A Nonstandard Proof of the Jordan Curve Theorem
نویسندگان
چکیده
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.
منابع مشابه
A Nonstandard Proof of the Jordan Curve Theorem
We give a nonstandard variant of Jordan’s proof of the Jordan curve theorem which is free of the defects his contemporaries criticized and avoids the epsilontic burden of the classical proof. The proof is selfcontained, except for the Jordan theorem for polygons taken for granted.
متن کاملA new proof for the Banach-Zarecki theorem: A light on integrability and continuity
To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...
متن کاملGroupoids, the Phragmen-brouwer Property, and the Jordan Curve Theorem
We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points.
متن کاملJordan’s Proof of the Jordan Curve Theorem
This article defends Jordan’s original proof of the Jordan curve theorem. The celebrated theorem of Jordan states that every simple closed curve in the plane separates the complement into two connected nonempty sets: an interior region and an exterior. In 1905, O. Veblen declared that this theorem is “justly regarded as a most important step in the direction of a perfectly rigorous mathematics”...
متن کاملDiscrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps
This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by structural or noetherian induction: Genus Theorem, Euler’s Formula, constructive planarity criteria. A notion of ring of faces is inductively defined and a ...
متن کامل