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Thomassen's proof and Filippov's proof of the Weak Jordan Theorem

Martin Klazar

Abstract

We present, in detail and with rigour, the two title proofs. The Weak Jordan Theorem states that the complement of any topological circuit in the plane is disconnected.

Thomassen's proof and Filippov's proof of the Weak Jordan Theorem

Abstract

We present, in detail and with rigour, the two title proofs. The Weak Jordan Theorem states that the complement of any topological circuit in the plane is disconnected.

Paper Structure

This paper contains 4 sections, 19 theorems, 49 equations.

Table of Contents

  1. JT, WJT and AT. Auxiliary results
  2. Thomassen's proof of the WJT
  3. Filippov's proof of the WJT
  4. Concluding remarks

Key Result

Theorem 1.1

For any circuit $f\colon I\to\mathbb{R}^2$, the complement --- it is a disjoint union of two nonempty open connected sets $A$ and $B$, where $A$ is bounded and $B$ is unbounded, such that $\partial A=\partial B=f[I]$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (19)

  • Theorem 1.1: Jordan, 1893
  • Theorem 1.2: WJT
  • Theorem 1.3: AT
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 9 more