The converse of the Jordan curve theorem and a characterization of planar maps
A region of a closed set F in \({\mathbb{R}}^ 2\) is an arcwise connected component of \({\mathbb{R}}^ 2\setminus F\). A point p in F is accessible from a region \(\Omega\) if there is a simple arc from any (hence all) point in \(\Omega\) to p having only p in common with F. First an elementary proof is given of a converse to the Jordan Curve Theorem due to Schönflies: If F is a compact set in \({\mathbb{R}}^ 2\) with precisely two regions such that every point in F is accessible from each of those regions, then F is a simple curve. The main result of the paper extends this result to the case where F has finitely many regions. Let F be compact with finitely many regions such that each point in F is on the boundary of at least two regions and is accessible from all regions of which it is a boundary point, then F is a finite planar graph.
scientific article; zbMATH DE number 4135040