Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology
It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-o...
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MDPI AG
2023-11-01
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| Series: | Mathematics |
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| author | Salem Saleh Tareq M. Al-shami A. A. Azzam M. Hosny |
| author_facet | Salem Saleh Tareq M. Al-shami A. A. Azzam M. Hosny |
| author_sort | Salem Saleh |
| collection | DOAJ |
| description | It is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-open sets, can contribute to the growth of fuzzy topology. In the present work, we create some classifications of fuzzy topologies which enable us to obtain several desirable features and relationships. At first, we introduce and analyze stronger forms of fuzzy pre-separation and regularity properties in fuzzy topology called fuzzy pre-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo></mrow></semantics></math></inline-formula> fuzzy pre-symmetric, and fuzzy pre-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> by utilizing the concepts of fuzzy pre-open sets and quasi-coincident relation. We investigate more novel properties of these classes and uncover their unique characteristics. By presenting a wide array of related theorems and interconnections, we structure a comprehensive framework for understanding these classes and interrelationships with other separation axioms in this setting. Moreover, the relations between these classes and those in some induced topological structures are examined. Additionally, we explore the hereditary and harmonic properties of these classes. |
| first_indexed | 2024-03-09T01:46:35Z |
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| id | doaj.art-0968daf05a5b49a3b62f73284002a7eb |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T01:46:35Z |
| publishDate | 2023-11-01 |
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| spelling | doaj.art-0968daf05a5b49a3b62f73284002a7eb2023-12-08T15:21:48ZengMDPI AGMathematics2227-73902023-11-011123480110.3390/math11234801Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy TopologySalem Saleh0Tareq M. Al-shami1A. A. Azzam2M. Hosny3Department of Computer Science, Cihan University-Erbil, Erbil P.O. Box 44001, IraqDepartment of Mathematics, Sana’a University, Sana’a P.O. Box 1247, YemenDepartment of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi ArabiaDepartment of Mathematics, College of Science, King Khalid University, Abha 61413, Saudi ArabiaIt is common knowledge that fuzzy topology contributes to developing techniques to address real-life applications in various areas like information systems and optimal choices. The building blocks of fuzzy topology are fuzzy open sets, but other extended families of fuzzy open sets, like fuzzy pre-open sets, can contribute to the growth of fuzzy topology. In the present work, we create some classifications of fuzzy topologies which enable us to obtain several desirable features and relationships. At first, we introduce and analyze stronger forms of fuzzy pre-separation and regularity properties in fuzzy topology called fuzzy pre-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo></mrow></semantics></math></inline-formula> fuzzy pre-symmetric, and fuzzy pre-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></semantics></math></inline-formula> by utilizing the concepts of fuzzy pre-open sets and quasi-coincident relation. We investigate more novel properties of these classes and uncover their unique characteristics. By presenting a wide array of related theorems and interconnections, we structure a comprehensive framework for understanding these classes and interrelationships with other separation axioms in this setting. Moreover, the relations between these classes and those in some induced topological structures are examined. Additionally, we explore the hereditary and harmonic properties of these classes.https://www.mdpi.com/2227-7390/11/23/4801fuzzy setsfuzzy pre-open setfuzzy quasi-coincidentfuzzy topologyfuzzy pre-<i>T<sub>i</sub></i> spacesfuzzy pre-symmetric |
| spellingShingle | Salem Saleh Tareq M. Al-shami A. A. Azzam M. Hosny Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology Mathematics fuzzy sets fuzzy pre-open set fuzzy quasi-coincident fuzzy topology fuzzy pre-<i>T<sub>i</sub></i> spaces fuzzy pre-symmetric |
| title | Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology |
| title_full | Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology |
| title_fullStr | Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology |
| title_full_unstemmed | Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology |
| title_short | Stronger Forms of Fuzzy Pre-Separation and Regularity Axioms via Fuzzy Topology |
| title_sort | stronger forms of fuzzy pre separation and regularity axioms via fuzzy topology |
| topic | fuzzy sets fuzzy pre-open set fuzzy quasi-coincident fuzzy topology fuzzy pre-<i>T<sub>i</sub></i> spaces fuzzy pre-symmetric |
| url | https://www.mdpi.com/2227-7390/11/23/4801 |
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