On Proximity Spaces Constructed on Rough Sets

Based on equivalence relation R on X, equivalence class x of a point and equivalence class A of a subset represent the neighborhoods of x and A, respectively. These neighborhoods play the main role in defining separation axioms, metric spaces, proximity relations and uniformity structures on an approximation space (X,R) depending on the lower approximation and the upper approximation of rough sets. The properties and the possible implications of these definitions are studied. The generated approximation topology τR on X is equivalent to the generated topologies associated with metric d, proximity δ and uniformity U on X. Separated metric spaces, separated proximity spaces and separated uniform spaces are defined and it is proven that both are associating exactly discrete topology τR on X.