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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279

Published online September 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.3.269

© The Korean Institute of Intelligent Systems

Soft Minimal Soft Sets and Soft Prehomogeneity in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to :
Samer Al Ghour (algore@just.edu.jo)

Received: July 4, 2021; Revised: August 16, 2021; Accepted: August 28, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology

No potential conflict of interest relevant to this article was reported.

Samer Al Ghour received the Ph.D. in Mathematics from University of Jordan, Jordan in 1999. Currently, he is a professor at the Department of Mathematics and Statistics, Jordan University of Science and Technology, Jordan. His research interests is include general topology, fuzzy topology, and soft set theory.

E-mail: algore@just.edu.jo


Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(3): 269-279

Published online September 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.3.269

Copyright © The Korean Institute of Intelligent Systems.

Soft Minimal Soft Sets and Soft Prehomogeneity in Soft Topological Spaces

Samer Al Ghour

Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

Correspondence to:Samer Al Ghour (algore@just.edu.jo)

Received: July 4, 2021; Revised: August 16, 2021; Accepted: August 28, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we give characterizations for soft minimal soft open sets in terms of the soft closure operator, and we conclude that soft subsets of soft minimal soft open sets are soft preopen sets. In addition to these, we define soft minimal soft sets and soft minimal soft preopen sets as two new classes of soft sets in soft topological spaces, and we define soft prehomogeneity as a new soft topological property. We give several relationships regarding these new notions and related known soft topological notions. We show that soft minimal soft preopen sets are soft points, and we prove that soft minimal soft sets with non-null soft interiors are soft minimal soft open sets. Moreover, we show that soft prehomogeneous soft topological space that has a soft minimal soft set is soft locally indiscrete. Also, we give several characterizations of soft locally indiscrete soft topological space in terms of soft minimal soft open sets, soft minimal soft sets, soft preopen sets, and soft prehomogeneity. We deal with correspondence between our new soft topological notions and their analogs topological ones. Finally, we raise six open questions.

Keywords: Soft minimal soft open sets, Soft preopen soft sets, Soft locally indiscrete, Prehomogeneity, Generated soft topology

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