# Definition:Discrete Subgroup

## Definition

Let $G$ be a topological group.

Let $H$ be a subgroup of $G$.

Then $H$ is called a discrete subgroup 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:

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

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