1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    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!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted