\begin{definition}[Definition:Open Set/Metric Space]
Let $M = \struct {A, d}$ be a metric space.
Let $U \subseteq A$.
Then $U$ is an '''open set in $M$''' {{iff}} it is a neighborhood of each of its points.
That is:
:$\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$
where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.
That is, for every point $y$ in $U$, we can find an $\epsilon \in \R_{>0}$, dependent on that $y$, such that the open $\epsilon$-ball of $y$ lies entirely inside $U$.
Another way of saying the same thing is that one can not get out of $U$ by moving an arbitrarily small distance from any point in $U$.
It is important to note that, in general, the values of $\epsilon$ depend on $y$.
That is, it is ''not'' required that:
:$\exists \epsilon \in \R_{>0}: \forall y \in U: \map {B_\epsilon} y \subseteq U$
\end{definition}