The complex torus as a Riemann surface

(A gap constant exists.) The map $T(s,t)=s{\omega }_1+t{\omega }_2$ is an $\mathbb{R}$-linear isomorphism ${\mathbb{R}}^2\to \mathbb{C}$, because ${\omega }_1,{\omega }_2$ are $\mathbb{R}$-linearly independent, and $\mathrm{\Lambda }=T({\mathbb{Z}}^2)$; applying [F3] to the inverse of $T$ gives $c>0$ with $\mathrm{\mid }T(s,t)\mathrm{\mid }\ge c\max (\mathrm{\mid }s\mathrm{\mid },\mathrm{\mid }t\mathrm{\mid })$, hence $\mathrm{\mid }\lambda \mathrm{\mid }\ge c$ for every nonzero $\lambda \in \mathrm{\Lambda }$.

($q$ is continuous and open.) The quotient topology makes $q$ continuous, and for open $W\subseteq \mathbb{C}$ the preimage $q^{-1}(q(W))={\bigcup }_{\lambda \in \mathrm{\Lambda }}(W+\lambda )$ is open as a union of translates of an open set, so $q(W)$ is open by the definition of the quotient topology.

(Lattice points in a disc are finite.) If $\lambda \mathrm{\ne }{\lambda }^{\mathrm{\prime }}$ are in $\mathrm{\Lambda }$ then $\mathrm{\mid }\lambda -{\lambda }^{\mathrm{\prime }}\mathrm{\mid }\ge c>0$ by step 1.1, so $\mathrm{\Lambda }$ is discrete: each $\{\lambda \}$ is open in $\mathrm{\Lambda }$; for every $R>0$ the set $\mathrm{\Lambda }\cap \overline{D(0,R)}$ is closed and bounded in $\mathbb{C}={\mathbb{R}}^2$, hence compact by [F5] and discrete as a subspace, hence finite by [F6].

(Connectedness.) The space $\mathbb{C}$ is convex, hence connected, and $q$ is continuous and surjective, so $\mathbb{C}\mathrm{/}\mathrm{\Lambda }=q(\mathbb{C})$ is connected by [F4].

(Compactness.) The map $F(s,t)=q(T(s,t))$ is continuous on the compact square $[0,1{]}^2$, so its image is compact by [F4] and [F5]; every $z\in \mathbb{C}$ equals $T(s,t)$ for some $(s,t)\in {\mathbb{R}}^2$, and subtracting the integer parts of $s$ and $t$ exhibits $z$ as an element of $T([0,1{]}^2)+\mathrm{\Lambda }$, so $F([0,1{]}^2)=\mathbb{C}\mathrm{/}\mathrm{\Lambda }$; hence $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is compact.

(Small discs give charts.) Fix any $z\in \mathbb{C}$ and put $W=D(z,c\mathrm{/}4)$; if $q(w)=q(w^{\mathrm{\prime }})$ for $w,w^{\mathrm{\prime }}\in W$ then $w-w^{\mathrm{\prime }}\in \mathrm{\Lambda }$ and $\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }<c\mathrm{/}2<c$, so $w=w^{\mathrm{\prime }}$ by step 1.1; thus $q$ restricted to $W$ is an injective continuous open map onto the open set $q(W)$ of $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$, and its inverse $\phi =q{\mathrm{\mid }}_W^{-1}$ is a homeomorphism $q(W)\to W\subseteq \mathbb{C}$, that is, a chart; the sets $q(D(z,c\mathrm{/}4))$ over $z\in \mathbb{C}$ cover $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$.

(Second countability.) Let $\mathcal{B}$ be a countable base of $\mathbb{C}$; the family $\{q(B):B\in \mathcal{B}\}$ is countable and consists of open sets by step 1.2, and it is a base of $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$: given $[z]\in V$ with $V$ open, the set $q^{-1}(V)$ is an open neighbourhood of $z$, so some $B\in \mathcal{B}$ satisfies $z\in B\subseteq q^{-1}(V)$, whence $[z]\in q(B)\subseteq V$.

(Hausdorffness.) Let $[z]\mathrm{\ne }[z^{\mathrm{\prime }}]$, so $z-z^{\mathrm{\prime }}\mathrm{\notin }\mathrm{\Lambda }$; the distance $d=\inf \{\mathrm{\mid }z-z^{\mathrm{\prime }}-\lambda \mathrm{\mid }:\lambda \in \mathrm{\Lambda }\}$ is positive, because otherwise points of $\mathrm{\Lambda }$ would accumulate at $z-z^{\mathrm{\prime }}$ inside some disc $\overline{D(0,R)}$ containing $z-z^{\mathrm{\prime }}$ and infinitely many distinct lattice points, contradicting the finiteness in step 2.1; with $r=d\mathrm{/}3$, the open sets $q(D(z,r))$ and $q(D(z^{\mathrm{\prime }},r))$ of step 1.2 are disjoint, since $u-v\in \mathrm{\Lambda }$ for $u\in D(z,r)$, $v\in D(z^{\mathrm{\prime }},r)$ would give $\mathrm{\mid }z-z^{\mathrm{\prime }}-(u-v)\mathrm{\mid }<2d\mathrm{/}3<d$.

(Transition functions are translations.) Let $\phi =q{\mathrm{\mid }}_W^{-1}$ and ${\phi }^{\mathrm{\prime }}=q{\mathrm{\mid }}_{W^{\mathrm{\prime }}}^{-1}$ be two charts as in step 2.4; on the overlap of their images, ${\phi }^{\mathrm{\prime }}({\phi }^{-1}(w))=w+\lambda (w)$ with $\lambda (w)={\phi }^{\mathrm{\prime }}(q(w))-w\in \mathrm{\Lambda }$, and $w\mapsto \lambda (w)$ is continuous because both ${\phi }^{-1}$ and $q$ are; since distinct lattice points are at distance at least $c$ by step 1.1, $\lambda$ is locally constant, so near each point the transition is $w\mapsto w+\lambda$ for a fixed $\lambda \in \mathrm{\Lambda }$, which is holomorphic.

(Conclusion.) The quotient $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is nonempty, connected by step 2.2, compact by step 2.3, Hausdorff by step 3.1 and second countable by step 2.5, and the charts of step 2.4 have the holomorphic transition functions $w\mapsto w+\lambda$ of step 3.2, so they form a holomorphic atlas; by [F1], $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is a compact Riemann surface.