Definition:Discrete Subgroup/Real Numbers

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

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

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