I was reading a website that said the boundary of a set's boundary is equal to the first boundary. Visually, this makes sense for subsets of R1 and R2 because the first boundary will not have an interior (no ball about the points will fall into the boundary). However, the reading went on to say that the boundary of the rationals Q is R. This seems wrong to me so I am questioning the entire site. Wouldn't the boundary of Q be Q? A ball of positive radius about any point in Q would contain both points from Q as well as irrationals from P. ddQ = dQ = Q (where d is boundary operator). The theorem at top does appear to hold but the example is messed up....no?