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In the book Introduction to Teichmüller Spaces, by Taniguchi and Imayoshi, we have the following definition for a Riemann Surface:

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At the following pages, the authors make a remark recalling some of the properties of a Riemann Surface, and then they cite that the topology of a Riemann Surface admits a countable basis.

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My doubt is pretty conceptual: isn't this a axiom in the definition of a Riemann Surface? I mean, shouldn't be in the definition: "A Riemann Surface is a topological space $R$, Hausdorff, with a countable basis, and bla bla bla"?

Otherwise, it should be possible to prove the "enumerable basis existence" only through the definition given above.

Thank you, guys!

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    $\begingroup$ Certainly "countable basis" must be assumed somewhere. I notice the word "manifold" is used without definition in the very first line you cited; maybe that's where the author hides "countable basis". $\endgroup$ – Lee Mosher Sep 1 '16 at 15:10
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    $\begingroup$ @LeeMosher slightly further down in the picture, there is a full definition (starting from a topological space), with no countability assumed. $\endgroup$ – Danu Sep 1 '16 at 15:40
  • $\begingroup$ math.stackexchange.com/questions/683452/… $\endgroup$ – Lee Mosher Sep 1 '16 at 16:59
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This theorem is proven in the book by Forster (as indeed listed in the references of your book). There is no assumption of countability, as far as I can tell. I'll give you the relevant definitions (and the theorem), verbatim from the book:

Definition 1: An $n$-dimensional manifold is a Hausdorff topological space such that every point in it has an open neighborhood homeomorphic to an open subset of $\Bbb R^n$.

Definition 2: A Riemann surface is a pair $(X,\Sigma)$ where $X$ is a connected 2-manifold and $\Sigma$ is a complex structure on $X$.

Theorem (Rado): Every Riemann surface has a countable topology.

The theorem seems to follow from the existence of solutions to the Dirichlet problem (discussed in section 22 of the book), with the proof of the theorem residing in section 23.

In particular, countability is not assumed anywhere. As you mention, countability is an oft-used assumption/axiom in differential geometry, but in the theory of Riemann surfaces it appears to be redundant.

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In his book (Lectures on Riemann surfaces), Forster define a Riemann surface as an Hausdorff space with holomorphic atlas.

This is a theorem of Rado, proved in the same book (page 186), that these conditions implies that a Riemann surface has a countable basis.

I have the same feeling as you though : in the definition of a manifold, the countable basis is usually required since it is a crucial tool for partitions of unity.

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