A universal covering of a Riemann surface inherits a unique complex structure

Every point of $W$ has arbitrarily small coordinate-disc neighbourhoods, and each coordinate disc $B$ is path-connected and simply connected: $B$ is homeomorphic to a Euclidean disc, a nonempty convex set, hence simply connected by [F12], and the homeomorphism transfers simple connectivity and path connectedness by [F13]. Consequently $W$ is locally path-connected and locally connected, and $W$ is semilocally simply connected in the sense of [F6]: a coordinate disc $U$ through $x$ satisfies ${\pi }_1(U,x)=1$, so the induced map to ${\pi }_1(W,x)$ is trivial.

The coordinate discs of the atlas of $W$ form an open cover of $W$; by [F10], applied to the second-countable space $W$ of [F1] under Countable Choice [F23], this cover has a countable subcover $\{B_n{\}}_{n\in \mathbb{N}}$. Each $B_n$ is a coordinate disc, and each $B_n$ is second countable as a subspace of $W$ [F1, F11].

Set $\mathcal{X}$ to be the set consisting of $x_0$, one chosen point of each $B_n$, and one chosen point of each path component of each intersection $B_m\cap B_n$ with $m,n\in \mathbb{N}$. This is a countable set: each intersection is an open subspace of the locally path-connected space $W$, so its components are open [F9] and are path-connected; distinct components are disjoint nonempty open sets and each contains a member of a countable base of $W$ [F1], so there are at most countably many of them. All choices are made from countably many nonempty sets of points, which is licensed by Countable Choice [F23].

$W$ is path-connected: it is connected by [F1] and locally path-connected by step 1.1, so [F7] applies. Hence [F3] provides the universal covering $p:\tilde{W}\to W$; by [F4] and [F5] the space $\tilde{W}$ is nonempty, path-connected and simply connected, so by [F8] it is connected and nonempty without further hypotheses.

$\tilde{W}$ is locally path-connected by [F14] and step 1.1. It is also Hausdorff: let $u\mathrm{\ne }v$ in $\tilde{W}$. If $p(u)\mathrm{\ne }p(v)$, choose disjoint open neighbourhoods $U\ni p(u)$, $V\ni p(v)$; the sheets of $p^{-1}(U)$ and $p^{-1}(V)$ containing $u$ and $v$ are disjoint open neighbourhoods. If $p(u)=p(v)=x$, choose an evenly covered neighbourhood $U$ of $x$; the distinct points $u,v$ of $p^{-1}(x)$ lie in distinct sheets of $p^{-1}(U)$, whose union is a disjoint union of open sets.

For every $n$ and every ordered pair $(y,y^{\mathrm{\prime }})$ of points of $\mathcal{X}\cap B_n$ choose a path $H_n(y,y^{\mathrm{\prime }})$ in $B_n$ from $y$ to $y^{\mathrm{\prime }}$, the constant path when $y=y^{\mathrm{\prime }}$; such paths exist because $B_n$ is path-connected (step 1.1). This is again a choice from countably many nonempty sets of paths, licensed by Countable Choice [F23].

Let $B$ be a coordinate disc of $W$. The restriction $p^{-1}(B)\to B$ of $p$ over the open set $B$ is a covering [F15], and $p^{-1}(B)$ is locally path-connected as a subspace of the locally path-connected space $\tilde{W}$ [F9, step 2.2]. Its components are therefore open and equal to its path components, and $p^{-1}(B)$ is their disjoint union [F9]. For such a component $S$ the restriction $p{\mathrm{\mid }}_S:S\to B$ is again a covering (it inherits evenly covered neighbourhoods inside $B$ from [F2]), and $B$ is path-connected and simply connected by step 1.1; [F16] therefore makes $p{\mathrm{\mid }}_S$ a homeomorphism onto $B$. We call these components the sheets over $B$; each sheet contains exactly one point of each fibre $p^{-1}(b)$, $b\in B$.

