Definition:Limit Point (Topology)

Definition

Let $T = \left({S, \vartheta}\right)$ be a topological space.

Limit Point of Set

Let $A \subseteq S$.

Definition from Open Set

A point $x \in S$ is called a limit point of $A$ if every open set $U \in \vartheta$ such that $x \in U$ contains some point of $A$ other than $x$.

Definition from Closure

$x$ is called a limit point of $A$ if $x$ belongs to the closure of $A$ but is not an isolated point of $A$.

Definition from Adherent Point

$x$ is called a limit point of $A$ if $x$ is an adherent point of $A$ but is not an isolated point of $A$.

Definition from Sequence

$x$ is called a limit point of $A$ if there is a sequence $\left\langle{x_n}\right\rangle$ in $A$ such that $x$ is a limit point of $\left\langle{x_n}\right\rangle$, considered as sequence in $S$.

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 called a limit point of $a$ if every open set $U \in \vartheta$ such that $x \in U$ contains $a$.

It can be seen that this is the same definition as for the definition of a limit point of a set, by requiring that the limit point for a point $a$ is defined as the limit point of the set $\left\{{a}\right\}$.

Limit Point of Sequence

Let $T = \left({S, \vartheta}\right)$ be a topological space.

Let $A \subseteq S$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $A$.

Let $\left \langle {x_n} \right \rangle$ converge to a value $\alpha \in A$.

Then $\alpha$ is known as a limit (point) of $\left \langle {x_n} \right \rangle$ (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 topology of $\R$.

• Every point of $\R$ is a limit point of $\R$ in the usual topology.

• In $\R$ under the usual 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 topology of $\R$.

See also

• Equivalence of Limit Point Definitions

• Condensation Point
• $\omega$-Accumulation Point
• Adherent Point

• Relationship between Limit Point Types

• Results about limit points can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points