1. Nov 29, 2011

AbelAkil

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?

2. Nov 29, 2011

micromass

Staff Emeritus
3. Nov 29, 2011

AbelAkil

How to modify it? Rational numbers are dense in R...

4. Nov 29, 2011

micromass

Staff Emeritus
Throw out a rational number at each step of the construction.

5. Nov 29, 2011

AbelAkil

yes, you are right...sorry, I made a mistake in my proof... but now I can understand it...thank U very much...

6. Nov 29, 2011

AbelAkil

We should choose the end points of each segments carefully to make sure that the end points are all irrational numbers and the set is still closed!