In this work, we introduce a comprehensive generalization of the kernel of a set within topological spaces equipped with a primal structure. The proposed notion of the primal kernel offers a powerful framework for redefining and analyzing generalized forms of open and closed sets. Leveraging this frame-work, we establish new, weaker separation axioms and construct a novel topology that is demonstrably incomparable with the classical topology derived from the primal structure. These results not only con-tribute to the refinement of topological concepts but also highlight the structural richness and potential applications of primal-based topologies.
Alshammari et al. (Wed,) studied this question.