\begin{definition}[Definition:Open Set/Normed Vector Space]
Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $U \subseteq X$.
Then $U$ is an '''open set in $V$''' {{iff}}:
:$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$
where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.
\end{definition}