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This is a continuation to my previous question here.

In Ahlfors' complex analysis text, page 299 he says (defines) that a global analytic function $\mathbf{f}$, which can be continued along all arcs in some punctured disk, has an algebraic singularity at $z=0$ whenever the Laurent development in terms of the "local uniformizing variable" $\zeta$ contains only a finite number of negative powers:

In special cases the Laurent development may contain only a finite number of negative powers. Then $F(\zeta)$ has either a removable singularity or a pole, and the multiple-valued function $f(z)$ (or, more correctly, the global analytic function obtained by continuing the given initial branch within a punctured disk) is said to have an algebraic singularity or branch point at $z = 0$, provided of course that $h > 1$. If $F(\zeta)$ has a removable singularity, the branch point is an ordinary algebraic singularity, in the opposite case it is an algebraic pole. In either case $f(z)$ tends to a definite limit $A_0$ or $\infty$ as $z$ tends to $0$ along an arbitrary arc.

On the next page (I guess) he wants to generalize the concept of a global analytic function by including such branch points:

In the case of an algebraic singularity it is desirable to complete the Riemann surface off so as to include a branch point with the projection $a$. The branch point itself is not a germ of $f$, but it is fully determined by a set of fractional power series developments $$f_\nu(z)=\sum_{n=n_0}^\infty A_n \omega^{n \nu} (z-a)^{n/h} \tag{3} $$ analogous to (2); for a singularity at infinity $z - a$ has to be replaced by $1/z$. The neighborhoods of the branch point shall include the branch point itself as well as, for some $\delta > 0$, all germs $(f_\nu,\zeta)$ with $|\zeta - a| < \delta $ obtained by substituting in (3) a single-valued branch of $(z - a)^{1/h}$ defined in a neighborhood of $\zeta$. The resulting topological space will be a surface in the sense that every point, including the branch points, has a neighborhood which is homeomorphic to a disk.

My question is: What is the precise definition of a branch point? From the first quote it appears that it should be the point $a \in \mathbb C$ itself - but that doesn't make sense to me when reading the next quote.

Also, if possible please illuminate how the resulting topological space becomes a surface. Thanks!

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What is the precise definition of a branch point?

It is an abstract point added to a topological space. If $X$ is a topological space and $d$ is your neighbor's dog (which is not an element of $X$), then $X\cup\{d\}$ becomes a topological space once we define what the neighborhoods of $d$ are. This is what Ahlfors does. Informally, we think of the branch point as something "hanging over" the point $a$, just as the Riemann surface "hangs over" $\mathbb C$.

how the resulting topological space becomes a surface

Let $A$ be the set of all germs $(f_\nu,\zeta)$ with $0<|\zeta-a|<\delta$ where $\delta$ is small enough so that we stay away from all other singularities of $f$. The key point is that $A$ is homeomorphic to an open annulus, equivalently, to a punctured disk. Then we think of punctured disk to which an abstract point is added to fill the puncture. The result is a disk. This is what Ahlfors means by the new space being a surface: every point has a neighborhood which is homeomorphic to a disk.

To see the key point above, observe that the germ of $f_\nu$ at $\zeta$ is determined by a choice of branch of $(\zeta-a)^{1/h}$. Let us write $\zeta -a= re^{it}$ with $r\in (0,\delta)$ and $t\in\mathbb R$. Then the neighborhood of $a$ in $\mathbb C$ is homeomorphic to $(0,\delta)\times (\mathbb R/2\pi \mathbb Z)$, because a point is determined by specifying its argument (in addition to modulus) up to a multiple of $2\pi$. Similarly, the set of germs of $f_\nu$ is homeomorphic to $(0,\delta)\times (\mathbb R/2\pi h\mathbb Z)$, because a branch of $(\zeta-a)^{1/h}$ is determined by specifying the argument of $\zeta-a$ up to a multiple of $2\pi h$.

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