Semi-Jordan curve theorem on the Marcus-Wyse topological plane.

The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., M W -topological plane or ( Z 2 , γ) for brevity. We first prove that while every simple closed M W -curve is semi-open in ( Z 2 , γ) , it may not be semi-closed. Given a simple closed M W -curve with l elements, denoted by S C γ l , after establishing a continuous analog of S C γ l denoted by A (S C γ l) , we initially show that A (S C γ l) is both semi-open and semi-closed in ( R 2 , U) , where ( R 2 , U) is the 2 -dimensional real plane R 2 with the usual topology U . Furthermore, we find a condition for A (S C γ l) to separate ( R 2 , U) into exactly two non-empty components, compared to a typical Jordan curve theorem on ( R 2 , U). Since not every S C γ l always separates ( Z 2 , γ) into two nonempty components, we find a condition for S C γ l , l ≠ 4 , to separate ( Z 2 , γ) into exactly two components. The semi-Jordan curve theorem on the M W -topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science. [ABSTRACT FROM AUTHOR]