Semi-topological properties of the Marcus-Wyse topological spaces

Abstract

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 $ \vert\, P\, \vert\geq 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.

Keywords

• semi-open
• semi-closed
• semi-t₃-space
• alexandroff space
• semi-regular
• semi-t₁-space
• regular open
• marcus-wyse topology