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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"? ;-) – Hans Lundmark Aug 7 '11 at 8:43
... or "donut" for that matter. – lhf Aug 7 '11 at 11:30
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. – Qiaochu Yuan Aug 7 '11 at 14:47
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." – Qiaochu Yuan Aug 7 '11 at 14:49
why "donut" and not a "bagel"? – PA6OTA May 22 '14 at 16:47

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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Actually, I wouldn't want to eat one of your two-dimensional donuts :) My coffee mugs are rather three-dimensional as well... – t.b. Aug 7 '11 at 10:59
Not even if it's filled with cream? :) – Andrea Mori Aug 7 '11 at 11:04
If it has a cream filling, it certainly cannot be two-dimensional anymore... :D – J. M. Aug 7 '11 at 11:10
The point is that the topological equivalence will not preserve any cream filling, as that will flow all over the place unless thoroughly beaten. – Tim Porter Aug 7 '11 at 11:33
The problem with the question is that neither the coffee cup nor the doughnut has been made that precise. In this case it is very difficult to give and explicit homeomorphism between them. On the other hand if you start with a mug without handle, specified as ever you like then it should be possible to produce a flat disc (thickness that of the mug) and then add a handle. (Is the torus a solid torus or just the surface... ditto for the mug?) – Tim Porter Aug 7 '11 at 11:39

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