Applications of δ-Open Sets via Separation Axioms, Covering Properties, and Rough Set Models

In this article, we make use of δ-open sets to establish some topological concepts related to separation axioms and covering properties and to propose novel topological rough set models. We first demonstrate that the classes of regular-open and δ-open subsets of a finite topological space are equivalent when this space has the property of ∂A∩∂B⊆∂A∪B for subsets A,B, where ∂ means the boundary topological operator. Then, we prove the equivalence between δTi and Ti in the cases of i=2,21/2 and construct an illustrative example showing that δ-topology is a proper class of an original topology in these two cases.With the help of counterexamples, we demonstrate that a normal (resp., a regular) space is a stronger condition than δ-normal (resp., δ-regular) and show that every Ti-space is δTi for each i=3,4. We also reveal the relationships between the covering properties studied herein (compactness, Lindelöf, local compactness, and local Lindelöf) and investigate some equivalences when the topological space is δT2. Furthermore, we discuss how the introduced concepts behave in the spaces of product topology, box topology, and sum topology. Finally, we set up a new framework for rough set models based on δ-open sets and examine its advantages and limitations in terms of retaining the characterizations of lower and upper approximations and increasing accuracy measures. An algorithm is designed to determine whether a subset is δ-definable or δ-rough.