= Solution
Two <holomorphic function> elements represent the same <germ of a holomorphic function> at $z$ if they agree on some neighbourhood of $z$. For a holomorphic $f$ on an open $D\subseteq G$, set $U(f,D)=\{[f]_z:z\in D\}$. These sets form a <basis of a topology> on the <space of germs of holomorphic functions>: every germ lies in one, and if a germ lies in two, equality of the representatives on a smaller neighbourhood supplies a <basis of a topology> set in their intersection. Declaring arbitrary unions of <basis of a topology> sets open therefore gives a topology.
Different base points are separated by disjoint small discs in $G$. If two different germs have the same base point, choose representatives on one connected small disc. Their sheets are disjoint: any common germ elsewhere on that disc would force equality throughout the disc by the <identity theorem>. This proves the Hausdorff property.
The projection $\pi([f]_z)=z$ maps each sheet $U(f,D)$ homeomorphically onto $D$. Use these maps as complex charts. On overlaps their transition functions are the identity, hence holomorphic; thus each connected component is a <Riemann surface>, and $\pi$ is a local <biholomorphism>, an <analytic germ projection>. For completeness, second countability of a component follows by following continuation chains through a countable <basis of a topology> of rational discs: starting at one germ, finite chains of such discs give only countably many continuation sheets, by uniqueness on overlaps, and they cover every germ in that component.
A <complete analytic function> consists of all germs reachable from one initial germ by <analytic continuation>. The supplied path-lifting equivalence identifies these with its path component. Since the germ space is locally path connected, connected components and path components agree. Thus its continuation surface is exactly one component $\mathcal G_{\mathcal F}$.
A local <biholomorphism> need not evenly cover the whole base; the germ projection is an analytic covering in this local sense. The examples below distinguish a genuine <regular covering map> from a surjective nonregular projection, with the meaning of regular stated explicitly.
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