Proposition 6

The mapping $p:x\mapsto e^{2\pi ix}$ of $\mathbf{R}$ into $\mathbf{S}_1$ makes $(\mathbf{R},0)$ a universal covering of $(\mathbf{S}_1,1)$. The group $\pi_1(\mathbf{S}_1,1)$ is isomorphic to $\mathbf{Z}$; the class of the loop $t\mapsto e^{2\pi it}$ is a generator of it. For every integer $n >0$, the mapping $z\mapsto z^n$ makes $\mathbf{S}_1$ a Galois covering of group $\mathbf{Z}/n\mathbf{Z}$ of $\mathbf{S}_1$. Every connected nonempty covering of $\mathbf{S}_1$ is isomorphic to one of these coverings.

Only the last assertion remains to be proved. The fibre $E_1$ over 1 of a connected nonempty covering E of $\mathbf{S}_1$ is endowed with a transitive operation of the group $\pi_1(§_1,1)$. Since the subgroups of $\mathbf{Z}$ are the sets $n\mathbf{Z}$, for $n\geqslant 0$, we see that $E_1$ is isomorphic to one of the homogeneous sets associated with the coverings described above. Since $\mathbf{S}_1$ is connected and locally arcwise connected, E is isomorphic to one of these coverings.