This wasn't originally a homework problem as such, so sorry if its confusing, but I thought I would ask it here;(adsbygoogle = window.adsbygoogle || []).push({});

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 ℝ^{n}by the relation x~y iff x-y [itex]\in[/itex] Z^{n}.

2) The n-torus given by the product of n circles T^{n}= S^{1}×S^{1}×..........×S^{1}.

2. Relevant equations

S^{1}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 S^{1}. This is clearly the same as the definition for S^{1}. 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,

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# Homework Help: Definition on an n-torus, two approaches.

Can you offer guidance or do you also need help?

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