Bo = union of subsets of B open in X. But they are only ∅. Ao= {1,3,4} since A is open.
ext(B)= union of open sets disjoint from B = {1}. ext(A)=∅.
b(B)= B∖Bo=B. Now {1}c={2,3,4}. So the latter is closed and contains B. Also B is not closed since Bc={1,4} is not open. Hence {2,3,4} is the smallest closed set containing B. Hence b(B)=B={2,3,4}. A=X since only other option is A itself and Ac={2} not open, hence A not closed. Hence b(A)=X∖Ao=X∖A={2}.
Comments