Let $\gamma$ be any loop at $x_0$. The sets ${\gamma }^{-1}(B_n)$ form an open cover of the compact interval $I$, so there are finitely many indices $n_1,\dots ,n_k$ and a partition $0=a_0<a_1<\cdots <a_k=1$ with $\gamma ([a_{i-1},a_i])\subseteq B_{n_i}$ for all $i$. Write ${\gamma }_i$ for the $i$-th restriction, a path from $\gamma (a_{i-1})$ to $\gamma (a_i)$ inside $B_{n_i}$. For $1\le i\le k-1$ the point $\gamma (a_i)$ lies in $B_{n_i}\cap B_{n_{i+1}}$, so by step 1.3 there is a point $x_i\in \mathcal{X}$ in the same component of this intersection, and a path $J_i$ in that component from $x_i$ to $\gamma (a_i)$; put $x_0$ for the initial and terminal basepoint and let $J_0$ and $J_k$ be the constant paths at $x_0$. Then $J_{i-1}$ joins $x_{i-1}$ to $\gamma (a_{i-1})$ and $J_i^{-1}$ joins $\gamma (a_i)$ to $x_i$, both inside $B_{n_i}$, so $F_i:=J_{i-1}\ast {\gamma }_i\ast J_i^{-1}$ is a path in $B_{n_i}$ from $x_{i-1}$ to $x_i$. Since $B_{n_i}$ is simply connected, every loop in it is null-homotopic, so $F_i$ is homotopic relative to endpoints to the chosen path $H_{n_i}(x_{i-1},x_i)$; and the telescoping homotopies $J_i\ast J_i^{-1}\simeq c_{x_i}$ show that $\gamma$ is homotopic relative to endpoints to $F_1\ast \cdots \ast F_k$, hence to $H_{n_1}(x_0,x_1)\ast H_{n_2}(x_1,x_2)\ast \cdots \ast H_{n_k}(x_{k-1},x_k)$.

Let $T$ be the set of all concatenations $H_{n_1}(y_0,y_1)\ast H_{n_2}(y_1,y_2)\ast \cdots \ast H_{n_k}(y_{k-1},y_k)$ with $k\ge 1$, $y_0=y_k=x_0$, and $y_i\in \mathcal{X}$ for all $i$. Such a concatenation is determined by the finite tuple $(n_1,\dots ,n_k;y_1,\dots ,y_{k-1})$ drawn from $\mathbb{N}$ and the countable set $\mathcal{X}$, so $T$ is countable and nonempty. By step 3.2 every loop at $x_0$ is homotopic relative to endpoints to a member of $T$, so the map $T\to {\pi }_1(W,x_0)$ sending a loop to its class is surjective; a set that is the image of a countable set is countable, hence ${\pi }_1(W,x_0)$ is countable.

Every fibre of $p$ is countable. Fix $x\in W$ and a path $\rho$ in $W$ from $x_0$ to $x$, which exists because $W$ is path-connected (step 2.1). Define $\mathrm{\Psi }:T\to p^{-1}(x)$ by letting $\mathrm{\Psi }(\sigma )$ be the endpoint of the unique lift of the path $\sigma \ast \rho$ that starts at ${\tilde{x}}_0$, where ${\tilde{x}}_0\in p^{-1}(x_0)$ is fixed; this is well defined by [F17]. To see that $\mathrm{\Psi }$ is onto, let $w\in p^{-1}(x)$ and choose a path $\tilde{\tau }$ in the path-connected space $\tilde{W}$ from ${\tilde{x}}_0$ to $w$ (step 2.1); then $\tau :=p\circ \tilde{\tau }$ is a path in $W$ from $x_0$ to $x$, and $\gamma :=\tau \ast {\rho }^{-1}$ is a loop at $x_0$ whose class is realised by some $\sigma \in T$ by step 3.2. Thus $\tau$ is homotopic relative to endpoints to $\sigma \ast \rho$, so by [F18] the lift of $\sigma \ast \rho$ from ${\tilde{x}}_0$ ends at the same point as $\tilde{\tau }$, namely $w$; hence $\mathrm{\Psi }(\sigma )=w$. Therefore $\mathrm{\mid }{p}^{-1}(x)\mathrm{\mid }\le \mathrm{\mid }T\mathrm{\mid }$ is countable.

$\tilde{W}$ is second countable. Fix $n$ and a point $b_n\in B_n$. By step 3.1 the sheets over $B_n$ are the components of $p^{-1}(B_n)$, and each contains exactly one point of $p^{-1}(b_n)$; conversely every point of $p^{-1}(b_n)$ lies in a sheet by [F2]. Hence there are at most $\mathrm{\mid }{p}^{-1}(b_n)\mathrm{\mid }$ sheets over $B_n$, a countable number by step 5.1, and each sheet is homeomorphic to $B_n$ (step 3.1), which is second countable by step 1.2; a countable disjoint union of second-countable spaces is second countable. Therefore each $p^{-1}(B_n)$ is second countable, and $\tilde{W}={\bigcup }_n{p}^{-1}(B_n)$ is covered by countably many open second-countable subspaces, so a countable union of countable bases is a countable base for $\tilde{W}$: the space $\tilde{W}$ is second countable.

