The germ neighborhoods form a Hausdorff, second-countable Riemann-surface atlas

Every point of $\mathcal{R}({\xi }_0,\mathrm{\Omega })$ is, by [L1], a germ $[f{]}_z$ coming from some function element $(f,U)$, and then $[f{]}_z\in N(f,U)$. So the family $\{N(f,U)\}$ covers the germ space.

Suppose $\xi =[f{]}_z$ lies in $N(f,U)\cap N(g,V)$. Then $\xi =[g{]}_z$ as well, so [L1] gives equality of the germs of $f$ and $g$ at $z$. Hence there is a disc $D$ centered at $z$ with $D\subseteq U\cap V$ and $f=g$ on $D$. For each $w\in D$ this implies $[f{]}_w=[g{]}_w$, so $N(f,D)=N(g,D)\subseteq N(f,U)\cap N(g,V)$. Thus [L2] makes the family $\{N(f,U)\}$ a basis for a topology on the germ space.

The space is Hausdorff. If $[f{]}_z$ and $[g{]}_w$ have $z\mathrm{\ne }w$, choose disjoint discs $D_z\ni z$ and $D_w\ni w$; then $N(f,D_z)$ and $N(g,D_w)$ are disjoint basis neighbourhoods. If $z=w$ but $[f{]}_z\mathrm{\ne }[g{]}_z$, choose discs $D_f\subseteq U$ and $D_g\subseteq V$ centered at $z$ so small that $D_f\cap D_g$ is connected. If $N(f,D_f)$ and $N(g,D_g)$ met, then $f$ and $g$ would agree as germs at some point of $D_f\cap D_g$, and [L3] would force $f=g$ on that connected overlap, hence near $z$, contradiction. So distinct germs have disjoint neighbourhoods, exactly as [L4] requires.

To prove second countability, let $\mathcal{D}$ be the countable family of rational open discs contained in $\mathrm{\Omega }$. For every finite chain $D_0,\dots ,D_m$ of discs in $\mathcal{D}$ with $a_0\in D_0$ and $D_{j-1}\cap D_j\mathrm{\ne }\mathrm{\varnothing }$, at most one branch of the complete analytic function is determined on $D_m$ by continuing the initial germ successively across that chain. So the basis sets arising from such rational-disc chains form a countable family.

On each basis element, ${\varphi }_{f,U}$ is bijective with inverse $z\mapsto [f{]}_z$. If $N(f,U)\cap N(g,V)\mathrm{\ne }\mathrm{\varnothing }$, then step 1.2 gives a disc $D\subseteq U\cap V$ on which $f=g$, so on ${\varphi }_{f,U}(N(f,D))=D$ the transition map ${\varphi }_{g,V}\circ {\varphi }_{f,U}^{-1}:D\to D$ is the identity. Therefore the charts are holomorphically compatible.

Let $[f{]}_z\in N(f,U)$. By [L5], there is a polygonal path in $\mathrm{\Omega }$ from $a_0$ to $z$. Cover its compact image by finitely many rational discs from $\mathcal{D}$ that lie inside the function-element neighborhoods of one continuation chain to $[f{]}_z$, and choose them in the order encountered along the path, with the last disc contained in $U$ and containing $z$. The resulting rational-disc chain determines the same terminal branch on that last disc, so it produces a countable-basis neighbourhood of $[f{]}_z$ contained in $N(f,U)$. Thus the topology has a countable basis, and [L4] makes the germ space second countable.