Vol. 36, No. 1, 1971

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A nonstandard proof of the Jordan curve theorem

Louis Edward Narens

Vol. 36 (1971), No. 1, 219–229

In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons.

The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the approximating sequence of polygons is often unwieldly. But by using nonstandard analysis, one can approximate a Jordan curve by a single polygon that is infinitesimally close to the curve. This allows types of reasoning which are extremely difficult and unnatural on sequences of polygons.

Mathematical Subject Classification
Primary: 54.75
Secondary: 02.00
Received: 6 April 1970
Published: 1 January 1971
Louis Edward Narens