# Homework Help: A Problem about perfect set

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

3. Nov 29, 2011

### AbelAkil

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

4. Nov 29, 2011

### micromass

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!