Semi-topological properties of the Marcus-Wyse topological spaces

Since the Marcus-Wyse ( $MW$-, for brevity) topological spaces play important roles in the fields of pure and applied topology (see Remark 2.2), the paper initially proves that the $MW$-topological space satisfies the semi- $T_3$-separation axiom. To do this work more efficiently, we first propose several techniques discriminating between the semi-openness or the semi-closedness of a set in the $MW$-topological space. Using this approach, we suggest the condition for simple $MW$-paths to be semi-closed, which confirms that while every $MW$-path $P$ with $|P|\ge 2$ is semi-open, it may not be semi-closed. Besides, for each point $p\in {\mathbb{Z}}^2$ the smallest open neighborhood of the point $p$ is proved to be a regular open set so that it is semi-closed. Note that the $MW$-topological space is proved to satisfy the semi- $T_3$-separation axiom, i.e., it is proved to be a semi- $T_3$-space so that we can confirm that it also satisfies an $s$- $T_3$-separation axiom. Finally, we prove that the semi- $T_3$-separation axiom is a semi-topological property.