0708-1300/Covering Product

Think about an infinite product of copies of $S^{1}$. Even more, think about the possibility of this space having a universal covering space. At some point it is probably needed that $\mathbb {Z} ^{n}\not \cong \mathbb {Z} ^{m}$ for $n\neq m$. It is interesting to produce a proof of this using the less knowledge you can. A possible idea is here.