Definition:Limit Point/Topology

Definition
Let $$X$$ 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 $$U$$ of $$X$$ 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:
 * $$x \in \bigcap \left\{{\overline V : V \in \mathcal F}\right\}$$

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

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 $$\R$$.