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Definition on an n-torus, two approaches.

  1. Apr 24, 2012 #1
    This wasn't originally a homework problem as such, so sorry if its confusing, but I thought I would ask it here;
    1. The problem statement, all variables and given/known data
    Show that the the two methods of creating the n-torus are equivalent.
    1) The n-torus as the quotient space obtained from ℝn by the relation x~y iff x-y [itex]\in[/itex] Zn.
    2) The n-torus given by the product of n circles Tn= S1×S1×..........×S1.

    2. Relevant equations

    S1 is the collection of equivalence classes [x]={y:y~x}.
    An equivalence relation x~y iff x-y [itex]\in[/itex] Z

    3. The attempt at a solution
    Well, I suppose we want an inductive proof? In our second definition, when n=1, we just have S1. This is clearly the same as the definition for S1. So then we want to then look at n = k and n = k + 1.......What's the best way to go about this?

    ( Note Z means the set of integers)

    Thanks,
     
  2. jcsd
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