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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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Apparently not: mathoverflow.net/questions/89835 – Martin Feb 2 '13 at 6:05
    
@Martin: The linked proof works only for smooth surfaces. If your surface is merely topological, that proof fails. On the other hand, Schoenflies theorem does hold in dimension 3 if you restrict to topologically tame subspheres. – studiosus Apr 24 at 17:09

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