Definition:Limit Point/Topology

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

Let $A \subseteq X$.

Definition from Neighborhood
A point $x \in X$ is called a limit point of $A$ if every neighborhood $U$ of $x$ satisfies $A \cap \left({U \setminus \left\{{x}\right\}}\right) \ne \varnothing$.

(Informally speaking, $x$ is a limit point of $A$ if there are points in $A$ that are different from $x$ but arbitrarily close to it.)

Definition from Open Set
A point $x \in X$ 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 $x$.

Note that this definition is the same as the previous one if the definition of neighborhood is the one which insists that the neighborhood must be open in $X$.

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 of Limit Points of Filters
Let $\mathcal F$ be a filter on $X$.

A point $x \in X$ is called a limit point of $\mathcal F$ if:
 * $\displaystyle x \in \bigcap \left\{{\overline V : V \in \mathcal F}\right\}$

where $\overline V$ is the complement of $V$ in $\mathcal F$.

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

Let $a \in X$.

A point $x \in X, x \ne a$ 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 from an open set by requiring that the limit point for a point $a$ is defined as the limit point of the set $\left\{{a}\right\}$.

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.


 * Any point $x \in \R$ is a limit point of the set of rational numbers $\Q$, because for any $\epsilon > 0$, there exists $y \in \Q: y \in \left({x . . x + \epsilon}\right)$ from Between Every Two Reals Exists a Rational. 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 $\R$.