Definition:Discrete Subgroup/Real Numbers

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Definition

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


Then $G$ is discrete if and only if:

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

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