= Discrete subgroup
{title2=$\Gamma\leq G$}
A subgroup $G$ of a <topological group> is discrete when the identity has a neighbourhood containing no other element of $G$. Translation then isolates every group element. In a Hausdorff <topological group>, such a subgroup is closed: any accumulation at a point would make differences of distinct nearby group elements approach the identity. Consequently its intersection with a <compact set> is finite. For isometries of <hyperbolic space>, this converts compactness of the set of isometries moving one point a bounded distance into a <properly discontinuous group action>.
For example, $(-1/2,1/2)\cap\mathbb Z=\{0\}$ isolates the <identity element> of the additive <discrete subgroup> $\mathbb Z\subset\mathbb R$. Its induced <subspace topology> is the <discrete topology>, while the surrounding <real numbers> retain their usual <topology>; see <closed subgroup> for why this <subgroup> is closed but not open.
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