The pullback atlas. For every chart $\phi :B\to \mathbb{C}$ of $W$ whose domain is a coordinate disc and every sheet $S$ over $B$ (step 3.1), put ${\psi }_{B,S}:=\phi \circ (p{\mathrm{\mid }}_S):S\to \phi (B)$, a homeomorphism onto the open set $\phi (B)\subseteq \mathbb{C}$. These domains cover $\tilde{W}$, because every point lies in some sheet over some coordinate disc. For two such charts $\psi =\phi \circ (p{\mathrm{\mid }}_S)$ and ${\psi }^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}\circ (p{\mathrm{\mid }}_{S^{\mathrm{\prime }}})$ with $S\cap S^{\mathrm{\prime }}\mathrm{\ne }\mathrm{\varnothing }$, the transition computed in $\phi (p(S\cap S^{\mathrm{\prime }}))$ is ${\psi }^{\mathrm{\prime }}\circ {\psi }^{-1}={\phi }^{\mathrm{\prime }}\circ {\phi }^{-1}$, the transition of two charts of $W$, hence holomorphic. Thus the family is a holomorphic atlas on the nonempty connected Hausdorff second-countable space $\tilde{W}$ (steps 2.1, 2.2, 6.1), so $\tilde{W}$ is a Riemann surface, and the projection is a covering map [F2], i.e. an unbranched covering.

The projection $p$ is holomorphic for this structure: near a point of a sheet $S$ over a coordinate disc $B$, take the chart ${\psi }_{B,S}$ upstairs and the chart $\phi$ downstairs; the chart expression is $\phi \circ p\circ {\psi }_{B,S}^{-1}={\mathrm{id}}_{\phi (B)}$, which is holomorphic.

Uniqueness. Let $\mathcal{A}$ be any complex structure on $\tilde{W}$ for which $p:\tilde{W}\to W$ is holomorphic, and let $\theta :V\to \mathbb{C}$ be a chart of $\mathcal{A}$ and $\psi =\phi \circ (p{\mathrm{\mid }}_S)$ a chart from step 7.1 with $V\cap S\mathrm{\ne }\mathrm{\varnothing }$. On $\theta (V\cap S)$ the map $g:=\phi \circ p\circ {\theta }^{-1}$ is holomorphic, being a chart expression of the holomorphic map $p$ [F20], and it is locally injective because it is the composite of the homeomorphism ${\theta }^{-1}$, the homeomorphism $p{\mathrm{\mid }}_S$ and the homeomorphism $\phi$. By [F21], $g$ is biholomorphic between neighbourhoods, so its inverse $\theta \circ (p{\mathrm{\mid }}_S{)}^{-1}\circ {\phi }^{-1}$ is holomorphic; hence $\theta \circ {\psi }^{-1}$ is holomorphic, while $\psi \circ {\theta }^{-1}$ is holomorphic because it equals $\phi \circ (p\circ {\theta }^{-1})$. Every chart of $\mathcal{A}$ is therefore compatible with every chart of the structure of step 7.1, so the union of the two atlases is an atlas and by [F1] the two complex structures on $\tilde{W}$ coincide.

Deck transformations. Let $h:\tilde{W}\to \tilde{W}$ be a deck transformation, so $h$ is a homeomorphism with $p\circ h=p$ [F22]. Given $y\in \tilde{W}$, choose a chart $\psi =\phi \circ (p{\mathrm{\mid }}_S)$ from step 7.1 near $y$ and a chart ${\psi }^{\mathrm{\prime }}={\phi }^{\mathrm{\prime }}\circ (p{\mathrm{\mid }}_{S^{\mathrm{\prime }}})$ near $h(y)$; shrinking $S$ we may assume $h(S)\subseteq S^{\mathrm{\prime }}$. On $\psi (S)$ the chart expression is ${\psi }^{\mathrm{\prime }}\circ h\circ {\psi }^{-1}={\phi }^{\mathrm{\prime }}\circ (p{\mathrm{\mid }}_{S^{\mathrm{\prime }}})\circ h\circ (p{\mathrm{\mid }}_S{)}^{-1}\circ {\phi }^{-1}={\phi }^{\mathrm{\prime }}\circ {\phi }^{-1}$, which is holomorphic by [F1]. Hence $h$ is holomorphic [F20], and the same argument applied to the deck transformation $h^{-1}$ shows that $h^{-1}$ is holomorphic. Therefore every deck transformation is biholomorphic.