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., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2022-09-01
|
| Series: | Electronic Research Archive |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2022220?viewType=HTML |
| _version_ | 1797983631397879808 |
|---|---|
| author | Sang-Eon Han |
| author_facet | Sang-Eon Han |
| author_sort | Sang-Eon Han |
| collection | DOAJ |
| description | The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science. |
| first_indexed | 2024-04-11T06:50:20Z |
| format | Article |
| id | doaj.art-85c71c9563a74eaea02c654c467bff33 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-04-11T06:50:20Z |
| publishDate | 2022-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-85c71c9563a74eaea02c654c467bff332022-12-22T04:39:14ZengAIMS PressElectronic Research Archive2688-15942022-09-0130124341436510.3934/era.2022220Semi-Jordan curve theorem on the Marcus-Wyse topological planeSang-Eon Han0Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk 54896, Republic of KoreaThe paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.https://www.aimspress.com/article/doi/10.3934/era.2022220?viewType=HTMLsemi-jordan curve theoremsemi-opensemi-closedalexandroff spacemarcus-wyse topologymarcus-wyse (mw-， for brevity) topological planesemi-homeomorphismcontinuous analog of a digital objectdigital-topological groupdigital topology |
| spellingShingle | Sang-Eon Han Semi-Jordan curve theorem on the Marcus-Wyse topological plane Electronic Research Archive semi-jordan curve theorem semi-open semi-closed alexandroff space marcus-wyse topology marcus-wyse (mw-， for brevity) topological plane semi-homeomorphism continuous analog of a digital object digital-topological group digital topology |
| title | Semi-Jordan curve theorem on the Marcus-Wyse topological plane |
| title_full | Semi-Jordan curve theorem on the Marcus-Wyse topological plane |
| title_fullStr | Semi-Jordan curve theorem on the Marcus-Wyse topological plane |
| title_full_unstemmed | Semi-Jordan curve theorem on the Marcus-Wyse topological plane |
| title_short | Semi-Jordan curve theorem on the Marcus-Wyse topological plane |
| title_sort | semi jordan curve theorem on the marcus wyse topological plane |
| topic | semi-jordan curve theorem semi-open semi-closed alexandroff space marcus-wyse topology marcus-wyse (mw-， for brevity) topological plane semi-homeomorphism continuous analog of a digital object digital-topological group digital topology |
| url | https://www.aimspress.com/article/doi/10.3934/era.2022220?viewType=HTML |
| work_keys_str_mv | AT sangeonhan semijordancurvetheoremonthemarcuswysetopologicalplane |