On topological covering-based rough spaces

Rough set theory, a mathematical tool to deal with vague concepts, has originally described the indiscernibility of elements by equivalence relations. Covering-based rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as subbase, neighborhood and separation axioms have been applied to study covering-based rough sets. However, the topological space on covering-based rough sets and the corresponding topological properties on the topological covering-based rough space are not studied. This paper studies some of these problems. We defined open sets, closed sets, rough inclusion, rough equality on covering-based rough sets and some of their properties are studied. Then, we give the definition of topology on covering-based rough sets. Finally, we study the properties of rough homeomorphisms. This research not only can form the theoretical basis for further applications of topology on covering-based rough sets but also lead to the development of the rough set theory and artificial intelligence.