Discrete subgroup

Remarks

Equivalently, there is an open neighbourhood $U$ of the identity $e$ in $G$ with $U \cap \Gamma = \set{e}$. Indeed, then $\gamma U$ is an open set with $\gamma U \cap \Gamma = \set{\gamma}$ for each $\gamma \in \Gamma$.

Example: $\Z$ is a discrete subgroup of $\R$, since $U = (-\tfrac12, \tfrac12)$ meets $\Z$ only in $0$. Likewise $\mathrm{SL}_n(\Z)$ is a discrete subgroup of $\mathrm{SL}_n(\R)$, since two distinct integer matrices differ by at least $1$ in some entry.

Non-example: $\Gamma = \set{a + b\sqrt2 : a, b \in \Z}$ is a subgroup of $\R$ that is not discrete. The numbers $(\sqrt2 - 1)^k$ lie in $\Gamma$, are nonzero, and tend to $0$, so every neighbourhood of $0$ meets $\Gamma$ in more than one point.

In a Hausdorff topological group, every discrete subgroup is closed.