Araştırma Makalesi
BibTex RIS Kaynak Göster

GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS

Yıl 2024, , 128 - 165, 31.07.2024
https://doi.org/10.33773/jum.1393185

Öz

In a recent paper (\textsc{Cf.} \cite{KHODABOCUS_2023_4}), we have introduced the definitions and studied the essential properties of the generalized topological operators $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ (\textit{$\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-derived} and \textit{$\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-coderived operators}) in a generalized topological space $\mathfrak{T}_{\mathfrak{g}} = \left(\Omega,\mathcal{T}_{\mathfrak{g}}\right)$ (\textit{$\mathcal{T}_{\mathfrak{g}}$-space}). Mainly, we have shown that $\left(\operatorname{\mathfrak{g}-Der_{\mathfrak{g}}},\operatorname{\mathfrak{g}-Cod_{\mathfrak{g}}}\right): \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right)$ is a pair of both \textit{dual and monotone $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-operators} that is \textit{$\left(\emptyset,\Omega\right)$, $\left(\cup,\cap\right)$-preserving}, and \textit{$\left(\subseteq,\supseteq\right)$-preserving} relative to $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-(open, closed) sets. We have also shown that $\left(\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}},\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}\right): \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right)$ is a pair of \textit{weaker} and \textit{stronger $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-operators}. In this paper, we define by transfinite recursion on the class of successor ordinals the $\delta^{\operatorname{th}}$-iterates $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}^{\left(\delta\right)}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ (\textit{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-derived} and \textit{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-coderived operators}) of $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$, respectively, and study their basic properties in a $\mathcal{T}_{\mathfrak{g}}$-space. Moreover, we establish the necessary and sufficient conditions for $\bigl(\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}^{\left(\delta\right)},\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}^{\left(\delta\right)}\bigr): \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)\times\mathcal{P}\left(\Omega\right)$ to be a pair of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-coderived operators in $\mathfrak{T}_{\mathfrak{g}}$. Finally, we diagram various relationships amongst $\operatorname{der}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{cod}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}^{\left(\delta\right)}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ and present a nice application to support the overall study.

Kaynakça

  • M. I. Khodabocus, N. -U. -H. Sookia, R. D. Somanah Generalized Topological Operator (g-Tg Operator) Theory in Generalized Topological Spaces (Tg-spaces): Part III. Generalized Derived (g-Tg-Derived) and Generalized Coderived (g-Tg-Coderived) Operators, Journal of Universal Mathematics, vol. 6, N. 2, pp. 183-220 (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part II. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, vol. x, N. y, pp. z-zz (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, vol. x, N. y, pp. z-zz (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Connectedness (g-Tg-Connectedness) in Generalized Topological Spaces (Tg-Spaces), Journal of Universal Mathematics,vol. 6, N. 1, pp. 1 -38 (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 2, pp. 98-118 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 1, pp. 26-45 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics, vol. 5, N. 1, pp. 1-23 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory, vol. 36, pp. 18-38 (2021).
  • M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R´eduit, Mauritius (2020).
  • Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science, vol. 345, pp. 63-76 (2019).
  • S. Ahmad, Absolutely Independent Axioms for the Derived Set Operator, The American Mathematical Monthly, vol. 73, N. 4, pp. 390-392 (1966).
  • A. Baltag and N. Bezhanishvili and A. ¨Ozg¨un and S. Smets, A Topological Apprach to Full Belief, Journal of Philosophical Logic, vol. 48 N. 2, pp. 205-244 (2019).
  • D. Cenzer and D. Mauldin, On the Borel Class of the Derived Set Operator, Bull. Soc. Math. France, vol. 110, pp. 357–380 (1982).
  • F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly, vol. 70, N. 10, pp. 1085-1086 (1963).
  • E. R. Hedrick, On Properties of a Domain for Which Any Derived Set is Closed, Transactions of the American Mathematical Society, vol. 12, N. 3, pp. 285-294 (1911).
  • R. M. Latif, Characterizations and Applications of γ-Open Sets, Soochow Journal of Mathematics, vol. 32, N. 3, pp. 1-10 (2006).
  • S. Modak, Some Points on Generalized Open Sets, Caspian Journal of Mathematical Sciences (CJMS), vol. 6, N. 2, pp. 99-106 (2017).
  • G. H. Moore, The Emergence of Open Sets, Closed Sets, and Limit Points in Analysis and Topology, Historia Mathematica, vol. 35, N. 3, pp. 220-241 (2008).
  • A. Al-Omari and M. S. M. Noorani, On b-Closed Sets, Bull. Malays. Sci. Soc., vol. 32, N. 1., pp. 19-30 (2009).
  • R. Rajendiran and M. Thamilselvan, Properties of g∗s∗-Closure, g∗s∗-Interior and g∗s∗-Derived Sets in Topological Spaces, Applied Mathematical Sciences, vol. 8, N. 140, pp. 6969-6978 (2014).
  • J. A. R. Rodgio and J. Theodore and H. Jansi, Notions via β∗-Open Sets in Topological Spaces, IOSR Journal of Mathematics (IOSR-JM), vol. 6, N. 3, pp. 25-29 (2013).
  • R. Spira, Derived-Set Axioms for Topological Spaces, Portugaliar Mathematica, vol. 26, pp. 165-167 (1967).
  • G. Cantor, ¨Uber die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen, Math. Ann., vol. 5, pp. 123-132 (1872).
  • G. Cantor, ¨Uber Unendliche, Lineaere Punktmannigfaltigkeiten, Ibid., vol. 20, N. III, pp.113-121 (1882).
  • N. E. Rutt, On Derived Sets, National Mathematics Magazine, vol. 18, N. 2, pp. 53–63 (1943).
  • J. Tucker, Concerning Consecutive Derived Sets, The American Mathematical Monthly, vol.74, N. 5, pp. 555-556 (1967).
  • D. Higgs, Iterating the Derived Set Function, The American Mathematical Monthly, vol. 90, N. 10, pp. 693-697 (1983).
Yıl 2024, , 128 - 165, 31.07.2024
https://doi.org/10.33773/jum.1393185

