1. Definition

1.1. Accumulation Points of a Set

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 ${\displaystyle T1}$ 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 ${\displaystyle S.}$^[1] 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 ${\displaystyle S.}$

Types of accumulation points

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 ${\displaystyle x}$ satisfies ${\displaystyle |U\cap S|=\left|S\right|,}$ then ${\displaystyle x}$ is a specific type of limit point called a complete accumulation point of ${\displaystyle S.}$

1.2. Accumulation Points of Sequences and Nets

A sequence enumerating all positive rational numbers. Each positive real number is a cluster point.

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 $\text{Undefined control sequence bull}$ 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 ${\displaystyle n\ge n_0}$ 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 $\text{Undefined control sequence bull}$ if and only if ${\displaystyle x}$ is a limit of some subsequence of $\text{Undefined control sequence bull}$ 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 ${\displaystyle f:(P,\le )\to X,}$ where ${\displaystyle (P,\le )}$ 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 ${\displaystyle p\ge p_0}$ 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.

2. Relation Between Accumulation Point of a Sequence and Accumulation Point of a Set

Every sequence $\text{Undefined control sequence bull}$ in ${\displaystyle X}$ is by definition just a map $\text{Undefined control sequence bull}$ so that its image $\text{Undefined control sequence operatorname}$ 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 $\text{Undefined control sequence operatorname}$ For example, if the sequence is the constant sequence with value ${\displaystyle x,}$ we have $\text{Undefined control sequence operatorname}$ and ${\displaystyle x}$ is an isolated point of $\text{Undefined control sequence operatorname}$ and not an accumulation point of $\text{Undefined control sequence operatorname}$

• 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 $\text{Undefined control sequence operatorname}$

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 $\text{Undefined control sequence bull}$ that will satisfy $\text{Undefined control sequence operatorname}$

• 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).

3. 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 $\text{Undefined control sequence operatorname}$ of a set ${\displaystyle S}$ is a disjoint union of its limit points ${\displaystyle L(S)}$ and isolated points ${\displaystyle I(S)}$: $\text{Undefined control sequence operatorname}$

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 ${\displaystyle S\setminus \{x\}.}$