Semi-topological properties of the Marcus-Wyse topological spaces

Sang-Eon Han; Sang-Eon Han

doi:10.3934/math.2022705

AIMS Mathematics

2022, Volume 7, Issue 7: 12742-12759. doi: 10.3934/math.2022705

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Research article

Semi-topological properties of the Marcus-Wyse topological spaces

Sang-Eon Han ^,

Department of Mathematics Education, Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju-City Jeonbuk, 54896, Korea

Received: 27 February 2022 Revised: 06 April 2022 Accepted: 08 April 2022 Published: 29 April 2022
MSC : 54A05, 54D10, 54F05, 54C08, 54C10, 54F65

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 $-separation axiom is a semi-topological property.
- semi-open,
- semi-closed,
- semi-$ T_3 $-space,
- Alexandroff space,
- semi-regular,
- semi-$ T_1 $-space,
- regular open,
- Marcus-Wyse topology
Citation: Sang-Eon Han. Semi-topological properties of the Marcus-Wyse topological spaces[J]. AIMS Mathematics, 2022, 7(7): 12742-12759. doi: 10.3934/math.2022705

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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.

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