Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points

Tareq M. Al-shami; El-Sayed A. Abo-Tabl; Tareq M. Al-shami; El-Sayed A. Abo-Tabl

doi:10.3934/math.2021335

AIMS Mathematics

2021, Volume 6, Issue 6: 5675-5694. doi: 10.3934/math.2021335

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Research article

Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points

Tareq M. Al-shami ^{1
,
,},
El-Sayed A. Abo-Tabl ^2,3

1.
Department of Mathematics, Sana'a University, Sana'a, Yemen
2.
Department of Mathematics, College of Arts and Science, Methnab, Qassim University, Buridah, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

Received: 06 December 2020 Accepted: 22 March 2021 Published: 24 March 2021
MSC : 54A05, 54C15, 54D10, 54D15, 47H10

Soft set theory is a theme of interest for many authors working in various areas because of its rich potential for applications in many directions. it received the attention of the topologists who always seeking to generalize and apply the topological notions on different structures. To contribute to this research area, in this paper, we formulate new soft separation axioms, namely $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and $ tt $-soft $ \alpha $-regular spaces. They are defined using total belong and total non-belong relations with respect to ordinary points. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft $ \alpha $-regular spaces. With the help of examples, we show the relationships between them as well as with soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and soft $ \alpha $-regular spaces. Also, we explore under what conditions they are kept between soft topological space and its parametric topological spaces. We characterize $ tt $-soft $ \alpha T_1 $ and $ tt $-soft $ \alpha $-regular spaces and give some conditions that guarantee the equivalence of $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2) $ and the equivalence of $ tt $-soft $ \alpha T_i\; (i = 1, 2, 3) $. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. In the end, we study the main properties of of $ \alpha $-fixed soft point theorem.
Citation: Tareq M. Al-shami, El-Sayed A. Abo-Tabl. Soft $ \alpha $-separation axioms and $ \alpha $-fixed soft points[J]. AIMS Mathematics, 2021, 6(6): 5675-5694. doi: 10.3934/math.2021335

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Abstract

Soft set theory is a theme of interest for many authors working in various areas because of its rich potential for applications in many directions. it received the attention of the topologists who always seeking to generalize and apply the topological notions on different structures. To contribute to this research area, in this paper, we formulate new soft separation axioms, namely $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and $ tt $-soft $ \alpha $-regular spaces. They are defined using total belong and total non-belong relations with respect to ordinary points. This way of definition helps us to generalize existing comparable properties via general topology and to remove a strict condition of the shape of soft open and closed subsets of soft $ \alpha $-regular spaces. With the help of examples, we show the relationships between them as well as with soft $ \alpha T_i\; (i = 0, 1, 2, 3, 4) $ and soft $ \alpha $-regular spaces. Also, we explore under what conditions they are kept between soft topological space and its parametric topological spaces. We characterize $ tt $-soft $ \alpha T_1 $ and $ tt $-soft $ \alpha $-regular spaces and give some conditions that guarantee the equivalence of $ tt $-soft $ \alpha T_i\; (i = 0, 1, 2) $ and the equivalence of $ tt $-soft $ \alpha T_i\; (i = 1, 2, 3) $. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces and sum of soft topological spaces. In the end, we study the main properties of of $ \alpha $-fixed soft point theorem.

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