Since x is a limit point of S, any open neighbourhood of x should have a non-trivial intersection with S. However, a set can not have a non-trivial intersection with its complement. Conversely, assume S contains all its limit points. We shall show that the complement of S is an open set. Let x be a point in the complement of S. By assumption, x is not a limit point, and hence there exists an open neighbourhood U of x that does not intersect S, and so U lies entirely in the complement of S. Since this argument holds for arbitrary x in the complement of S, the complement of S can be expressed as a union of open neighbourhoods of the points in the complement of S. Hence the complement of S is open.

No isolated point is a limit point of any set.

Proof

If x is an isolated point, then \{x\} is a neighbourhood of x that contains no points other than x.

A space X is discrete if and only if no subset of X has a limit point.

Proof

If X is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if X is not discrete, then there is a singleton \{x\} that is not open. Hence, every open neighbourhood of \{x\} contains a point y\neq x, and so x is a limit point of X.

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.

Proof

As long as S\setminus \{x\} is nonempty, its closure will be X. It is only empty when S is empty or x is the unique element of S.

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Sources and credits

This article is adapted from the Wikipedia article “Accumulation point”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Adherent point

In mathematics, an adherent point of a subset A {\displaystyle A} of a topological space X , {\displaystyle X,} is a point x {\displaystyle x} in X {\displaystyle X} such that every neighbourhood of x {\displaystyle x} (or equivalently, every open neighborhood of x {\displaystyle x} ) contains at least one point of A . {\displaystyle A.} A point x ∈ X {\displaystyle x\in X} is an adherent point for A {\displaystyle A} if and only if x {\displaystyle x} is in the closure of A , {\displaystyle A,} thus x ∈ Cl X ⁡ A {\displaystyle x\in \operatorname {Cl} _{X}A} if and only if for all open subsets U ⊆ X , {\displaystyle U\subseteq X,} if x ∈ U then U ∩ A ≠ ∅ .

Condensation point

In mathematics, a condensation point p of a subset S of a topological space is any point p such that every neighborhood of p contains uncountably many points of S. Thus "condensation point" is synonymous with " ℵ 1 {\displaystyle \aleph _{1}} -accumulation point".

Filters in topology

In mathematics, a filter on a set is a collection of nonempty subsets which is upward closed and closed under finite intersections. An example of a filter is the collection of neighborhoods of a point in a topological space.