\begin{definition}[Definition:Open Set/Real Analysis/Real Euclidean Space]
Let $n \ge 1$ be a natural number.
Let $U \subseteq \R^n$ be a subset.
Then $U$ is '''open (in $\R^n$)''' {{iff}}:
:$\forall x \in U : \exists R \in \R_{>0}: \map B {x, R} \subset U$
where $\map B {x, R}$ denotes the open ball of radius $R$ centered at $x$.
\end{definition}