Definition:Limit Point/Metric Space

Definition
Let $M = \left({S, d}\right)$ be a metric space.

Let $A \subseteq S$ be a subset of $S$.

Let $\alpha \in S$.

Let $N_\epsilon \left({\alpha}\right)$ be the $\epsilon$-neighborhood of $\alpha$ for a given $\epsilon \in \R$ such that $\epsilon > 0$.

Then $\alpha$ is a limit point of $A$ iff every $N_\epsilon \left({\alpha}\right)$ contains a point in $A$ other than $\alpha$:
 * $\forall \epsilon \in \R, \epsilon > 0: N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} \ne \varnothing$

that is:
 * $\forall \epsilon \in \R, \epsilon > 0: \left\{{x \in A: 0 < d \left({x, \alpha}\right) < \epsilon}\right\} \ne \varnothing$

Note that $\alpha$ does not have to be an element of $A$ to be a limit point.