Properties of Supra Equivalent Subsets and Their Applications via Supra R0 Spaces

The extension of a topology by weakening its axioms is highly significant, as it generates new frameworks that preserve certain topological properties while allowing for the modeling of several practical problems. In this paper, we introduce the concept of supra equivalent subsets in the framework of a supra topological space and look at the main properties of supra equivalent subsets. We also explore how they interact with other ideas and notions in supra topological spaces, such as supra compactness, supra connectedness, generalizations of supra open sets, supra T0, and supra T1. We determine under which conditions we obtain some results concerning supra equivalent sets and provide examples to clarify key relationships as the need arises. Then, we characterize the supra R0 axiom using the supra equivalence between singletons and their closure. Finally, we examine the properties of supra connectedness, supra regularity, supra normality, and supra paracompactness between supra subspaces of a supra R0 space that are defined on supra equivalent subsets.