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Does anyone know a reference for a relatively elementary proof that every compact connected surface can be triangulated? By "elementary," I mean that I could present at least a sketch to undergraduates taking a first semester topology course.

When one proves the classification of surfaces, invariably one assumes that the surface is triangulable or smooth, but I'd like to justify this missing step.

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    @lhf: initially it may seem obvious that all compact manifolds should be triangulable - just cover them with disks, triangulate each one, and refine things along the boundary so they match up. It then becomes clear that the problem lies in how wildly the boundaries of those disks may intersect. It feels intuitively right that circles can only cause so much trouble, but higher-dimensional spheres may exhibit such wild behavior that it may be impossible to fix.2011-04-16

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This paper by Carsten Thomassen (snatched from the MO thread I mentioned in the comments) may contain the most elementary proof.

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    This looks very accessible. Thanks. I wish I had known about the MO thread before I posted the question.2011-04-16