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Perfect set with no rationals

  1. Oct 29, 2009 #1
    1. The problem statement, all variables and given/known data
    Is there a perfect set that contains no rationals?


    2. Relevant equations
    A set is perfect if it is closed and contains no isolated points.


    3. The attempt at a solution
    Why not just take the set of irrational numbers as your whole space? This set is certainly closed as it's the whole metric space, and it contains no isolated points, as the irrationals are dense in themselves.

    What's wrong with this? I'm not confident it's correct since this was given as a bonus problem on an assignment, but I can't see what's wrong with my example.

    Edit: the prof probably meant for us to find a perfect subset of R with no rationals, 'cause that seems a lot harder and I don't even know where to start. so how might I go about doing that, then?
     
    Last edited: Oct 29, 2009
  2. jcsd
  3. Oct 29, 2009 #2
    I'm pretty sure what is meant is a set perfect in R. The answer is affirmative. The simplest example is [itex]\emptyset[/itex]. If you require the set to be non-empty, then you can emulate the construction of the cantor set (consider irrational numbers a,b and enumerate the rationals in [a,b] and then inductively remove an interval around each of them). If you need further help or ideas see the solution at (I suggest you try to construct it yourself before you look):
    http://planetmath.org/encyclopedia/ANonemptyPerfectSubsetOfMathbbRThatContainsNoRationalNumber.html [Broken]
     
    Last edited by a moderator: May 4, 2017
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