\begin{definition}[Definition:Neighborhood (Metric Space)]
Let $M = \struct {A, d}$ be a metric space.
Let $S \subseteq A$ be a subset of $A$.
Let $x \in S$.
Let there exist $\epsilon \in \R_{>0}$ such that the open $\epsilon$-ball at $x$ lies completely in $S$, that is:
:$\map {B_\epsilon} x \subseteq S$
Then $S$ is a '''neighborhood of $x$ in $M$'''.
\end{definition}