The quotient $\mathbb{C}\mathrm{/}\mathrm{\Lambda }$ is a compact Riemann surface

Put $A:=\mathrm{\mid }{\omega }_1{\mathrm{\mid }}^2$, $B:=\mathrm{Re}({\omega }_1\overline{{\omega }_2})$, $C:=\mathrm{\mid }{\omega }_2{\mathrm{\mid }}^2$; then $A,C>0$ and expanding with [F3] gives $\mathrm{\mid }t{\omega }_1+s{\omega }_2{\mathrm{\mid }}^2=A{t}^2+2Bts+C{s}^2$ for all real $t,s$, while $AC-B^2=(\mathrm{Im}({\omega }_1\overline{{\omega }_2}){)}^2>0$, because $\mathrm{Im}({\omega }_1\overline{{\omega }_2})=0$ would make ${\omega }_2=(\mathrm{Re}({\omega }_1\overline{{\omega }_2})\mathrm{/}\mathrm{\mid }{\omega }_1{\mathrm{\mid }}^2){\omega }_1$ a real multiple of ${\omega }_1$, contradicting real-linear independence.

The map $\phi (t,s):=t{\omega }_1+s{\omega }_2$ is surjective: the list ${\omega }_1,{\omega }_2$ is real-linearly independent by [F1], and it must span $\mathbb{C}$ over $\mathbb{R}$, since otherwise a complex number $z\mathrm{\notin }{\mathrm{span}}_{\mathbb{R}}\{{\omega }_1,{\omega }_2\}$ would make ${\omega }_1,{\omega }_2,z$ real-linearly independent (a relation with nonzero coefficient of $z$ would exhibit $z$ as a real combination of ${\omega }_1,{\omega }_2$, so that coefficient vanishes, and then the other two vanish), an independent set of three elements in a space spanned by the two-element set $\{1,i\}$ by [F8], contradicting [F9].

The map $\phi :{\mathbb{R}}^2\to \mathbb{C}$ is continuous: by [F3], $\mathrm{\mid }\phi (t,s)-\phi (t^{\mathrm{\prime }},s^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\mid }{\omega }_1\mathrm{\mid }\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }{\omega }_2\mathrm{\mid }\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid }\le 2\max (\mathrm{\mid }{\omega }_1\mathrm{\mid },\mathrm{\mid }{\omega }_2\mathrm{\mid })\max (\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid },\mathrm{\mid }s-s^{\mathrm{\prime }}\mathrm{\mid })$, so $\phi$ is Lipschitz for the max norm on ${\mathbb{R}}^2$ and is continuous for it; by [F4] the max norm gives the same topology as $d_2$, which is the topology of $\mathbb{C}$ by [F2].

$T_{\mathrm{\Lambda }}$ is nonempty and connected: $\mathbb{C}$ is ${\mathbb{R}}^2$ by [F2], hence contractible by [F10], hence path-connected and connected by [F11] and [F12]; a continuous image of a connected space is connected by [F13], and $\pi$ is continuous and surjective by [F1].

Completing the square in each variable gives $\mathrm{\mid }t{\omega }_1+s{\omega }_2{\mathrm{\mid }}^2=A(t+\frac{B}{A}s{)}^2+\frac{AC-B^2}{A}{s}^2=C(s+\frac{B}{C}t{)}^2+\frac{AC-B^2}{C}{t}^2$ for all real $t,s$, so with $\delta :=\sqrt{(AC-B^2)\mathrm{/}\max (A,C)}>0$ one has $\mathrm{\mid }t{\omega }_1+s{\omega }_2\mathrm{\mid }\ge \delta \max (\mathrm{\mid }t\mathrm{\mid },\mathrm{\mid }s\mathrm{\mid })$; hence every nonzero $\lambda \in \mathrm{\Lambda }$ satisfies $\mathrm{\mid }\lambda \mathrm{\mid }\ge \delta$, and distinct $\lambda ,{\lambda }^{\mathrm{\prime }}\in \mathrm{\Lambda }$ satisfy $\mathrm{\mid }\lambda -{\lambda }^{\mathrm{\prime }}\mathrm{\mid }\ge \delta$.

