Definition:Open Set/Metric Space

Definition
Let $M = \left({A, d}\right)$ be a metric space.

Let $U \subseteq A$.

Then $U$ is: iff:
 * an open set in $M$
 * open in $M$
 * a $d$-open set
 * $d$-open
 * $\forall y \in U: \exists \epsilon \in \R_{>0}: B_\epsilon \left({y}\right) \subseteq U$

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:
 * $\exists \epsilon \in \R_{>0}: \forall y \in U: B_\epsilon \left({y}\right) \subseteq U$

Also see

 * Open Ball is Open Set
 * Open Set may not be Open Ball
 * Induced Topology (Metric Space), the set of all open sets in a given metric space