In this note we prove some results on the intersection properties of maximal subfields of division algebras which are finite dimensional over their centers. These results indicate that we can get very small intersections with any subalgebra if we use the appropriate maximal subfields. As a consequence of our first theorem, we obtain some theorems which are known and some which can be obtained from these known theorems (see, for instance, Theorem 3, Chapter VII in [3]). The proofs of these known results given here are very elementary and are quite different from the ones in the literature.