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

TL;DR

This paper provides a detailed, rigorously modern presentation of Thomassen's and Filippov's proofs of the Weak Jordan Theorem, which states that the complement of a topological circuit in the plane is disconnected.

Contribution

It offers a comprehensive, rigorously modernized exposition of two classical proofs of the Weak Jordan Theorem.

Findings

Both proofs confirm the disconnectedness of the complement of topological circuits in the plane.

The paper clarifies and formalizes the original arguments with modern mathematical rigor.

It enhances understanding of classical topological results through detailed exposition.

Abstract

We present, in detail and with a modern rigor, the two title proofs. The Weak Jordan Theorem (WJT) states that the complement of any topological circuit in the plane is disconnected.