Öz

Kaynakça

  • M. I. Khodabocus, N. -U. -H. Sookia, R. D. Somanah Generalized Topological Operator (g-Tg Operator) Theory in Generalized Topological Spaces (Tg-spaces): Part III. Generalized Derived (g-Tg-Derived) and Generalized Coderived (g-Tg-Coderived) Operators, Journal of Universal Mathematics, vol. 6, N. 2, pp. 183-220 (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part II. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, vol. x, N. y, pp. z-zz (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Generalized Topological Operator Theory in Generalized Topological Spaces: Part I. Generalized Interior and Generalized Closure, Proceedings of International Mathematical Sciences, vol. x, N. y, pp. z-zz (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Connectedness (g-Tg-Connectedness) in Generalized Topological Spaces (Tg-Spaces), Journal of Universal Mathematics,vol. 6, N. 1, pp. 1 -38 (2023).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part II. Countable, Sequential and Local Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 2, pp. 98-118 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Compactness in Generalized Topological Spaces: Part I. Basic Properties, Fundamentals of Contemporary Mathematical Sciences, vol. 3, N. 1, pp. 26-45 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Separation Axioms in Generalized Topological Spaces, Journal of Universal Mathematics, vol. 5, N. 1, pp. 1-23 (2022).
  • M. I. Khodabocus, N. -U. -H. Sookia, Theory of Generalized Sets in Generalized Topological Spaces, Journal of New Theory, vol. 36, pp. 18-38 (2021).
  • M. I. Khodabocus, A Generalized Topological Space endowed with Generalized Topologies, PhD Dissertation, University of Mauritius, R´eduit, Mauritius (2020).
  • Y. Lei and J. Zhang, Generalizing Topological Set Operators, Electronic Notes in Theoretical Science, vol. 345, pp. 63-76 (2019).
  • S. Ahmad, Absolutely Independent Axioms for the Derived Set Operator, The American Mathematical Monthly, vol. 73, N. 4, pp. 390-392 (1966).
  • A. Baltag and N. Bezhanishvili and A. ¨Ozg¨un and S. Smets, A Topological Apprach to Full Belief, Journal of Philosophical Logic, vol. 48 N. 2, pp. 205-244 (2019).
  • D. Cenzer and D. Mauldin, On the Borel Class of the Derived Set Operator, Bull. Soc. Math. France, vol. 110, pp. 357–380 (1982).
  • F. R. Harvey, The Derived Set Operator, The American Mathematical Monthly, vol. 70, N. 10, pp. 1085-1086 (1963).
  • E. R. Hedrick, On Properties of a Domain for Which Any Derived Set is Closed, Transactions of the American Mathematical Society, vol. 12, N. 3, pp. 285-294 (1911).
  • R. M. Latif, Characterizations and Applications of γ-Open Sets, Soochow Journal of Mathematics, vol. 32, N. 3, pp. 1-10 (2006).
  • S. Modak, Some Points on Generalized Open Sets, Caspian Journal of Mathematical Sciences (CJMS), vol. 6, N. 2, pp. 99-106 (2017).
  • G. H. Moore, The Emergence of Open Sets, Closed Sets, and Limit Points in Analysis and Topology, Historia Mathematica, vol. 35, N. 3, pp. 220-241 (2008).
  • A. Al-Omari and M. S. M. Noorani, On b-Closed Sets, Bull. Malays. Sci. Soc., vol. 32, N. 1., pp. 19-30 (2009).
  • R. Rajendiran and M. Thamilselvan, Properties of g∗s∗-Closure, g∗s∗-Interior and g∗s∗-Derived Sets in Topological Spaces, Applied Mathematical Sciences, vol. 8, N. 140, pp. 6969-6978 (2014).
  • J. A. R. Rodgio and J. Theodore and H. Jansi, Notions via β∗-Open Sets in Topological Spaces, IOSR Journal of Mathematics (IOSR-JM), vol. 6, N. 3, pp. 25-29 (2013).
  • R. Spira, Derived-Set Axioms for Topological Spaces, Portugaliar Mathematica, vol. 26, pp. 165-167 (1967).
  • G. Cantor, ¨Uber die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen, Math. Ann., vol. 5, pp. 123-132 (1872).
  • G. Cantor, ¨Uber Unendliche, Lineaere Punktmannigfaltigkeiten, Ibid., vol. 20, N. III, pp.113-121 (1882).
  • N. E. Rutt, On Derived Sets, National Mathematics Magazine, vol. 18, N. 2, pp. 53–63 (1943).
  • J. Tucker, Concerning Consecutive Derived Sets, The American Mathematical Monthly, vol.74, N. 5, pp. 555-556 (1967).
  • D. Higgs, Iterating the Derived Set Function, The American Mathematical Monthly, vol. 90, N. 10, pp. 693-697 (1983).
Toplam 27 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Topoloji
Bölüm Araştırma Makalesi
Yazarlar

