@article{Han2022, 
author = {Sang-Eon Han},
title = {Semi-Jordan curve theorem on the Marcus-Wyse topological plane},
year = {2022},
journal = {Electronic Research Archive},
volume = {30},
number = {12},
pages = {4341-4365},
keywords = {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.sciopen.com/article/10.3934/era.2022220},
doi = {10.3934/era.2022220},
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.}
}