Definition:Discrete Subgroup
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Definition
Let $G$ be a topological group.
Let $H$ be a subgroup of $G$.
Then $H$ is called a discrete subgroup if and only if it is a discrete group for the induced topology.
Real Numbers
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 $G$.