Constructing a Perfect Set in R Without Rational Numbers

[AbelAkil]

Homework Statement

Is there any nonempty perfect set in R which contains no rational number?

Homework Equations

A set E is perfect iff E is closed and every point of E is a limit point of E

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?

 

[micromass]

The construction of such a set is quite tricky. You will have to modify the construction of the Cantor set in such a way that it misses the rationals. See http://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set for general information about such a set.

 

[AbelAkil]

The construction of such a set is quite tricky. You will have to modify the construction of the Cantor set in such a way that it misses the rationals. See http://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set for general information about such a set.

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

 

[micromass]

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

 

[AbelAkil]

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

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

 

[AbelAkil]

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

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!