Symmetric Extensions of Separation Axioms via GP-Operators and Their Applications

This study is motivated by the need to investigate the largest possible collection of classical separation axioms within a newly introduced triple structure of generalized primal topological spaces, and to understand how primal collections influence these familiar notions. The purpose of the paper is to extend several classical concepts by introducing new classes of separation axioms, including $(\mathfrak{g},\mathcal{P})$-$D_i$, $(\mathfrak{g},\mathcal{P})$-$T_i$, and $(\mathfrak{g},\mathcal{P})$-$R_i$ for $i=0,1,2$. Within the same framework, we also define $(\mathfrak{g},\mathcal{P})$-$G_{\delta }$ sets and $(\mathfrak{g},\mathcal{P})$-$F_{\sigma }$ sets, which naturally lead to new symmetric variants of separation axioms such as $(\mathfrak{g},\mathcal{P})$-$R_{\delta }$, $(\mathfrak{g},\mathcal{P})$-weakly regular, $(\mathfrak{g},\mathcal{P})$-$R_{D_{\delta }}$, and $(\mathfrak{g},\mathcal{P})$-$R_{D^{★}}$. The main contribution of this work lies in establishing the relationships among these newly introduced axioms and demonstrating how primal collections affect their behavior. Several illustrative examples based on simple graphs are provided to highlight the structure and significance of the results. Overall, the findings offer a broader perspective on separation phenomena in generalized primal settings and deepen the understanding of symmetry within these spaces.