$T_{\mathrm{\Lambda }}$ is compact: the box $[0,1{]}^2$ is compact in ${\mathbb{R}}^2$ by [F5], its image $F:=\phi ([0,1{]}^2)=\{t{\omega }_1+s{\omega }_2:0\le t,s\le 1\}$ is compact by [F6] and step 1.3, and $\pi (F)=T_{\mathrm{\Lambda }}$: by step 1.2 every $z\in \mathbb{C}$ is $\phi (t,s)$ for real $t,s$, and writing $t=m+t^{\mathrm{\prime }}$, $s=n+s^{\mathrm{\prime }}$ with $m,n\in \mathbb{Z}$ and $t^{\mathrm{\prime }},s^{\mathrm{\prime }}\in [0,1)$ by the division algorithm for real numbers gives $z-(m{\omega }_1+n{\omega }_2)=\phi (t^{\mathrm{\prime }},s^{\mathrm{\prime }})\in F$ with $m{\omega }_1+n{\omega }_2\in \mathrm{\Lambda }$; hence $T_{\mathrm{\Lambda }}$ is a continuous image of the compact set $F$, so it is compact by [F6].

$\mathrm{\Lambda }$ is uniformly discrete and closed in $\mathbb{C}$: by step 2.1 the ball $B(0,\delta )$ contains no nonzero lattice point, so every point of $\mathrm{\Lambda }$ is isolated, and if ${\lambda }_n\in \mathrm{\Lambda }$ converges to $z\in \mathbb{C}$ then $\mathrm{\mid }{\lambda }_n-{\lambda }_m\mathrm{\mid }<\delta$ for all large $n,m$, which forces ${\lambda }_n={\lambda }_m$ for all large $n,m$ by the uniform gap, and then $z={\lambda }_N\in \mathrm{\Lambda }$; moreover, since $\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }m{\omega }_1+n{\omega }_2\mathrm{\mid }\ge \delta \max (\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }n\mathrm{\mid })$ for $\lambda =m{\omega }_1+n{\omega }_2\in \mathrm{\Lambda }$, the lattice points in any bounded set have bounded parameters $m,n$ and are therefore finite, so for $p\mathrm{\notin }\mathrm{\Lambda }$ the distance $r:=\mathrm{dist}(p,\mathrm{\Lambda })={\inf }_{\lambda \in \mathrm{\Lambda }}\mathrm{\mid }p-\lambda \mathrm{\mid }$ is positive.

The group $\mathrm{\Lambda }$ acts on $\mathbb{C}$ by translations $\lambda \cdot z:=z+\lambda$, which are homeomorphisms of $\mathbb{C}$ by [F2], and this is a covering-space action in the sense of [F14]: for $z\in \mathbb{C}$ put $U:=B(z,\delta \mathrm{/}2)$; if $w\in U\cap (\lambda +U)$ with $\lambda \in \mathrm{\Lambda }\setminus \{0\}$, then $w=\lambda +u$ with $u\in U$ and $\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\mid }w-u\mathrm{\mid }\le \mathrm{\mid }w-z\mathrm{\mid }+\mathrm{\mid }z-u\mathrm{\mid }<\delta$, contradicting step 2.1.

By [F15] the orbit map of the action of step 3.2 is a covering map, and its orbit space is $T_{\mathrm{\Lambda }}$ with orbit map $\pi$ by [F1]; hence $\pi$ is a covering map, so $\pi$ is continuous and locally injective; moreover $\pi$ is open, because for open $W\subseteq \mathbb{C}$ one has ${\pi }^{-1}(\pi (W))={\bigcup }_{\lambda \in \mathrm{\Lambda }}(W+\lambda )$, a union of open translates, so $\pi (W)$ is open in $T_{\mathrm{\Lambda }}$ by [F1].

Take all open balls $U\subseteq \mathbb{C}$ on which $\pi$ is injective. These include $B(z,\delta \mathrm{/}2)$ for every $z$, since two points in such a ball with the same class differ by a lattice element of modulus $<\delta$. For each such $U$, the restriction $\pi {\mathrm{\mid }}_U$ is continuous, open and bijective onto the open set $\pi (U)$ by step 4.1. Thus ${\phi }_U:=(\pi {\mathrm{\mid }}_U{)}^{-1}:\pi (U)\to U$ is a homeomorphism onto an open subset of $\mathbb{C}$, hence a chart in the sense of [F16].

