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I was wondering what would the quoteint group $\mathbb R/\mathbb Z$ look like geometrically ? Any two elements in this group will be related only if their difference is an integer.

I was reading about the geometry of a torus, wherein it was stated that $\mathbb R^n/\mathbb Z^n$ was equivalent to a torus in n-dimension, so i was thinking that $\mathbb R/\mathbb Z$ should probably be a circle,but if that is correct how would i get that kind of a construction?

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Yes, $\mathbb{R}/\mathbb{Z} \cong S^1$ as a group as well as a manifold, with the isomorphism given by $x\to exp(2\pi ix)$.

In fact in many cases a torus is defined as the $k-fold$ product of $S^1$, for example the usual $2-d$ torus is $S^1\times S^1$.

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