In this research, we introduce and analyze the concepts of \ (G\) -compactness, \ (G\) -Lindelofness, and \ (G\) -countably compactness within the framework of topological spaces, which are characterized by more rigorous conditions compared to those governing compact and Lindelof spaces. We formulate a series of theorems and present a variety of examples to clarify the relationships among \ (G\) -compactness, \ (G\) -Lindelofness, compactness, and Lindelofness. Additionally, we define the \ (G\) -separation axioms utilizing G_ sets and explore the interrelations among these concepts.
Shatnawi et al. (Fri,) studied this question.