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An Elementary Proof of the Jordan Curve Theorem and its Application to the Poincaré-Bendixson Theorem

Zhu, Yining LU (2021) In Bachelor's Theses in Mathematical Sciences MATK11 20211
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Abstract
This thesis is mainly focused on a proof of the Jordan Curve Theorem. We follow Tverberg’s paper: A PROOF OF THE JORDAN CURVE THEOREM, but add some more details and graphics in order to make this topological proof more understandable. Later in this thesis, the proof of the Poincare-Bendixson Theorem will also be discussed, as an application of the Jordan Curve Theorem.
Popular Abstract
Imagine that you draw a curve on a paper. If the curve forms a closed loop without self-intersecting, then it will divide the paper into two parts, the interior and the exterior. This is what the Jordan curve theorem says. While it may seem self-evident, the proof is actually not that simple. Mathematicians have developed several different ways of proving it. This thesis follows Tverberg’s paper "A PROOF OF THE JORDAN CURVE THEOREM" but using graphics to demonstrate and make the proof easier to understand. The Poincaré-Bendixson theorem states the long-term behavior of an orbit of a continuous dynamical system in the plane. The Jordan curve theorem plays an important role in proving this theorem and the thesis explains how.
Please use this url to cite or link to this publication:
author
Zhu, Yining LU
supervisor
organization
course
MATK11 20211
year
type
M2 - Bachelor Degree
subject
keywords
Jordan Curve Theorem, Poincaré-Bendixson Theorem
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4123-2021
ISSN
1654-6229
other publication id
2021:K33
language
English
id
9063529
date added to LUP
2024-04-15 16:24:16
date last changed
2024-04-15 16:24:16
@misc{9063529,
  abstract     = {{This thesis is mainly focused on a proof of the Jordan Curve Theorem. We follow Tverberg’s paper: A PROOF OF THE JORDAN CURVE THEOREM, but add some more details and graphics in order to make this topological proof more understandable. Later in this thesis, the proof of the Poincare-Bendixson Theorem will also be discussed, as an application of the Jordan Curve Theorem.}},
  author       = {{Zhu, Yining}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{An Elementary Proof of the Jordan Curve Theorem and its Application to the Poincaré-Bendixson Theorem}},
  year         = {{2021}},
}