# Definition:Limit Point

## Real Analysis

Let $S \subseteq \R$ be a subset of the real numbers.

Let $\xi \in \R$ and let $S_\xi$ be the set defined as:

- $S_\xi := \set {x: x \in S, x \ne \xi}$

Then $\xi$ is a **limit point** of $S$ if and only if $\xi$ is at zero distance from $S_\xi$.

## Complex Analysis

Let $S \subseteq \C$ be a subset of the set of complex numbers.

Let $z_0 \in \C$.

Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$.

Then $z_0$ is a **limit point of $S$** if and only if *every* deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$:

- $\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$

that is:

- $\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$

## Metric Space

Let $M = \struct {S, d}$ be a metric space.

Let $\tau$ be the topology induced by the metric $d$.

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.

### Definition 1

$\alpha$ is a **limit point of $A$** if and only if *every* deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$:

- $\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$

that is:

- $\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$

Note that $\alpha$ does not have to be an element of $A$ to be a **limit point**.

### Definition 2

$\alpha$ is a **limit point of $A$** if and only if there is a sequence $\sequence{\alpha_n}$ in $A \setminus \set \alpha$ such that $\sequence{\alpha_n}$ converges to $\alpha$, that is, $\alpha$ is a limit of the sequence $\sequence{\alpha_n}$ in $S$.

### Definition 3

$\alpha$ is a **limit point of $A$** if and only if $\alpha$ is a limit point in the topological space $\struct{S, \tau}$.

## Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $Y \subseteq X$ be a subset of $X$.

### Limit Point of Set

Let $\alpha \in X$.

Then $\alpha$ is a **limit point of $Y$** if and only if *every* deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $Y$:

- $\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap Y \ne \O$

that is:

- $\forall \epsilon \in \R_{>0}: \set {x \in Y: 0 < \norm {x - \alpha} < \epsilon} \ne \O$

Note that $\alpha$ does not have to be an element of $A$ to be a **limit point**.

### Limit Point of Sequence

Let $L \in Y$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $Y \setminus \set L$.

Let $\sequence {x_n}_{n \mathop \in \N}$ converge to $L$.

Then $L$ is a **limit of $\sequence {x_n}_{n \mathop \in \N}$ as $n$ tends to infinity** which is usually written:

- $\ds L = \lim_{n \mathop \to \infty} x_n$

## Topology

Let $T = \struct {S, \tau}$ be a topological space.

### Limit Point of Set

Let $A \subseteq S$.

#### Definition from Open Neighborhood

A point $x \in S$ is a **limit point of $A$** if and only if every open neighborhood $U$ of $x$ satisfies:

- $A \cap \paren {U \setminus \set x} \ne \O$

That is, if and only if every open set $U \in \tau$ such that $x \in U$ contains some point of $A$ distinct from $x$.

#### Definition from Closure

A point $x \in S$ is a **limit point of $A$** if and only if

- $x$ belongs to the closure of $A$ but is not an isolated point of $A$.

#### Definition from Adherent Point

A point $x \in S$ is a **limit point of $A$** if and only if $x$ is an adherent point of $A$ but is not an isolated point of $A$.

#### Definition from Relative Complement

A point $x \in S$ is a **limit point of $A$** if and only if $\left({S \setminus A}\right) \cup \left\{{x}\right\}$ is *not* a neighborhood of $x$.

### Limit Point of Point

The concept of a **limit point** can be sharpened to apply to individual points, as follows:

Let $a \in S$.

A point $x \in S, x \ne a$ is a **limit point of $a$** if and only if every open neighborhood of $x$ contains $a$.

That is, it is a limit point of the singleton $\set a$.

## Limit Point of Filter

Let $T = \struct {S, \tau}$ be a topological space.

Let $\FF$ be a filter on $S$.

A point $x \in S$ is called a **limit point of $\FF$** if and only if $\FF$ is finer than the neighborhood filter of $x$.

## Limit Point of Filter Basis

Let $T = \struct {S, \tau}$ be a topological space.

Let $\FF$ be a filter on the underlying set $S$ of $T$.

Let $\BB$ be a filter basis of $\FF$.

### Definition 1

A point $x \in S$ is called a **limit point of $\BB$** if and only if $\FF$ converges on $x$.

$\BB$ is likewise said to converge on $x$.

### Definition 2

A point $x \in S$ is called a **limit point of $\BB$** if and only if every neighborhood of $x$ contains a set of $\BB$.

## Also known as

A **limit point** is also known as a **cluster point**, or a **point of accumulation**.

However, note that an accumulation point is also seen with a subtly different definition from that of a **limit point**, so be careful.

## Examples

### End Points of Real Interval

The real number $a$ is a limit point of both the open real interval $\openint a b$ as well as of the closed real interval $\closedint a b$.

It is noted that $a \in \closedint a b$ but $a \notin \openint a b$.

### Union of Singleton with Open Real Interval

Let $\R$ be the set of real numbers.

Let $H \subseteq \R$ be the subset of $\R$ defined as:

- $H = \set 0 \cup \openint 1 2$

Then $0$ is not a limit point of $H$.

### Real Number is Limit Point of Rational Numbers in Real Numbers

Let $\R$ be the set of real numbers.

Let $\Q$ be the set of rational numbers.

Let $x \in \R$.

Then $x$ is a limit point of $\Q$.

### Zero is Limit Point of Integer Reciprocal Space

Let $A \subseteq \R$ be the set of all points on $\R$ defined as:

- $A := \set {\dfrac 1 n : n \in \Z_{>0} }$

Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology.

Then $0$ is the only limit point of $A$ in $\R$.

## Also see

- Results about
**limit points**can be found**here**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**limit point**