Convergence frameworks for dynamic rough set approximations: Set-theoretic and metric perspectives

We investigate the convergence behavior of lower and upper approximation spaces and boundary regions in dynamic rough set theory under two frameworks. The first is a set-theoretic framework, where convergence is defined through the eventual stability of sets. In this setting, we show that the convergence of the lower and upper approximations always leads to the convergence of the boundary regions, and that the convergence of the boundary regions also ensure the convergence of the lowerupper approximation spaces. The second framework adopts a metric-based approach, where convergence is determined by a distance between sets. In general, this relationship between the lower approximations, upper approximations and the boundary regions do not hold. However, when the distance is defined by the symmetric difference between sets, the convergence relationships among the lower, upper and boundary regions are preserved. These results provide a unified understanding of convergence in evolving rough set systems.