I have read that most proofs of the triangulability of surfaces require the use of the Jordan-Schönflies Theorem. However, is such high-tech machinery really needed? The problem is that 3-manifolds can also be triangulated, yet the analogue of the Jordan-Schönflies Theorem in dimension 3 fails (I believe that the Alexander horned-sphere yields a counterexample). Hence, can the main idea behind the success of being able to triangulate surfaces be elucidated without fiddling with the Jordan-Schönflies Theorem?
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