The converse of the Jordan curve theorem and a characterization of planar maps
type: Item
instance of: scholarly article
title: The converse of the Jordan curve theorem and a characterization of planar maps
zbMATH Open document ID: 0693.57009
DOI: 10.1007/BF00181436
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 4135040
zbMATH Keywords: Jordan curve theorem
zbMATH Keywords: planar maps
author: Carsten Thomassen
reviewed by: George E. Lang jun.
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4135040