[A question I remember from many years ago.]


Definition



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> A **Riemann surface** is a connected complex manifold $X$ of complex dimension one. This means that $X$ is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point $x \in X$ there is a neighbourhood of $x$ that is homeomorphic to the open unit disk of the complex plane, and the transition maps between two overlapping charts are required to be holomorphic.  
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> [source](https://en.wikipedia.org/wiki/Riemann_surface)
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When we define ordinary manifolds, we have to add the assumption "paracompact" if we want to rule out strange manifolds like the [long line](https://en.wikipedia.org/wiki/Long_line_(topology)). But that assumption is not included in the definition above. What is a reference for the (I think I remember) fact that a Riemann surface as defined above is necessarily paracompact?



A Riemann surface is automatically paracompact
This is precisely the statement of [Radó's theorem](https://en.wikipedia.org/wiki/Rad%C3%B3%27s_theorem_(Riemann_surfaces)) (modulo the standard equivalence between paracompactness and second-countability for connected manifolds). I believe there are multiple proofs available, one of which is in section 1.3 of [Hubbard's book](http://matrixeditions.com/TeichmullerVol1.html) (referenced on Wikipedia.)


