\begin{definition}[Definition:Limit Point/Metric Space]
Let $M = \struct {S, d}$ be a metric space.
Let $A \subseteq S$ be a subset of $S$.
Let $\alpha \in S$. 
Then $\alpha$ is a '''limit point of $A$''' {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$:
:$\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap A \ne \O$
that is:
:$\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$
Note that $\alpha$ does not have to be an element of $A$ to be a '''limit point'''.
(Informally speaking, $\alpha$ is a limit point of $A$ if there are points in $A$ that are different from $\alpha$ but arbitrarily close to it.)
\end{definition}