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

Abstract

<abstract><p>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.</p></abstract>