# Coffee Mug and Doughnut are Homeomorphic

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## Theorem

A doughnut and a mug coffee mug are homeomorphic.

## Proof

They are both solid objects with one hole.

A classical doughnut is in the shape of a torus.

That is, it is a solid figure with one hole.

The classic coffee-cup shape is a receptacle with a closed loop of china forming a handle.

The hole through which you put your fingers is homeomorphic to the hole in the doughnut.

$\blacksquare$

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.6$: Homeomorphisms: Examples $3.6.2 \ \text{(e)}$