## Jordan Curve Theorem, Proof

- JCT - Topology Toolbox
- JCT - Prologue
- JCT - Simple Cases
- JCT - Abstract Graphs and Euler’s Formula
- JCT - K3,3 and the Crossed Arcs Lemma
- JCT - Jordan Separation - the general case
- JCT - Boundaries of the components of the complement of a Jordan curve - I
- JCT - The Jordan Arc theorem
- JCT - Boundaries of the components of the complement of a Jordan curve II
- JCT - Uniqueness of the bounded component of the complement
- JCT - K3,3 on a Torus or Moebius Strip
- JCT - Sources

# JCT - Sources

Scott E. Brodie, MD, PhD

Icahn School of Medicine at Mount Sinai

New York

Our proof of Jordan Separation for Polygons follows Courant and Robbins, "What is Mathematics? An Elementary Approach to Ideas and Methods."

The connection between $K_{3,3}\;$ and JCT is derived from Thomassen (1992), "The Jordan-Schonflies Theorem and the Classification of Surfaces"

The treatment of the Crossed Arcs Lemma simplifies the proof of Maehara (1984), "The Jordan Curve theorem via the Brouwer Fixed Point Theorem." (Maehara deduces the Crossed Ars Lemma and JAT from the Brouwer Fixed Point Theorem and the Tietze Extension Theorem.)

The treatment of the Jordan Arc Theorem is derived from Moise, "Geometric topology in dimensions 2 and 3," (hopefully) modified for clarity.

## Jordan Curve Theorem, Proof

- JCT - Topology Toolbox
- JCT - Prologue
- JCT - Simple Cases
- JCT - Abstract Graphs and Euler’s Formula
- JCT - K3,3 and the Crossed Arcs Lemma
- JCT - Jordan Separation - the general case
- JCT - Boundaries of the components of the complement of a Jordan curve - I
- JCT - The Jordan Arc theorem
- JCT - Boundaries of the components of the complement of a Jordan curve II
- JCT - Uniqueness of the bounded component of the complement
- JCT - K3,3 on a Torus or Moebius Strip
- JCT - Sources

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