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Equivalence relation

  1. Sep 24, 2007 #1
    1. The problem statement, all variables and given/known data
    If Z acts on R by n.x=n+x then R/Z is just S^1. CLaims the book

    But I think R/Z is (0,1)


    3. The attempt at a solution
    Any number greater than or equal to 1 is dealt with by the equivalence relation. How does the unit circle come into it? We are dealing only with one dimensional space here.
     
  2. jcsd
  3. Sep 24, 2007 #2
    Because the points x=0 and x'=1 are identified as the same point in R/Z?
     
  4. Sep 24, 2007 #3

    HallsofIvy

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    That's the whole point- the equivalence relation makes 0 and 1 equivalent- you are bending [0,1] (not (0,1)) back on itself so it becomes a circle.
     
  5. Sep 24, 2007 #4
    Actually the interval should be [0,1) so if you include 1 then 1 is 0 so it bends back on itself. However they described it as S^1. Why S^1? That is the unit circle with radius 1. So has circumference 2pi. But our interval has length 2pi?
     
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