Definition:Open Set/Metric Space

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

Let $U \subseteq M$.

Then $U$ can be described as iff:
 * an open set in $M$
 * open in $M$
 * a $d$-open set
 * $d$-open
 * $\forall y \in U: \exists \epsilon \left({y}\right) > 0: B \left({y; \epsilon \left({y}\right)}\right) \subseteq U$

That is, for every point $y$ in $U$, we can find an $\epsilon > 0$, dependent on that $y$, such that the open $\epsilon$-ball of that point 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 the necessary value of $\epsilon$ may be different for each $y$.

Also see

 * Open Ball is Open Set
 * Open Set may not be Open Ball