I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?
Moreover, is there a theorem which classifies the 2-manifolds with boundary?
I recently learned about the Classification Theorem for compact 2-manifolds. Is there a similar classification theorem for ALL 2-manifolds, not just the compact ones?
Moreover, is there a theorem which classifies the 2-manifolds with boundary?
Yes, there's a classification theorem for non-compact 2-manifolds.
This paper gives the classification for triangulable 2-manifolds:
http://www.jstor.org/stable/1993768
That an arbitrary (2nd countable, Hausdorff) topological 2-manifold admits a triangulation is fairly classical. Ahlfors book "Riemann Surfaces" has a proof. There are others available, see for example this list:
https://mathoverflow.net/questions/17578/triangulating-surfaces
If all you're interested in is compact manifolds with boundary, you get that classification immediately from the closed manifold case. Because if you have a compact manifold with boundary, its boundary is a disjoint union of circles. So cap those circles off with discs to produce a closed manifold. So compact manifolds with boundary are classified by the closed manifold you get by "capping off" and the number of boundary circles you started with.
Non-compact manifolds have a more delicate classification -- think for example about the complement of a Cantor set in a compact surface.
Amazingly, the complete classification of noncompact 2-manifolds with boundary was not completed until 2007. Here's the reference:
A. O. Prishlyak and K. I. Mischenko, Classification of noncompact surfaces with boundary, Methods Funct. Anal. Topology 13 (2007), no. 1, 62–66.
EDIT: It turns out, as pointed out in the comment by Moishe Cohen below, that the classification had actually been completed much earlier by E. Brown and R. Messer (The classification of two-dimensional manifolds, Trans. Amer. Math. Soc. 255 (1979), 377–402).