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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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    What's your definition of "coffee mug"? ;-)2011-08-07
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    ... or "donut" for that matter.2011-08-07
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    This isn't really the kind of statement one can attach a rigorous proof to until the meaning of the terms "topological equivalence," "donut," and "coffee mug" are fixed. In my experience the first term can mean homotopy, ambient isotopy, or homeomorphism and it is never really clear which one is meant (sometimes intentionally so). "Donut" could refer either to a torus or a solid torus. And a coffee mug... is it thickened? Do you care about the surface? Etc.2011-08-07
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    As far as I'm concerned you don't need a more rigorous proof of this kind of statement (which is clearly meant more as an easily-grasped intuitive example than anything else) than "if you made a coffee mug out of clay, you could reshape it into a donut without breaking or attaching anything."2011-08-07
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    why "donut" and not a "bagel"?2014-05-22
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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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