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 ( M W -, for brevity) topology. The Alexandroff compactification of the M W -topological plane is called the infinite M W -topological sphere up to homeomorphism. We first prove that under the M W -topology on Z 2 the connectedness of X (⊂ Z 2) with X ♯ ≥ 2 implies the semi-openness of X. Besides, for the infinite M W -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 M W -topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite M W -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 M W -topological plane preserves the semi- T 3 separation axiom. [ABSTRACT FROM AUTHOR]