Various Types of Supra $epsilon$-separation Axioms and Relationships

Abstract

 This manuscript presents a new weaker version of supra septarian axioms based on supra $\epsilon$-open sets, along with its essential features, which are called supra-$\epsilon$-$T_{j}$-space, $j=0,1,2$, in the framework of supra topological spaces (or STSs). We give comprehensive explanations of each type of them, backed up by several examples and counterexamples that highlight the significance of our original approaches. We also provide a diagram that outlines these relationships. Additionally, we present the supra $\epsilon$-symmetric property and supra difference property and study their effects on these version of supra-$\epsilon$-septarian axioms. In especial, we show that the two concepts of supra-$\epsilon$-$T_{0}$-space and supra-$\epsilon$-$T_{1}$-space are the same for any STS that fulfills supra $\epsilon$-symmetric property. Finally, we study the supra topological and supra hereditary properties for each of the previously discussed approaches. In particular, we show that the property of being a supra-$\epsilon$-$T_{j}$-space, where $j=0,1,2$, is a supra-hereditary (topological) property.