Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many wellknown 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 wellknown, I am no geometer.

6$\begingroup$ Yes. You can use the Riemann uniformization theorem to prove that. See en.wikipedia.org/wiki/… $\endgroup$ – André Henriques Feb 8 '16 at 22:37
There is in fact a classification of noncompact surfaces, though it is more involved; see https://en.wikipedia.org/wiki/Surface#Noncompact_surfaces for a decent explanation. This classification does indeed imply that a noncompact simplyconnected surface is homeomorphic to the plane.

1$\begingroup$ One does not need this classification to deal with simply connected case. $\endgroup$ – Alexandre Eremenko Feb 9 '16 at 3:50

1$\begingroup$ The reference missing in the Wikipedia article is: Brown and Messer "The classification of twodimensional manifolds", Transactions of AMS, 1979. The link is here: ams.org/journals/tran/197925500/S00029947197905428871/… $\endgroup$ – 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 noncompact 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 simplyconnected compact surface is a sphere, but it would seem amiss not to mention it. $\endgroup$ – Kevin Casto Feb 9 '16 at 4:50