Open Real Interval is Open Ball

Theorem
Let $\R$ denote the real number line with the usual (Euclidean) metric.

Let $I := \openint a b \subseteq \R$ be an open real interval.

Then $I$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$ of some $\alpha \in \R$.

Proof
Let:

Then:

Thus:
 * $\openint a b = \openint {\alpha - \epsilon} {\alpha + \epsilon}$

From Open Ball in Real Number Line is Open Interval:
 * $\openint {\alpha - \epsilon} {\alpha + \epsilon} = \map {B_\epsilon} \alpha$

where $\map {B_\epsilon} \alpha$ is the open $\epsilon$-ball of $\alpha$ in $\R$.

Hence the result

Also see

 * Open Ball in Real Number Line is Open Interval