# Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set ${\displaystyle S}$ in a topological space ${\displaystyle X}$ is a point ${\displaystyle x}$ that can be "approximated" by points of ${\displaystyle S}$ in the sense that every neighbourhood of ${\displaystyle x}$ with respect to the topology on ${\displaystyle X}$ also 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 ${\displaystyle S}$.

Limit points should not be confused with boundary points. For example, ${\displaystyle 0}$ is a boundary point (but not a limit point) of set ${\displaystyle \{0\}}$ in ${\displaystyle \mathbb {R} }$ with standard 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). [1] [2] [3]

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.

With respect to the usual Euclidean topology, the sequence of rational numbers ${\displaystyle x_{n}=(-1)^{n}{\frac {n}{n+1}}}$ 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 ${\displaystyle \{x_{n}\}}$.

There is also a closely related concept for sequences. A cluster point (or accumulation point) of a sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ in a topological 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 concept generalizes to nets and filters.

## Definition

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 ${\displaystyle S}$  if every neighbourhood of ${\displaystyle x}$  contains at least one point of ${\displaystyle S}$  different from ${\displaystyle x}$  itself.

Note that it doesn't 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 (which all metric spaces are), 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 ${\displaystyle S}$ . 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 limit point

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

## For 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 ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$  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 cluster point of ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$  if and only if ${\displaystyle x}$  is a limit of some subsequence of ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ . 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 the 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 the related topic of filters.

### Properties

• Every limit of a non-constant sequence is an accumulation point of the sequence.

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

To each sequence ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$  in a topological space ${\displaystyle X}$  we can associate the set ${\displaystyle A=\{x_{n}:n\in \mathbb {N} \}}$  consisting of all the elements in the sequence.

• If there a 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 ${\displaystyle A}$ . For example, if the sequence is the constant sequence with value ${\displaystyle x}$ , we have ${\displaystyle A=\{x\}}$  and ${\displaystyle x}$  is an isolated point of ${\displaystyle A}$  and not an accumulation point of ${\displaystyle A}$ .
• 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 ${\displaystyle A}$ .

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 that will have ${\displaystyle A}$  as associated set of elements.

• 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 (even none) points of ${\displaystyle A}$  and that neighborhood can only contain finitely many terms of such sequences).

## Selected facts

• We have the following characterization of limit points: ${\displaystyle x}$  is a limit point of ${\displaystyle S}$  if and only if it is in the closure of ${\displaystyle S\setminus \{x\}}$ .
• 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, ${\displaystyle x}$  is a limit point of ${\displaystyle S}$ , if and only if every neighborhood of ${\displaystyle x}$  contains a point of ${\displaystyle S}$  other than ${\displaystyle x}$ , if and only if every neighborhood of ${\displaystyle x}$  contains a point of ${\displaystyle S\setminus \{x\}}$ , if and only if ${\displaystyle x}$  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.
• Proof: ("Left subset") Suppose ${\displaystyle x}$  is in the closure of ${\displaystyle S}$ . If ${\displaystyle x}$  is in ${\displaystyle S}$ , we are done. If ${\displaystyle x}$  is not in ${\displaystyle S}$ , then every neighbourhood of ${\displaystyle x}$  contains a point of ${\displaystyle S}$ , and this point cannot be ${\displaystyle x}$ . In other words, ${\displaystyle x}$  is a limit point of ${\displaystyle S}$  and ${\displaystyle x}$  is in ${\displaystyle L(S)}$ . ("Right subset") If ${\displaystyle x}$  is in ${\displaystyle S}$ , then every neighbourhood of ${\displaystyle x}$  clearly meets ${\displaystyle S}$ , so ${\displaystyle x}$  is in the closure of ${\displaystyle S}$ . If ${\displaystyle x}$  is in ${\displaystyle L(S)}$ , then every neighbourhood of ${\displaystyle x}$  contains a point of ${\displaystyle S}$  (other than ${\displaystyle x}$ ), so ${\displaystyle x}$  is again in the closure of ${\displaystyle S}$ . This completes the proof.
• 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.
• Proof: ${\displaystyle S}$  is closed if and only if ${\displaystyle S}$  is equal to its closure if and only if ${\displaystyle S=S\cup L(S)}$  if and only if ${\displaystyle L(S)}$  is contained in ${\displaystyle S}$ .
• Another proof: Let ${\displaystyle S}$  be a closed set and ${\displaystyle x}$  a limit point of ${\displaystyle S}$ . If ${\displaystyle x}$  is not in ${\displaystyle S}$ , then the complement to ${\displaystyle S}$  comprises an open neighbourhood of ${\displaystyle x}$ . Since ${\displaystyle x}$  is a limit point of ${\displaystyle S}$ , any open neighbourhood of ${\displaystyle x}$  should have a non-trivial intersection with ${\displaystyle S}$ . However, a set can not have a non-trivial intersection with its complement. Conversely, assume ${\displaystyle S}$  contains all its limit points. We shall show that the complement of ${\displaystyle S}$  is an open set. Let ${\displaystyle x}$  be a point in the complement of ${\displaystyle S}$ . By assumption, ${\displaystyle x}$  is not a limit point, and hence there exists an open neighbourhood U of ${\displaystyle x}$  that does not intersect ${\displaystyle S}$ , and so ${\displaystyle U}$  lies entirely in the complement of ${\displaystyle S}$ . Since this argument holds for arbitrary ${\displaystyle x}$  in the complement of ${\displaystyle S}$ , the complement of ${\displaystyle S}$  can be expressed as a union of open neighbourhoods of the points in the complement of ${\displaystyle S}$ . Hence the complement of ${\displaystyle S}$  is open.
• No isolated point is a limit point of any set.
• Proof: If ${\displaystyle x}$  is an isolated point, then ${\displaystyle \{x\}}$  is a neighbourhood of ${\displaystyle x}$  that contains no points other than ${\displaystyle x}$ .
• The closure ${\displaystyle cl(S)}$  of a set ${\displaystyle S}$  is a disjoint union of its limit points ${\displaystyle L(S)}$  and isolated points ${\displaystyle I(S)}$ :
${\displaystyle cl(S)=L(S)\cup I(S),L(S)\cap I(S)=\emptyset .}$
• A space ${\displaystyle X}$  is discrete if and only if no subset of ${\displaystyle X}$  has a limit point.
• Proof: If ${\displaystyle X}$  is discrete, then every point is isolated and cannot be a limit point of any set. Conversely, if ${\displaystyle X}$  is not discrete, then there is a singleton ${\displaystyle \{x\}}$  that is not open. Hence, every open neighbourhood of ${\displaystyle \{x\}}$  contains a point ${\displaystyle y\neq x}$ , and so ${\displaystyle x}$  is a limit point of ${\displaystyle X}$ .
• 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 ${\displaystyle S}$ .
• Proof: As long as ${\displaystyle S\setminus \{x\}}$  is nonempty, its closure will be ${\displaystyle X}$ . It's only empty when ${\displaystyle S}$  is empty or ${\displaystyle x}$  is the unique element of ${\displaystyle S}$ .
• By definition, every limit point is an adherent point.