An elementary geometric nonstandard proof of the Jordan curve theorem |
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Authors: | Néstor Bertoglio Rolando Chuaqui |
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Institution: | (1) Facultad de Matemáticas, P. Universidad Cátolica de Chile, Casilla 306, Santiago 22, Chile |
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Abstract: | We give an elementary proof, using nonstandard analysis, of the Jordan curve theorem. We also give a nonstandard generalization of the theorem. The proof is purely geometrical in character, without any use of topological concepts and is based on a discrete finite form of the Jordan theorem, whose proof is purely combinatorial.Some familiarity with nonstandard analysis is assumed. The rest of the paper is self-contained except for the proof a discrete standard form of the Jordan theorem. The proof is based on hyperfinite approximations to regions on the plane.Research of the first author partially supported by FONDECYT Grant # 91-1208 and of the second author, by FONDECYT Grant # 90-0647. |
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Keywords: | 54J05 57N05 26E35 |
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