A Note on the Discrete Jordan Curve Theorem (Revised)

Abstract

According to a general definition of discrete curves, surfaces, and manifolds. This paper focuses on the Jordan curve theorem in 2D discrete spaces. The Jordan curve theorem says that a (simply) closed curve separates a simply connected surface into two components. Based on the definition of discrete surfaces, we give three reasonable definitions of simply connected spaces. Theoretically, these three definition shall be equivalent. We have proved the Jordan curve theorem under the third definition of simply connected spaces. The Jordan theorem shows the relationship among an object, its boundary, and its outside area. After the publication of the first version of the paper ({\it L. Chen, Note on the discrete Jordan Curve Theorem. In: SPIE Conf. on Vision Geometry VIII, vol. 3811, pp. 82-94. (1999).}), we found some statements in the original proof of the Jordan Curve Theorem were not explained well. One case was not proven in details. In this revision, we added two more minor definitions and make the proof more solid and sound when it is needed for embedding a discrete surface into a Euclidean space. In this revision, we also proved that the third definition of simply connected spaces equivalent to the second definition of simply connected spaces.