AI Summary & Key Takeaways
AI InsightTo render this work well-founded, diagrams are inserted to exhibit several relationships amongst the $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.
📌 3 Key Takeaways
Abstract
It is the intent of this paper to present the definitions of new families of generalized maps ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps) and generalized continuous ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous), generalized irresolute (briefly, $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute) and generalized homeomorphism ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism) maps between generalized topological spaces ($\mathcal{T}_{\mathfrak{g}}$-spaces) in terms of generalized sets ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets) and study their fundamental properties. For each type of such $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps, necessary and sufficient conditions are determined. These classes are found to be superclasses of those whose elements are defined in terms of the classes of open, closed sets, semi-open, semi-closed sets, preopen, preclosed sets, and semi-preopen, semi-preclosed sets. The investigation reveals that the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps is a superclass of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps and both classes of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps are subclasses of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps. To render this work well-founded, diagrams are inserted to exhibit several relationships amongst the $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.
Özet
It is the intent of this paper to present the definitions of new families of generalized maps (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps}) and generalized continuous (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous}), generalized irresolute (briefly, \emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute}) and generalized homeomorphism (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-homeomorphism}) maps between generalized topological spaces (\emph{$\mathcal{T}_{\mathfrak{g}}$-spaces}) in terms of generalized sets (\emph{$\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-sets}) and study their fundamental properties. For each type of such $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps, necessary and sufficient conditions are determined. These classes are found to be superclasses of those whose elements are defined in terms of the classes of open, closed sets, semi-open, semi-closed sets, preopen, preclosed sets, and semi-preopen, semi-preclosed sets. The investigation reveals that the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous maps is a superclass of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps and both classes of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-continuous and $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-irresolute maps are subclasses of the class of $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps. To render this work well-founded, diagrams are inserted to exhibit several relationships amongst the $\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.