A surface is simply a 2-manifold. The classification theorem for compact connected surfaces (with boundary) is commonly regarded in the categories **TOP**, **DIFF** and **PL**. Well known proofs (e.g. via triangulations, or Morse theory) yield the *same* classification because of results that *connect* these categories for surfaces. Informally speaking, here is what I know to be true for compact connected surfaces 

1) [**TOP** & **PL**]. *Topological surfaces always admit a triangulation, and any two triangulations of a surface are piecewise-linear equivalent* ([Hauptvermutung][1] for surfaces)

2) [**DIFF** & **PL** (without using 1.)]. *Every smooth surface admits a PL-structure, as every smooth manifold does* (See the paper "*On $C^{1}$ Complexes*", by J.H.C. Whitehead).

Next is where I seek to be enlightened:

3) [**TOP** & **DIFF**, (without using either 1. or 2.)]. *Two smooth surfaces are diffeomorphic iff they are homeomorphic, and a topological surface always admits a smoothing.*

**Where can I find a formal statement, and a complete proof of 3.?**

Finally, consider non-compact connceted surfaces (with boundary). There seems to be a complete classification of non-compact connected triangulable surfaces with boundary (See the paper ["*Classification of Noncompact Surfaces with Boundary*"][2], by A.O. Prishlyak and K.I. Mischenko).**What about the TOP and DIFF categories? That is, do the results 1-3 above hold for non-compact surfaces?** 

NOTE: I want to mention the post [Classification problem for non-compact manifolds][3] for a related, yet different discussion. The paper: "*On the Classification of Noncompact Surfaces*", by Ian Richards is mentioned there in a comment. This paper considers the case of non-compact triangulable surfaces without boundary.

Thank you!


  [1]: http://en.wikipedia.org/wiki/Hauptvermutung
  [2]: http://www.imath.kiev.ua/~mfat/html/papers/2007/1/pri_mis/art.pdf
  [3]: http://mathoverflow.net/questions/4155/classification-problem-for-non-compact-manifolds