Maximal Soft Compact and Maximal Soft Connected Topologies
In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the pr...
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| Vydané v: | Applied Computational Intelligence and Soft Computing Ročník 2022; s. 1 - 7 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Hindawi
08.02.2022
John Wiley & Sons, Inc Wiley |
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| ISSN: | 1687-9724, 1687-9732 |
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| Abstract | In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft T1, while a maximal soft connected topology is soft T0. Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal. |
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| AbstractList | In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft
T
1
, while a maximal soft connected topology is soft
T
0
. Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal. In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft T1, while a maximal soft connected topology is soft T0. Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal. In various articles, it is said that the class of all soft topologies on a common universe forms a complete lattice, but in this paper, we prove that it is a complete lattice. Some soft topologies are maximal, and some are minimal with respect to specific soft topological properties. We study the properties of soft compact and soft connected topologies that are maximal. Particularly, we prove that a maximal soft compact topology has identical families of soft compact and soft closed sets. We further show that a maximal soft compact topology is soft T[sub.1], while a maximal soft connected topology is soft T[sub.0]. Lastly, we establish that each soft connected relative topology to a maximal soft connected topology is maximal. |
| Audience | Academic |
| Author | Al Ghour, Samer Ameen, Zanyar A. |
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| Cites_doi | 10.1155/2020/3807418 10.1142/s1793005711002025 10.1016/j.camwa.2011.05.016 10.1016/s0898-1221(03)00016-6 10.1016/j.camwa.2008.11.009 10.1007/s00521-011-0722-3 10.1016/0165-0114(94)90331-x 10.1155/2019/6903809 10.1007/s00500-017-2824-z 10.1142/9789814261302_0021 10.1016/s0898-1221(99)00056-5 10.1109/21.229476 10.22436/jmcs.017.04.02 10.1016/j.camwa.2011.09.051 10.1145/219717.219791 10.3390/math9172153 10.1016/j.camwa.2011.02.006 10.1016/s0165-0114(86)80034-3 10.2298/fil1813755e 10.30948/afmi.2017.13.1.145 10.1007/s13370-020-00783-4 |
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| Copyright | Copyright © 2022 Samer Al Ghour and Zanyar A. Ameen. COPYRIGHT 2022 John Wiley & Sons, Inc. Copyright © 2022 Samer Al Ghour and Zanyar A. Ameen. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. https://creativecommons.org/licenses/by/4.0 |
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| References | 22 23 G. Ilango (29) 2013; 4 25 27 L. A. Zadeh (1) 1996 N. Çağman (21) 2011; 62 İ. Zorlutuna (24) 2012; 3 S. Bayramov (11) 2018; 9 S. Nazmul (16) 2013; 6 10 13 R. A. Mohammed (12) 14 15 17 18 19 R. Vaidya-Nātha-Svāmi (28) 1947 2 3 4 5 6 7 A. Aygünoğlu (8) 2012; 21 Z. A. Ameen (26) 2021 F. Lin (9) 2013; 7 20 |
| References_xml | – ident: 13 doi: 10.1155/2020/3807418 – start-page: 241 ident: 12 article-title: Soft J S-spaces and strong soft J S-spaces – ident: 23 doi: 10.1142/s1793005711002025 – volume: 62 start-page: 351 issue: 1 year: 2011 ident: 21 article-title: Soft topology publication-title: Computers & Mathematics with Applications doi: 10.1016/j.camwa.2011.05.016 – year: 2021 ident: 26 article-title: Extensions of soft topologies – ident: 18 doi: 10.1016/s0898-1221(03)00016-6 – ident: 20 doi: 10.1016/j.camwa.2008.11.009 – volume: 21 start-page: 113 issue: 1 year: 2012 ident: 8 article-title: Some notes on soft topological spaces publication-title: Neural Computing and Applications doi: 10.1007/s00521-011-0722-3 – volume: 9 start-page: 82 issue: 21 year: 2018 ident: 11 article-title: A new approach to separability and compactness in soft topological spaces publication-title: TWMS Journal of Pure and Applied Mathematics – ident: 4 doi: 10.1016/0165-0114(94)90331-x – ident: 15 doi: 10.1155/2019/6903809 – ident: 19 doi: 10.1007/s00500-017-2824-z – start-page: 394 volume-title: Fuzzy Sets, in Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A Zadeh year: 1996 ident: 1 doi: 10.1142/9789814261302_0021 – ident: 6 doi: 10.1016/s0898-1221(99)00056-5 – volume: 4 start-page: 399 year: 2013 ident: 29 article-title: On soft preopen sets in soft topological spaces publication-title: International Journal of Mathematics Research – ident: 3 doi: 10.1109/21.229476 – ident: 10 doi: 10.22436/jmcs.017.04.02 – ident: 22 doi: 10.1016/j.camwa.2011.09.051 – volume-title: Treatise on Set Topology year: 1947 ident: 28 – volume: 6 start-page: 1 issue: 1 year: 2013 ident: 16 article-title: Neighbourhood properties of soft topological spaces publication-title: Annals of Fuzzy Mathematics and Informatics – ident: 5 doi: 10.1145/219717.219791 – ident: 14 doi: 10.3390/math9172153 – ident: 7 doi: 10.1016/j.camwa.2011.02.006 – ident: 2 doi: 10.1016/s0165-0114(86)80034-3 – ident: 25 doi: 10.2298/fil1813755e – ident: 17 doi: 10.30948/afmi.2017.13.1.145 – volume: 3 start-page: 171 issue: 2 year: 2012 ident: 24 article-title: Remarks on soft topological spaces publication-title: Annals of fuzzy Mathematics and Informatics – volume: 7 start-page: 277 issue: 2 year: 2013 ident: 9 article-title: Soft connected spaces and soft paracompact spaces publication-title: International Journal of Mathematics and Computer Science – ident: 27 doi: 10.1007/s13370-020-00783-4 |
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| Title | Maximal Soft Compact and Maximal Soft Connected Topologies |
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