Upper half-plane (reviewed)

The upper half-plane $\mathcal{H}$ is the set of complex numbers whose imaginary part is positive, endowed with the topology induced from $\C$.

The completed upper half-plane $\mathcal{H}^*$ is \[ \mathcal{H} \cup \Q \cup \{ \infty\},\] endowed with the topology such that the disks tangent to the real line at $r \in \Q$ form a fundamental system of neighbourhoods of $r$, and strips $\{ z \in \mathcal{H} \ \vert \ \Im z > y \} \cup \{ \infty\}$, $y>0$, form a fundamental system of neighbourhoods of $\infty$, which should therefore be thought of as $i \infty$.

The modular group $\SL_2(\Z)$ acts properly discontinuously on $\mathcal{H}$ and $\mathcal{H}^*$ by the formula \[ \left( \begin{matrix} a & b \\ c& d \end{matrix} \right) \cdot z = \frac{az+b}{cz+d}, \] with the obvious conventions regarding $\infty$.