This is not an answer that points to a more recent and more accessible account of the triangulability of surfaces, but rather a way to make the account in the first chapter of Ahlfors' & Sario's book more accessible, if sufficient time is available. It should be noted that the proof given by Ahlfors & Sario works for *all* (connected, 2nd countable) surfaces: compact or noncompact; with or without boundary. I will describe what I found to be the difficulties with Ahlfors' & Sario's presentation and how one can mitigate these difficulties, especially if the learning of this material is by self-study, as was the case for me, and not in the context of a university course. *Disclaimer: I am a mathematician but not a topologist.*

I found there were three main difficulties, all stemming from Ahlfors' & Sario's terse style of writing. The first difficulty is the absence of any references for background material. I found that the classic, self-contained book, *Elements of the Topology of Plane Sets of Points* (2nd ed.), by M.H.A. Newman, and the first two sections of the third chapter of the book, *Algebraic Topology*, by E. Spanier (for the basics of the theory of abstract simplicial complexes), provide sufficient background. The second difficulty is that almost every sentence resembles the statement of a lemma whose proof is left to the reader. The third difficulty is an intentionally omitted proof of a rather difficult result, "46C".

Regarding the second and third difficulties: after filling in the missing details, I decided to write them up in the form of a list of notes (as opposed to an article). I then created a <a href="https://iiscoe.wordpress.com">website</a>, on which I posted these notes. Included in them, is a proof of the result "46C" that relies heavily on the material in the cited book by Newman. Although I did not strive for either optimum mathematical efficiency or elegance, perhaps my notes will be useful to others, in making Ahlfors' & Sario's account more accessible.

While I was at it, I also posted some details for a proof of Schoenflies' Theorem via Complex Analysis; these are details for the presentation in the book, *Boundary Behaviour of Conformal Maps*, by C. Pommerenke. Note that this proof assumes the Riemann Mapping Theorem, proofs of which are more widely available. (These notes were written before I was aware of Newman's book, which happens to also contain a proof of Schoenflies' Theorem -- a proof that is purely topological in nature. As mentioned in Allen Hatcher's answer, an historical account of proofs of Schoenflies' Theorem, including a new one at the time, appeared in the <a href="https://iopscience.iop.org/article/10.1070/RM2005v060n04ABEH003672/pdf">paper</a> and its <a href="https://iopscience.iop.org/article/10.1070/RM2005v060n05ABEH004295">errata</a>. A preprint of the paper is freely available <a href="https://www.researchgate.net/publication/230994655_The_Osgood-Schoenflies_theorem_revisited">here</a>.)

A careful accounting of all the prerequisite material for Ahlfors' & Sario's approach, reveals that it is a very substantial amount. Although such a proof of the triangulability of surfaces can be made accessible to undergraduates, it is not clear whether an undergraduate who embarks on such a project of self-study will still be an undergraduate upon completion of the project.