Example 1

Let X be a simply connected topological space and let $f$ be a continuous function from X into $\mathbf{C}^*$. Recall (I, p. 101, example 6) that the mapping $z\mapsto e^z$ makes $\mathbf{C}$ a covering of $\mathbf{C}^*$. By prop. 3, there exists a continuous function $h: X\rightarrow \mathbf{C}$ such that $f(x) =e^{h(x)}$ for every $x\in X$. If $h': X\rightarrow \mathbf{C}$ is another continuous function such that $f(x) =e^{h'(x)}$ for every $x\in X$, there exists an integer $q\in \mathbf{Z}$ such that $h'=h+ 2\pi iq$.

Analogously, let $n$ be an integer $>0$; the mapping $z\mapsto z^n$ makes $\mathbf{C}^*$ into a covering of itself (I, p. 101, example 5). There therefore exists a continuous function $k$ from X into $\mathbf{C}^*$ such that $k(x)^n=f(x)$ for every $x\in X$, for example the function $k(x) =e^{h(x)/n}$. If $k': X\rightarrow \mathbf{C}^*$ is another continuous function such that $k'(x)^n=f(x)$ for every $x\in X$, there exists an $n^e$ root of unity $\mu\in \mathbf{C}$ such that $k'=\mu k$.