# Definition:Limit Point/Topology

## Definition

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 $x$.

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

$\forall U\in \tau :x\in U \implies A \cap \paren {U \setminus \set x} \ne \O\text{.}$

#### 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 Sequence

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

Let $A \subseteq S$.

Let $\sequence {x_n}$ be a sequence in $A$.

Let $\sequence {x_n}$ converge to a value $\alpha \in S$.

Then $\alpha$ is known as a **limit (point) of $\sequence {x_n}$ (as $n$ tends to infinity)**.

## 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 $\R$.

- Every point of $\R$ is a limit point of $\R$ in the usual (Euclidean) topology.

- The set $\Z$ has no limit points in the usual (Euclidean) topology of $\R$.

## Also

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