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Semi-Jordan curve theorem on the Marcus-Wyse topological plane.
- Source :
- Electronic Research Archive; 2022, Vol. 30 Issue 12, p1-25, 25p
- Publication Year :
- 2022
-
Abstract
- 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]
- Subjects :
- MATHEMATICS
COMPUTER science
COGNITIVE science
JORDAN curves
JORDAN algebras
Subjects
Details
- Language :
- English
- ISSN :
- 26881594
- Volume :
- 30
- Issue :
- 12
- Database :
- Complementary Index
- Journal :
- Electronic Research Archive
- Publication Type :
- Academic Journal
- Accession number :
- 178362135
- Full Text :
- https://doi.org/10.3934/era.2022220