Semi-topological Properties of the K-Topological Version of the Jordan Curve Theorem

Abstract

The paper initially investigates semi-topological properties of the Khalimsky (K-, for brevity) topological version of the classical Jordan curve theorem. Assume a simple closed K-curve with l elements in $({\mathbb {Z}}^2, \kappa ^2)$ that is the K-topological plane, denoted by $C_K^l$ for brevity. Then every $C_K^l$ separates (${\mathbb {Z}}^2, \kappa ^2)$ into exactly two nonempty components that may be neither open nor closed in (${\mathbb {Z}}^2, \kappa ^2)$. Hence we need to investigate semi-topological features of $C_K^l$ and ${\mathbb {Z}}^2 {\smallsetminus } C_K^l$ in $({\mathbb {Z}}^2, \kappa ^2)$. We first show that not every $C_K^l$ is always semi-open or semi-closed in $({\mathbb {Z}}^2, \kappa ^2)$. Second, we find a condition for $C_K^l$ to be either semi-open or semi-closed in $({\mathbb {Z}}^2, \kappa ^2)$. After establishing a continuous analog of $C_K^l$ denoted by $\mathcal {A}(C_K^l)\,(\subset {\mathbb {R}}^2)$, we show that $\mathcal {A}(C_K^l)$ is both semi-open and semi-closed in $({\mathbb {R}}^2, \mathcal {U})$ that is the 2-dimensional real plane with the usual topology. Besides, we show that $\mathcal {A}(C_K^l)$ always separates $({\mathbb {R}}^2, \mathcal {U})$ into two non-empty components, denoted by C and D, that are both semi-open and semi-closed to obtain a partition of ${\mathbb {R}}^2$, i.e., $\{C, D, \mathcal {A}(C_K^l)\}$. Finally, we obtain a partition of ${\mathbb {Z}}^2$, i.e., $\{I(C_K^l), O(C_K^l), C_K^l\}$ and prove that each of $I(C_K^l)$ and $O(C_K^l)$ is semi-closed and it need not be semi-open, where $I(C_K^l)$ and $O(C_K^l)$ are called an inside and outside of $C_K^l$, respectively.