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

Abstract

In a generalized topological space $\mathfrak{T}_{\mathfrak{g}} = \left(\Omega,\mathcal{T}_{\mathfrak{g}}\right)$ ($\mathcal{T}_{\mathfrak{g}}$-space), $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-interior and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-closure operators) and $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-coderived operators) are pairs of generalized topological operators which may be employed to topologize the underlying set $\Omega$ or to give characterizations of generalized operations in the generalized sense. Generalized exterior and generalized frontier operators $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-exterior and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-frontier operators), respectively, are other generalized topological operators by means of which characterizations of generalized operations under $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ can be given without even realizing generalized interior and generalized closure operations first in order to topologize $\Omega$ in the generalized sense. In two papers, we introduced the definitions and studied the essential properties and commutativity of $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-interior} and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-closure operators) in $\mathfrak{T}_{\mathfrak{g}}$. In another two papers, we introduced the definitions and studied the essential properties of $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-coderived operators) in $\mathfrak{T}_{\mathfrak{g}}$. Moreover, we also defined by transfinite recursion on the class of successor ordinals the $\delta^{\operatorname{th}}$-iterates $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}^{\left(\delta\right)}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-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 $\mathfrak{T}_{\mathfrak{g}}$. In this paper, we present novel definitions of generalized exterior and generalized frontier operators $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$, respectively, study their essential properties, and establish further characterizations of generalized operations under $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ in $\mathfrak{T}_{\mathfrak{g}}$.

Keywords

• Generalized topological space ($\mathcal{T}_{\mathfrak{g}}$-space)
• generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets)
• generalized exterior operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-exterior operator)
• generalized frontier operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-frontier operator)

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

Abstract

In a generalized topological space $\mathfrak{T}_{\mathfrak{g}} = \left(\Omega,\mathcal{T}_{\mathfrak{g}}\right)$ ($\mathcal{T}_{\mathfrak{g}}$-space), $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-interior and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-closure operators) and $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-coderived operators) are pairs of generalized topological operators which may be employed to topologize the underlying set $\Omega$ or to give characterizations of generalized operations in the generalized sense. Generalized exterior and generalized frontier operators $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-exterior and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-frontier operators), respectively, are other generalized topological operators by means of which characterizations of generalized operations under $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ can be given without even realizing generalized interior and generalized closure operations first in order to topologize $\Omega$ in the generalized sense. In two papers, we introduced the definitions and studied the essential properties and commutativity of $\operatorname{\mathfrak{g}-Int}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cl}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-interior} and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-closure operators) in $\mathfrak{T}_{\mathfrak{g}}$. In another two papers, we introduced the definitions and studied the essential properties of $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}}_{\mathfrak{g}}$-coderived operators) in $\mathfrak{T}_{\mathfrak{g}}$. Moreover, we also defined by transfinite recursion on the class of successor ordinals the $\delta^{\operatorname{th}}$-iterates $\operatorname{\mathfrak{g}-Der}_{\mathfrak{g}}^{\left(\delta\right)}$, $\operatorname{\mathfrak{g}-Cod}_{\mathfrak{g}}^{\left(\delta\right)}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-derived and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}^{\left(\delta\right)}}$-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 $\mathfrak{T}_{\mathfrak{g}}$. In this paper, we present novel definitions of generalized exterior and generalized frontier operators $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$, respectively, study their essential properties, and establish further characterizations of generalized operations under $\operatorname{\mathfrak{g}-Ext}_{\mathfrak{g}}$, $\operatorname{\mathfrak{g}-Fr}_{\mathfrak{g}}: \mathcal{P}\left(\Omega\right) \longrightarrow \mathcal{P}\left(\Omega\right)$ in $\mathfrak{T}_{\mathfrak{g}}$.

Keywords

• Generalized topological space ($\mathcal{T}_{\mathfrak{g}}$-space)
• generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets)
• generalized exterior operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-exterior operator)
• generalized frontier operator ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-frontier operator)

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