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Homework Help: Closed set with rationals question

  1. Dec 24, 2015 #1
    1. The problem statement, all variables and given/known data
    If A is a closed set that contains every rational number in the closed interval [0,1], show that [0,1] is a subset of A.

    2. Relevant equations

    3. The attempt at a solution
    I'm confused because for the set A = all rationals in [0,1], every point is a boundary point so the set is closed. but clearly [0,1] is not a subset of A. This question is from Spivaks Calculus on Manifolds, question 1-19
  2. jcsd
  3. Dec 24, 2015 #2


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    Though the boundary of a set is closed, that doesn't mean that a set A consisting only of boundary points is necessarily closed (the boundary of A could contain points outside of A too).
    The set of all rationals in [0,1] is not closed.

    You could try to prove that any (irrational) number in [0,1] is in the closure of the rationals in [0,1].
    Last edited: Dec 24, 2015
  4. Dec 24, 2015 #3
    oh right I didn't realize the rationals was not closed, I think I got it now, Thanks!
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