On the topology via kernel sets and primal spaces

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 framework, 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 contribute to the refinement of topological concepts but also highlight the structural richness and potential applications of primal-based topologies.