Definition:Discrete Subgroup/Real Numbers

Definition

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

Then $G$ is discrete if and only if:

$\forall g \in G: \exists \epsilon \in \R_{>0}: \openint {g - \epsilon} {g + \epsilon} \cap G = \set g$

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