[A question I remember from many years ago.]


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.

When we define ordinary manifolds, we have to add the assumption "paracompact" if we want to rule out strange manifolds like the long line. 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?

  • 3
    $\begingroup$ en.wikipedia.org/wiki/Rad%C3%B3%27s_theorem_(Riemann_surfaces) $\endgroup$
    – Wojowu
    Sep 24 at 16:20
  • $\begingroup$ @Wojowu, that seems like an answer, not just a comment …. $\endgroup$
    – LSpice
    Sep 24 at 16:21
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    $\begingroup$ A proof of Radó's theorem can also be found in Forster and Gilligan's "Lectures on Riemann Surfaces", chapter 23 to be precise. $\endgroup$
    – M.G.
    Sep 24 at 16:27

This is precisely the statement of Radó's theorem (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 (referenced on Wikipedia.)


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