The cusp chart and compactness of X(1)

The action extends to ${\mathfrak{H}}^{\ast }$: for $\gamma \in G$ and a cusp ${\gamma }_0\cdot \mathrm{\infty }=m\mathrm{/}n$ with $m,n$ coprime, Bézout gives $b,d\in \mathbb{Z}$ with $md-nb=1$, so $\left(\begin{array}{cc}m& b\\ n& d\end{array}\right)\in S{L}_2(\mathbb{Z})$ and $\gamma {\gamma }_0\cdot \mathrm{\infty }=\gamma \cdot (m\mathrm{/}n)$; Möbius maps are homeomorphisms of $\widehat{\mathbb{C}}$ carrying $\mathbb{Q}\cup \{\mathrm{\infty }\}$ to itself and the basic cusp neighbourhoods of ${\gamma }_0\cdot \mathrm{\infty }$ to basic cusp neighbourhoods of $\gamma {\gamma }_0\cdot \mathrm{\infty }$, so the extended action is continuous and well defined. Every rational $m\mathrm{/}n$ in lowest terms equals $\gamma \cdot \mathrm{\infty }$ for the matrix above with $\gamma =\left(\begin{array}{cc}m& b\\ n& d\end{array}\right)$, and $\mathrm{\infty }$ itself is in the orbit, so the cusps form the single orbit $\mathbb{Q}\cup \{\mathrm{\infty }\}=G\cdot \mathrm{\infty }$.

Fix $N>1$ and put $B_N:=\{\mathrm{\Im }\tau >N\}\cup \{\mathrm{\infty }\}$. If $\gamma \in S{L}_2(\mathbb{Z})$ has $c\mathrm{\ne }0$ and $z,\gamma z\in \mathfrak{H}$ with $\mathrm{\Im }z,\mathrm{\Im }\gamma z>N$, then $\mathrm{\Im }(\gamma z)=\mathrm{\Im }z\mathrm{/}\mathrm{\mid }cz+d{\mathrm{\mid }}^2\le \mathrm{\Im }z\mathrm{/}(c\mathrm{\Im }z{)}^2=1\mathrm{/}(c^2\mathrm{\Im }z)<1\mathrm{/}N<N$, a contradiction; hence every $\gamma$ mapping a point of $B_N$ back into $B_N$ has $c=0$, i.e. lies in $⟨T⟩$. Consequently the $G$-orbit of a point of $B_N$ meets $B_N$ exactly in its $⟨T⟩$-orbit, and $B_N\mathrm{/}⟨T⟩$ is identified with its image $p(B_N)$. By [F4] the map $q(\tau )=e^{2\pi i\tau }$ realises $\mathfrak{H}\mathrm{/}⟨T⟩\cong \{0<\mathrm{\mid }q\mathrm{\mid }<1\}$, so it descends to a homeomorphism of $p(B_N)$ onto $\{\mathrm{\mid }q\mathrm{\mid }<e^{-2\pi N}\}$ sending the cusp class to $0$; the topology at the cusp was defined exactly so that $\{\mathrm{\Im }\tau >N^{\mathrm{\prime }}\}$ corresponds to $\{\mathrm{\mid }q\mathrm{\mid }<e^{-2\pi N^{\mathrm{\prime }}}\}$, so this is a homeomorphism onto the open disc and provides the cusp chart.

Put $K:=\bar{D}\cup \{\mathrm{\infty }\}$ with its subspace topology in ${\mathfrak{H}}^{\ast }$. Given an intrinsic open cover $\mathcal{U}$ of $K$, take $U_{\mathrm{\infty }}\in \mathcal{U}$ containing $\mathrm{\infty }$; the subspace topology and the cusp neighbourhood basis give $N>1$ with $K\cap B_N\subseteq U_{\mathrm{\infty }}$. The set $L:=\bar{D}\cap \{\mathrm{\Im }\tau \le N\}$ is closed and bounded in $\mathbb{C}$, with imaginary part at least $\sqrt{3}\mathrm{/}2$, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line and the metric/topological compactness agreement in [F3]. Since $\mathfrak{H}$ has its usual topology inside ${\mathfrak{H}}^{\ast }$, the topology induced on $L$ from $K$ is its usual subspace topology. Thus $\{U\cap L:U\in \mathcal{U}\}$ is an intrinsic open cover of $L$ and has a finite subcover. The corresponding finitely many members of $\mathcal{U}$, together with $U_{\mathrm{\infty }}$, cover $K$, proving its compactness. The quotient map $p$ is continuous, so $p(\bar{D}\cup \{\mathrm{\infty }\})$ is compact by [F3]; it equals $X(1)$ because every point of $\mathfrak{H}$ is $G$-equivalent to a point of $\bar{D}$ by [F1] and every cusp lies in the orbit of $\mathrm{\infty }$ by 1.1. Hence $X(1)$ is compact.

To separate the cusp from an interior point $p(\tau )$, choose a relatively compact open neighbourhood $U$ of $\tau$ with $0<y_0\le \mathrm{\Im }z\le Y$ on $U$. For every $\gamma \in G$, its height on $U$ is at most $M=\max (Y,1\mathrm{/}{y}_0)$: if $c=0$ height is unchanged, while if $c\mathrm{\ne }0$, $\mathrm{\Im }(\gamma z)\le 1\mathrm{/}(c^2\mathrm{\Im }z)\le 1\mathrm{/}{y}_0$. For $N>\max (1,M)$ the open sets $p(U)$ and $p(B_N)$ are disjoint. The quotient map on ${\mathfrak{H}}^{\ast }$ is open, since the saturation of each open set is the union of its translates; hence these are open neighbourhoods in $X(1)$. Two interior points are separated by [F2], so $X(1)$ is Hausdorff. The interior quotient is open and dense, since every cusp neighbourhood meets $\mathfrak{H}$; its complement is the unique cusp class. Its atlas [F2] is compatible with the cusp coordinate: for $N>1$ stabilisers on $B_N\cap \mathfrak{H}$ are trivial and $q^{\mathrm{\prime }}=2\pi iq\mathrm{\ne }0$, so the transition to any local lift chart and its inverse are holomorphic. The inherited countable interior basis plus $p(B_n)$, $n\ge 2$, gives second countability; connectedness follows from the connected dense interior. Together with 2.1 this makes $X(1)$ a compact Riemann surface.