Let A be a subspace of X and D be a subset of A. By Int_A(D) and Int_X(D) we mean, respecibely, the interior of D in the subspace topology on A and the interior of D in the topology on X. Similarly, we define CL_A(D) and CL_X(D) for the closure, and d_A(D) and d_x(D) for the boundary.
a) Explore the relationship between Cl_A(D) and A∩CL_X(D). For each containment, either prove that it holds or find a counterexample.
b) Do the same thing for d_A(D) and A∩d_x(D).