Theory of Generalized Maps in Generalized Topological Spaces

It is the intent of this paper to present the definitions of new families of generalized maps (g-Tg-maps) and generalized continuous (g-Tg-continuous), generalized irresolute (briefly, g-Tg-irresolute) and generalized homeomorphism (g-Tg-homeomorphism) maps between generalized topological spaces (Tg-spaces) in terms of generalized sets (g-Tg-sets) and study their fundamental properties. For each type of such g-Tg-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 g-Tg-continuous maps is a superclass of the class of g-Tg-irresolute maps and both classes of g-Tg-continuous and g-Tg-irresolute maps are subclasses of the class of g-Tg-maps. To render this work well-founded, diagrams are inserted to exhibit several relationships amongst the g-Tg-maps studied in this paper and in the literature, and an illustrative application is given to support some theoretical facts.