Definition:Discrete Subgroup/Real Numbers

Definition
Let $G$ be a subgroup of the additive group of real numbers.

Then $G$ is discrete :
 * $\forall g \in G : \exists \epsilon > 0: \left({g - \epsilon \,.\,.\, g + \epsilon}\right) \cap G = \left\{ {g}\right\}$

That is, there exists a neighborhood of $g$ which contains no other elements of $G$.