1. The problem statement, all variables and given/known data Is there any nonempty perfect set in R which contains no rational number? 2. Relevant equations A set E is perfect iff E is closed and every point of E is a limit point of E 3. The attempt at a solution We should avoid rational numbers to become limit points, so we have to kick out countable segments with rational numbers...But how can I construct the set so that the remaining irrational numbers are still limit points and no isolated point exists?