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.
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.