# Definition:Limit Point/Topology

## Contents

## Definition

Let $T = \left({S, \tau}\right)$ 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:

- $A \cap \left({U \setminus \left\{{x}\right\}}\right) \ne \varnothing$

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 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 $\alpha \in A$.

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

## Simple Examples

- $0$ is the only limit point of the set $\left\{{1/n: n \in \N}\right\}$ in the usual (Euclidean) topology of $\R$.

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

- In $\R$ under the usual (Euclidean) topology, $a$ is a limit point of the open interval $\left({a \,.\,.\, b}\right)$ and also of the closed interval $\left[{a \,.\,.\, b}\right]$. Thus it can be seen that a limit point of a set may or may not be part of that set.

- From Rationals Dense in Reals, it is shown that any point $x \in \R$ is a limit point of the set of rational numbers $\Q$. This is an interesting case, because $\Q$ is countable but its set of limit points in $\R$ is $\R$ itself, which is uncountable.

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

## Also

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