Some New Notions of Continuity in Generalized Primal Topological Space

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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Bibliographic Details
Main Authors:	Muhammad Shahbaz, Tayyab Kamran, Umar Ishtiaq, Mariam Imtiaz, Ioan-Lucian Popa, Fethi Mohamed Maiz
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Mathematics
Subjects:	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
Online Access:	https://www.mdpi.com/2227-7390/12/24/3995
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Summary:	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.
ISSN:	2227-7390