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.
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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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Bertoglio, N., Chuaqui, R. An elementary geometric nonstandard proof of the Jordan curve theorem. Geom Dedicata 51, 15–27 (1994). https://doi.org/10.1007/BF01264098
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DOI: https://doi.org/10.1007/BF01264098