1. The problem statement, all variables and given/known data Is there a perfect set that contains no rationals? 2. Relevant equations A set is perfect if it is closed and contains no isolated points. 3. The attempt at a solution Why not just take the set of irrational numbers as your whole space? This set is certainly closed as it's the whole metric space, and it contains no isolated points, as the irrationals are dense in themselves. What's wrong with this? I'm not confident it's correct since this was given as a bonus problem on an assignment, but I can't see what's wrong with my example. Edit: the prof probably meant for us to find a perfect subset of R with no rationals, 'cause that seems a lot harder and I don't even know where to start. so how might I go about doing that, then?