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

Mohammad Irshad Khodabocus; Noor-ul-hacq Sookıa; Radhakhrishna Dinesh Somanah

doi:10.33773/jum.1393185

EN

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

Abstract

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.

Keywords

References

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Details

Primary Language

English

Subjects

Topology

Journal Section

Research Article

Authors

Mohammad Irshad Khodabocus ^*
0000-0003-2252-4342
Mauritius

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

Radhakhrishna Dinesh Somanah
0000-0001-6202-7610
Mauritius

Publication Date

July 31, 2024

Submission Date

November 20, 2023

Acceptance Date

July 31, 2024

Published in Issue

Year 2024 Volume: 7 Number: 2

DOI

https://doi.org/10.33773/jum.1393185

IZ

https://izlik.org/JA59AA57MN

Cite

RIS / Bibtex

APA

Khodabocus, M. I., Sookıa, N.- ul- hacq, & 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

1.Khodabocus MI, Sookıa N ul hacq, 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):128-165. doi:10.33773/jum.1393185

Chicago

Khodabocus, Mohammad Irshad, Noor-ul-hacq Sookıa, and Radhakhrishna Dinesh Somanah. 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): 128-65. https://doi.org/10.33773/jum.1393185.

EndNote

Khodabocus MI, Sookıa N- ul- hacq, Somanah RD (July 1, 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

[1]M. I. Khodabocus, N.- ul- hacq Sookıa, and 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, vol. 7, no. 2, pp. 128–165, July 2024, doi: 10.33773/jum.1393185.

ISNAD

Khodabocus, Mohammad Irshad - Sookıa, Noor-ul-hacq - Somanah, Radhakhrishna Dinesh. “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 (July 1, 2024): 128-165. https://doi.org/10.33773/jum.1393185.

JAMA

1.Khodabocus MI, Sookıa N- ul- hacq, 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, et al. “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, vol. 7, no. 2, July 2024, pp. 128-65, doi:10.33773/jum.1393185.

Vancouver

1.Mohammad Irshad Khodabocus, Noor-ul-hacq Sookıa, 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. JUM. 2024 Jul. 1;7(2):128-65. doi:10.33773/jum.1393185

Cited By

GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-Spaces)\\ Part V. GENERALIZED EXTERIOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-EXTERIOR) AND GENERALIZED FRONTIER ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-FRONTIER) OPERATORS

Journal of Universal Mathematics

https://doi.org/10.33773/jum.1913242