Let K be the set defined in the interval 01 consisting of al

Let K be the set defined in the interval [0,1] consisting of all points that can be written with a decimal representation that does not have a 3 nor a 5. Then K is perfect. Is this true or false? Provide a justification.

Solution

It is false. It can not be perfect. As the set is not closed. e.g o.3 can not be a member of the set by construction. But ).3 is a limit point of the set. Take any epsilon greater than 0 we will find some finite number (say p)of 9 in the expression 0.299999999.......p times , such that the element belongs inside the epsilon nbd of 0.3. Therefore the set is not closed as it does not contain all of its limit points. Hence can not be perfect too.