I've read in several places that the boundary of the rational numbers is the empty set. I feel I must be misinterpreting the definition of a boundary, because this doesn't seem right to me. My understanding of the boundary of a set S is that it is the set of all elements which can be approached from both the inside and the outside. That is, the set of all r such that r is the limit of a sequence in S and also the limit of a sequence outside of S. We know of course that every real number is the limit of a sequence of rational numbers. We know also that every real number r is the limit of the constant sequence (r). So shouldn't the boundary of the rationals be the set of all irrational numbers?