Some New Notions of Continuity in Generalized Primal Topological Space
This study analyzes the characteristics and functioning of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantic...
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2024-12-01
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| author | Muhammad Shahbaz Tayyab Kamran Umar Ishtiaq Mariam Imtiaz Ioan-Lucian Popa Fethi Mohamed Maiz |
| author_facet | Muhammad Shahbaz Tayyab Kamran Umar Ishtiaq Mariam Imtiaz Ioan-Lucian Popa Fethi Mohamed Maiz |
| author_sort | Muhammad Shahbaz |
| collection | DOAJ |
| description | This study analyzes the characteristics and functioning of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-homeomorphisms, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mrow><mo>∗</mo><mo>#</mo></mrow></msubsup></semantics></math></inline-formula>-homeomorphisms in generalized topological spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">GTS</mi><mo>)</mo></mrow></semantics></math></inline-formula>. A few important points to emphasize are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-irresolute functions, perfectly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous, and strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous functions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">GTS</mi></mrow></semantics></math></inline-formula> and generalized primal topological spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">GPTS</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Some specific kinds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-open mappings and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-closed mappings, are discussed. We also analyze the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">GPTS</mi></mrow></semantics></math></inline-formula>, providing a thorough look at the way these functions work in this specific context. The goal here is to emphasize the concrete implications of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions and to further the theoretical understanding of them by merging different viewpoints. This work advances the area of topological research by providing new perspectives on the behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions and their applicability in various topological settings. The outcomes reported here contribute to our theoretical understanding and establish a foundation for further research. |
| format | Article |
| id | doaj-art-83ed8a1137ba4d94af093315629f6557 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-83ed8a1137ba4d94af093315629f65572025-08-20T02:43:20ZengMDPI AGMathematics2227-73902024-12-011224399510.3390/math12243995Some New Notions of Continuity in Generalized Primal Topological SpaceMuhammad Shahbaz0Tayyab Kamran1Umar Ishtiaq2Mariam Imtiaz3Ioan-Lucian Popa4Fethi Mohamed Maiz5Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, PakistanDepartment of Mathematics, Quaid-I-Azam University, Islamabad 45320, PakistanOffice of Research, Innovation and Commercialization, University of Management and Technology, Lahore 54770, PakistanDepartment of Mathematics, The Islamia University of Bahawalpur, Bahawalnagar Campus, Bahawalpur 06314, PakistanDepartment of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, RomaniaPhysics Department, Faculty of Science, King Khalid University, Abha P.O. Box 9004, Saudi ArabiaThis study analyzes the characteristics and functioning of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-homeomorphisms, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mrow><mo>∗</mo><mo>#</mo></mrow></msubsup></semantics></math></inline-formula>-homeomorphisms in generalized topological spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">GTS</mi><mo>)</mo></mrow></semantics></math></inline-formula>. A few important points to emphasize are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-irresolute functions, perfectly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous, and strongly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-continuous functions in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">GTS</mi></mrow></semantics></math></inline-formula> and generalized primal topological spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">GPTS</mi><mo>)</mo></mrow></semantics></math></inline-formula>. Some specific kinds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-open mappings and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula>-closed mappings, are discussed. We also analyze the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">GPTS</mi></mrow></semantics></math></inline-formula>, providing a thorough look at the way these functions work in this specific context. The goal here is to emphasize the concrete implications of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions and to further the theoretical understanding of them by merging different viewpoints. This work advances the area of topological research by providing new perspectives on the behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>S</mi><mi>g</mi><mo>∗</mo></msubsup></semantics></math></inline-formula> functions and their applicability in various topological settings. The outcomes reported here contribute to our theoretical understanding and establish a foundation for further research.https://www.mdpi.com/2227-7390/12/24/3995generalized primal topological space<i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8600"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function<i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8601"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism<i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8602"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>#</mml:mo> </mml:mrow> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism<named-content content-type="equation"><inline-formula> <mml:math id="mm8403"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8603"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function<named-content content-type="equation"><inline-formula> <mml:math id="mm8401"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8604"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism |
| spellingShingle | Muhammad Shahbaz Tayyab Kamran Umar Ishtiaq Mariam Imtiaz Ioan-Lucian Popa Fethi Mohamed Maiz Some New Notions of Continuity in Generalized Primal Topological Space Mathematics generalized primal topological space <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8600"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8601"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8602"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>#</mml:mo> </mml:mrow> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism <named-content content-type="equation"><inline-formula> <mml:math id="mm8403"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8603"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function <named-content content-type="equation"><inline-formula> <mml:math id="mm8401"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8604"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism |
| title | Some New Notions of Continuity in Generalized Primal Topological Space |
| title_full | Some New Notions of Continuity in Generalized Primal Topological Space |
| title_fullStr | Some New Notions of Continuity in Generalized Primal Topological Space |
| title_full_unstemmed | Some New Notions of Continuity in Generalized Primal Topological Space |
| title_short | Some New Notions of Continuity in Generalized Primal Topological Space |
| title_sort | some new notions of continuity in generalized primal topological space |
| topic | generalized primal topological space <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8600"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8601"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism <i>τ<sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8602"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>∗</mml:mo> <mml:mo>#</mml:mo> </mml:mrow> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism <named-content content-type="equation"><inline-formula> <mml:math id="mm8403"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8603"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-continuous function <named-content content-type="equation"><inline-formula> <mml:math id="mm8401"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:semantics> </mml:math> </inline-formula></named-content><i><sub>g</sub></i>–<named-content content-type="inline-formula"><inline-formula> <mml:math id="mm8604"> <mml:semantics> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>g</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-homeomorphism |
| url | https://www.mdpi.com/2227-7390/12/24/3995 |
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