A ring version of Mazur's conjecture on topology of rational points

Barry Mazur has conjectured that the topological closure of any variety V(ℚ) in V(ℝ) possesses at most a finite number of connected components. We consider this conjecture in a ring setting. Given an integrally closed subring R of a number field, we investigate the number of connected components of the topological closure of V(R) in ℝ or ℂ.