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Determine if this set is compact or not

  1. Jul 24, 2009 #1
    1. The problem statement, all variables and given/known data
    Let [tex]E[/tex] be the set of all [tex]x\in [0,1][/tex] whose decimal expansion contains only the digits 4 and 7. Is [tex]E[/tex] compact? Is [tex]E[/tex] perfect?


    2. Relevant equations



    3. The attempt at a solution
    My answer is: [tex]E[/tex] is compact and perfect.

    By Heine-Borel theorem ([tex]E[/tex] is compact equivalent to [tex]E[/tex] is closed and bounded), since [tex]E[/tex] is bounded, and [tex]\forall x\in E[/tex], and [tex]\epsilon > 0[/tex], we always have that the neighborhood of [tex]x[/tex], [tex]N_\epsilon (x)[/tex] contains infinitely many elements of [tex]E[/tex], in other words, any number in [tex]E[/tex] is a limit point of [tex]E[/tex]. Conversely, I assume [tex]E[/tex] is not closed, that is, there exists at least one number which is not in [tex]E[/tex], but it is a limit point of [tex]E[/tex]. I denote this number by [tex]y[/tex]. Suppose the i-th digit of [tex]y[/tex], say [tex]y_i[/tex] is the first digit which is not in [tex]\{4,7\}[/tex]. Then we always can find some other numbers which are between [tex]y[/tex] and the closest number in [tex]E[/tex]. So we can conclude that if [tex]y\not\in E[/tex], [tex]y[/tex] must not be a limit point of [tex]E[/tex]. So we can conclude that [tex]E[/tex] is closed and perfect. By the aforementioned Heine-Borel theorem, it is also compact.

    However, I am still worrying about this proof, since I am not very sure on the correctness of the proof. Can you guys help me to see if both my answer and my proof are correct? Thanks a lot!
     
  2. jcsd
  3. Jul 24, 2009 #2

    Dick

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    Homework Helper

    Your answer is correct. It's a Cantor type set. Your proof leaves a lot to be desired. And it is a bit complicated since you have to prove two things, that E is closed and perfect. Start with closed. As you said, if a point x is not in E, then it has a digit yi that is not 4 or 7. You have to show x is a positive distance away from E. Can you give an estimate for a positive lower bound on how far away it might be? Suppose the digit is say '3'?
     
    Last edited: Jul 24, 2009
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