Mohammad Irshad Khodabocus 0000-0003-2252-4342

Noor-ul-hacq Sookıa 0000-0002-3155-0473

Radhakhrishna Dinesh Somanah 0000-0001-6202-7610

Yayımlanma Tarihi 31 Temmuz 2024
Gönderilme Tarihi 20 Kasım 2023
Kabul Tarihi 31 Temmuz 2024
Yayımlandığı Sayı Yıl 2024

Kaynak Göster

APA Khodabocus, M. I., Sookıa, N.-u.-h., & Somanah, R. D. (2024). GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS. Journal of Universal Mathematics, 7(2), 128-165. https://doi.org/10.33773/jum.1393185
AMA Khodabocus MI, Sookıa Nuh, Somanah RD. GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS. JUM. Temmuz 2024;7(2):128-165. doi:10.33773/jum.1393185
Chicago Khodabocus, Mohammad Irshad, Noor-ul-hacq Sookıa, ve Radhakhrishna Dinesh Somanah. “GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS”. Journal of Universal Mathematics 7, sy. 2 (Temmuz 2024): 128-65. https://doi.org/10.33773/jum.1393185.
EndNote Khodabocus MI, Sookıa N-u-h, Somanah RD (01 Temmuz 2024) GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS. Journal of Universal Mathematics 7 2 128–165.
IEEE M. I. Khodabocus, N.-u.-h. Sookıa, ve R. D. Somanah, “GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS”, JUM, c. 7, sy. 2, ss. 128–165, 2024, doi: 10.33773/jum.1393185.
ISNAD Khodabocus, Mohammad Irshad vd. “GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS”. Journal of Universal Mathematics 7/2 (Temmuz 2024), 128-165. https://doi.org/10.33773/jum.1393185.
JAMA Khodabocus MI, Sookıa N-u-h, Somanah RD. GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS. JUM. 2024;7:128–165.
MLA Khodabocus, Mohammad Irshad vd. “GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS”. Journal of Universal Mathematics, c. 7, sy. 2, 2024, ss. 128-65, doi:10.33773/jum.1393185.
Vancouver Khodabocus MI, Sookıa N-u-h, Somanah RD. GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS. JUM. 2024;7(2):128-65.