TITLE:
A Proof of the Jordan Curve Theorem
AUTHORS:
Xing Zhang
KEYWORDS:
Jordan Curve Theorem, JCT, Tverberg, Jordan Polygon, Sausage Domain, Jordan Curve
JOURNAL NAME:
Natural Science,
Vol.11 No.12,
December
20,
2019
ABSTRACT: The
proof of the Jordan Curve Theorem (JCT) in this paper is focused on a graphic
illustration and analysis ways so as to make the topological proof more understandable, and is based on the Tverberg’s
method, which is acknowledged as being quite esoteric with no graphic
explanations. The preliminary constructs a parametrisation model for Jordan Polygons. It takes quite a length to introduce four lemmas
since the proof by Jordan Polygon is the approach we want to concern about.
Lemmas show that JCT holds for Jordan polygon and Jordan curve could be
approximated uniformly by a sequence of Jordan polygons. Also, lemmas provide a certain metric description of Jordan polygons to help
evaluate the limit. The final part is the proof of the theorem on the premise
of introduced preliminary and lemmas.