| 1. | This observation may be mathematically proved using the Jordan curve theorem.
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| 2. | To see this you may use Jordan curve theorem again.
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| 3. | The Jordan curve theorem implies that there is exactly one such coloring.
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| 4. | Since this is true for all closed Jordan curves, ? must be exact.
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| 5. | Page 7 introduces the Jordan curve theorem.
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| 6. | As Algebraist mentioned, the Jordan curve theorem is the one you're looking for.
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| 7. | He did research on topology, especially the theory of polyhedra and the Jordan curve theorem.
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| 8. | :: Conversely suppose ? is a Jordan curve that does not separate the Riemann surface.
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| 9. | If ? is a closed Jordan curve on the surface, then it separates the surface.
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| 10. | Let ? be closed Jordan curve.
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