0708-1300/Covering Product

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