#### Question

Let $T=S^{1} \times S^{1}$, the torus. There is an isomorphism of $\pi_{1}\left(T, b_{0} \times b_{0}\right)$ with $\mathbb{Z} \times \mathbb{Z}$ induced by projections of T onto its two factors. (a) Find a covering space of T corresponding to the subgroup of $\mathbb{Z} \times \mathbb{Z}$ generated by the element $m \times 0$, where m is a positive integer. (b) Find a covering space of T corresponding to the trivial subgroup of $\mathbb{Z} \times \mathbb{Z}$. (c) Find a covering space of T corresponding to the subgroup of $\mathbb{Z} \times \mathbb{Z}$ generated by $m \times 0$ and $0 \times n$ where m and n are positive integers.

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