Semi-separation axioms associated with the Alexandroff compactification of the $MW$-topological plane

The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ( $MW$-, for brevity) topology. The Alexandroff compactification of the $MW$-topological plane is called the infinite $MW$-topological sphere up to homeomorphism. We first prove that under the $MW$-topology on ${\mathbb{Z}}^2$ the connectedness of $X(\subset {\mathbb{Z}}^2)$ with $X^{\mathrm{♯}}\ge 2$ implies the semi-openness of $X$. Besides, for the infinite $MW$-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $MW$-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $MW$-topological sphere is a semi-regular space; thus, it is a semi- $T_3$-space because it is a semi- $T_1$-space. Hence we finally conclude that an Alexandroff compactification of the $MW$-topological plane preserves the semi- $T_3$ separation axiom.