$T_{\mathrm{\Lambda }}$ is Hausdorff: if $[z]\mathrm{\ne }[w]$, then $p:=w-z\mathrm{\notin }\mathrm{\Lambda }$, and with $r=\mathrm{dist}(p,\mathrm{\Lambda })>0$ from step 3.1 the open sets $\pi (B(z,r\mathrm{/}2))$ and $\pi (B(w,r\mathrm{/}2))$ are disjoint, since $a\in B(z,r\mathrm{/}2)$ and $b\in B(w,r\mathrm{/}2)$ with $\pi (a)=\pi (b)$ would give $b-a\in \mathrm{\Lambda }$ and $\mathrm{\mid }p-(b-a)\mathrm{\mid }\le \mathrm{\mid }w-b\mathrm{\mid }+\mathrm{\mid }z-a\mathrm{\mid }<r$, contradicting $r=\mathrm{dist}(p,\mathrm{\Lambda })$.

$T_{\mathrm{\Lambda }}$ is second countable: by [F7] and [F2] the topology of $\mathbb{C}$ has a countable basis $\mathcal{B}$ of rational boxes, and $\{\pi (B):B\in \mathcal{B}\}$ is a countable family of open subsets of $T_{\mathrm{\Lambda }}$ by step 4.1; it is a basis, because for open $W\subseteq T_{\mathrm{\Lambda }}$ and $x\in W$ one picks $z\in {\pi }^{-1}(x)$, then a box $B\in \mathcal{B}$ with $z\in B\subseteq {\pi }^{-1}(W)$ (possible since ${\pi }^{-1}(W)$ is open by [F1]), and then $x\in \pi (B)\subseteq W$.

The domains of the charts of step 5.1 cover $T_{\mathrm{\Lambda }}$, since the family includes the charts from $B(z,\delta \mathrm{/}2)$ for every $z\in \mathbb{C}$, so the charts ${\phi }_z$ of step 5.1 form an atlas; any two are compatible: for charts ${\phi }_U,{\phi }_V$ of this family put $\mathrm{\Omega }:={\phi }_U(\pi (U)\cap \pi (V))$, an open subset of $\mathbb{C}$, and for $u\in \mathrm{\Omega }$ let $v(u):={\phi }_V(\pi (u))$, so that $u-v(u)\in \mathrm{\Lambda }$; fixing $u_0\in \mathrm{\Omega }$ and ${\lambda }_0:=u_0-v(u_0)$, continuity of $u\mapsto v(u)$ (a composite of the continuous maps $\pi$, ${\phi }_V$) and step 2.1 give a neighbourhood of $u_0$ on which $\mathrm{\mid }(u-v(u))-{\lambda }_0\mathrm{\mid }<\delta$, and since $(u-v(u))-{\lambda }_0\in \mathrm{\Lambda }$ and all nonzero lattice elements have modulus $\ge \delta$ by step 2.1, there $u-v(u)={\lambda }_0$; hence ${\phi }_V\circ {\phi }_U^{-1}$ equals the translation $u\mapsto u-{\lambda }_0$ near each point of its open domain $\mathrm{\Omega }$, and is holomorphic.

$\pi$ is holomorphic: use the identity chart on $\mathbb{C}$ from [F18]. For every chart ${\phi }_U$ of step 5.1, its expression ${\phi }_U\circ \pi$ is the identity on $U$, hence holomorphic. The balls in that family cover $\mathbb{C}$, so [F16] gives holomorphy of $\pi$ at every point.

Collecting: the charts of step 5.1 are pairwise compatible by step 6.1 and cover $T_{\mathrm{\Lambda }}$; $T_{\mathrm{\Lambda }}$ is nonempty and connected (step 1.4), Hausdorff (step 5.2) and second countable (step 5.3), so $T_{\mathrm{\Lambda }}$ is a Riemann surface by [F16], and it is compact by step 2.2; $\pi$ is a covering map by step 4.1 and holomorphic by step 6.2, so it is a holomorphic covering map. The construction uses only the set $\mathrm{\Lambda }$ and the metric and quotient structures attached to it: a change of representatives of a class does not change $\pi$, and the family of all open balls on which $\pi$ is injective is determined by $\mathrm{\Lambda }$ alone. The basis-dependent constant $\delta$ only proves that this family covers the quotient; it does not restrict the family defining the atlas. Thus changing the oriented basis leaves this atlas unchanged.