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

Year 2024, Volume: 7 Issue: 2, 128 - 165, 31.07.2024

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

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

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

Generalized topological space ($\mathcal{T}_{\mathfrak{g}}$-space), generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets), $\delta^{\operatorname{th}}$-order generalized derived operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-derived operator), $\delta^{\operatorname{th}}$-order generalized coderived operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-coderived operator)

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