THEORY OF GENERALIZED MAPS IN GENERALIZED TOPOLOGICAL SPACES

Source 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.