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 vertical bar P vertical bar >= 2 is semi-open, it may not be semi-closed. Besides, for each point p is an element of 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.

Automated summary

This article translates the importance of MW-topological spaces and proves that they satisfy the semi-T-3 separation axiom. By proposing several techniques, the article distinguishes between the semi-openness and semi-closedness of sets in MW-topological spaces and proves some important properties.