\begin{definition}[Definition:Open Ball/Normed Vector Space]
Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space.
Let $x \in X$.
Let $\epsilon \in \R_{>0}$ be a strictly positive real number.
The '''open $\epsilon$-ball of $x$ in $\struct {X, \norm {\,\cdot\,} }$''' is defined as:
:$\map {B_\epsilon} x = \set {y \in X: \norm{x - y} < \epsilon}$
\end{definition}