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[Cardinality] Prove there is no bijection between two sets

  1. Feb 27, 2012 #1
    1. The problem statement, all variables and given/known data
    prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R


    2. Relevant equations



    3. The attempt at a solution

    is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between those two sets

    R has the cardinality of continuum

    the unit circle is defined on [-1,1]x[-1,1] and since [a,b] has same cardinality as R for all a,b, cardinality of the unit circle would be c*c = c^2 but c^2=c, but this can't be since then there would be a bijection between the unit circle and R

    ???
     
  2. jcsd
  3. Feb 27, 2012 #2
    How about something easier?

    Is R compact? Is the unit circle compact?

    Can you have continuous map from a compact set to a non compact set?
     
  4. Feb 27, 2012 #3
    nope

    i just realized it after posting this thread but i dont know how to delete it now

    thanks
     
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