The Jordan Curve Theorem

Date
2014-01-15
Authors
Hiura, Evelyn
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University of Hawaii at Manoa
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In mathematics, those statements that are intuitively obvious are often the most difficult ones to prove. One classic example is the Jordan Curve Theorem, which states, “A simple closed curve in the open plane has two complementary domains, of each of which it is the complete frontier.” Part of the difficulty derives from defining the terms involved in precise mathematical language. The more difficult task is to find a means by which the above statement can be verified once the proper definitions have been made. One such verification is given by Newman in Elements of the Topology of Plane Sets of Points. His proof is based on the concept of a “rectangular grating,” which he defines, and theorems established as topological “facts.” This paper is an elaboration of the proof given by Newman of the Jordan Curve Theorem. An attempt is made to incorporate a few modern concepts and terms.
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24 pages
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