Definition:Limit Point/Topology

Topology
Let $$X$$ be a topological space.

Let $$A \subseteq X$$.

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

(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.)

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

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.
 * The set $$\Z$$ has no limit points in the usual topology of $$\R$$.