Several specific types of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) have been defined and investigated for various purposes from time to time in the literature of Tg-spaces. Our recent research in the field of a new class of g-Tg-sets in Tg-spaces is reported herein as a starting point for more generalized classes. It is shown that the class of g-Tg-sets is a superclass of those whose elements are called open, closed, semi-open, semi-closed, pre-open, pre-closed, semi-pre-open, and semi-pre-closed sets in a Tg-space. A subclass of the Tg-subspace corresponds to the class of g-Tg-sets of a Tg-space. A class of g-Tg-sets of the Cartesian product of these Tg-spaces corresponds to the Cartesian product of a finite number of classes of g-Tg-sets, each of which belongs to a Tg-space. Diagrams establish the various relationships amongst the classes presented here and in the literature, and an ad hoc application supports the overall theory.

In this paper, a novel class of generalized compact sets (briefly, g-Tg-compact sets ) in generalized topological spaces (briefly, Tg-spaces ) is studied. The study reveals that g-Tg -compactness implies ordinary compactness (briefly, Tg -compactness ) in Tg-spaces, and such statement implies its analogue in ordinary topological spaces (briefly, T -spaces ). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in relation to other types of g-T -compactness in T-spaces presented in the literature of Tg -spaces, and a nice application supports the overall theory.

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In this paper, a new class of generalized separation axioms (briefly, g-Tg-separation axioms) whose elements are called g-T g,K , g-T g,F , g-T g,H , g-T g,R , g-T g,N -axioms is defined in terms of generalized sets (briefly, g-Tg-sets) in generalized topological spaces (briefly, Tg-spaces) and the properties and characterizations of a Tg-space endowed with each such g-T g,K , g-T g,F , g-T g,H , g-T g,R , g-T g,N -axioms are discussed. The study shows that g-T g,F -axiom implies g-T g,K -axiom, g-T g,H -axiom implies g-T g,F -axiom, g-T g,R -axiom implies g-T g,H -axiom, and g-T g,N -axiom implies g-T g,R -axiom. Considering the T g,K , T g,F , T g,H , T g,R , T g,N -axioms as their analogues but defined in terms of corresponding elements belonging to the class of open, closed, semi-open, semiclosed, preopen, preclosed, semi-preopen, and semi-preclosed sets, the study also shows that the statement Tg,α-axiom implies g-T g,α -axiom holds for each α ∈ {K, F, H, R, N}. Diagrams expose the various implications amongst the classes presented here and in the literature, and a nice application supports the overall theory.

In a recent paper, a novel class of generalized compact sets (briefly, g-Tg -compact sets) in generalized topological spaces (briefly, Tg -spaces) has been studied. In this paper, the concept is further studied and, other derived concepts called countable, sequential, and local generalized compactness (countable, sequential, local g-Tg -compactness) in Tg -spaces are also studied relatively. The study reveals that g-Tg -compactness implies local g-Tg -compactness and countable g-Tg-compactness, sequential g-Tg -compactness implies countable g-Tg -compactness and, g-Tg -compactness is a generalized topological property (briefly, Tg -property). Diagrams establish the various relationships amongst these types of g-Tg -compactness presented here and in the literature, and a nice application supports the overall theory.