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 be semi-closed. Given a simple closed $MW$-curve with $l$ elements, denoted by $S{C}_{\gamma }^l$, after establishing a continuous analog of $S{C}_{\gamma }^l$ denoted by $\mathcal{A}(S{C}_{\gamma }^l)$, we initially show that $\mathcal{A}(S{C}_{\gamma }^l)$ 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 $\mathcal{A}(S{C}_{\gamma }^l)$ 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 $S{C}_{\gamma }^l$ always separates ( ${\mathbb{Z}}^2,\gamma )$ into two nonempty components, we find a condition for $S{C}_{\gamma }^l,l\ne 4,$ 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.