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A Problem about perfect set

  1. Nov 29, 2011 #1
    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?
     
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  3. Nov 29, 2011 #2

    micromass

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  4. Nov 29, 2011 #3
    How to modify it? Rational numbers are dense in R...
     
  5. Nov 29, 2011 #4

    micromass

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    Throw out a rational number at each step of the construction.
     
  6. Nov 29, 2011 #5
    yes, you are right...sorry, I made a mistake in my proof... but now I can understand it...thank U very much...
     
  7. Nov 29, 2011 #6
    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!
     
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