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Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact surfaces bu I can't find the same kind of results in the noncompact case. I am sorry if it is well-known, I am no geometer.

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There is in fact a classification of non-compact surfaces, though it is more involved; see https://en.wikipedia.org/wiki/Surface#Non-compact_surfaces for a decent explanation. This classification does indeed imply that a non-compact simply-connected surface is homeomorphic to the plane.

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    $\begingroup$ One does not need this classification to deal with simply connected case. $\endgroup$ – Alexandre Eremenko Feb 9 '16 at 3:50
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    $\begingroup$ The reference missing in the Wikipedia article is: Brown and Messer "The classification of two-dimensional manifolds", Transactions of AMS, 1979. The link is here: ams.org/journals/tran/1979-255-00/S0002-9947-1979-0542887-1/… $\endgroup$ – Moishe Kohan Feb 9 '16 at 3:51
  • $\begingroup$ @AlexandreEremenko Sure, I agree this is overkill for answering the specific question asked. Still, Thomas seemed to be asking in the context of comparing the situation of non-compact surfaces to the classification of compact surfaces, in which case the answer seemed relevant. E.g., you wouldn't need the full classification of compact surfaces to prove that a simply-connected compact surface is a sphere, but it would seem amiss not to mention it. $\endgroup$ – Kevin Casto Feb 9 '16 at 4:50

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