Example 6

Endowed with the mapping $z\mapsto e^{2\pi iz}$ (TG, VIII, p. 8, remark), the space $\mathbf{C}$ is a principal covering of $\mathbf{C}^*$ with group $\mathbf{Z}$. The mapping $x\mapsto e^{2\pi ix}=\mathbf{e}(x)$ of $\mathbf{R}$ onto $\mathbf{S}_1$ makes $\mathbf{R}$ into a principal covering of $\mathbf{S}_1$ with group $\mathbf{Z}$.