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I've seen this example given numerous times, but have never seen a real proof in a textbook.

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    So what's the answer? Suppose I had a topological definition of a coffee mug. I want to show that the mug is homeomorphic to a doughnut (torus). In order to do that, I need to PROVE (if not explicitly give the formula) the existence of a function that's a homeomorphism between those two topological spaces. "It intuitively exists" is not a valid argument in mathematics in my opinion. Or is it in topology?2015-02-13

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I am assuming that the questioner knows that the question is about either the surfaces or the 3-manifolds in question. A rigorous, yet diagrammatic proof is in our book Knotted Surfaces and Their Diagrams.

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They are both homeomorphic to a sphere with one handle attached. This is quite clear for the coffee mug (where the handle is precisely the handle.....) and it is easily obtained for the donut (a.k.a. torus) by playing a bit with its representation as a square with opposite edges identified.

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    So what's the answer? Suppose I had a topological definition of a coffee mug. I want to show that the mug is homeomorphic to a doughnut (torus). In order to do that, I need to PROVE (if not explicitly give the formula) the existence of a function that's a homeomorphism between those two topological spaces. "It intuitively exists" is not a valid argument in mathematics in my opinion. Or is it in topology?2015-02-14