Proposition 1

Let B be a topological space and let E be a covering of B. Let $B'$ be a topological space and let $f_0$ and $f_1$ be continuous mappings of $B'$ into B. If the mappings $f_0$ and $f_1$ are homotopic, the coverings $f_0^*(E)$ and $f_1^*(E)$ of $B'$ are isomorphic.

Let $\sigma : B'\times \mathbf{I}\rightarrow B$ be a homotopy joining $f_0$ to $f_1$. Let us denote by $i_0$ and $i_1$ the mappings $x\mapsto (x,0)$ and $x\mapsto (x,1)$ of $B'$ into $B'\times \mathbf{I}$. If $t\in \{0,1\}$, we have $f_t=\sigma \circ i_t$ and the coverings $f_t^*(E)$ and $i^*_t(\sigma^*(E))$ of $B'$ are therefore isomorphic (I, p. 15). Since the topological space $\mathbf{I}$ is locally connected and simply connected (I, p. 127, corollary), the coverings $i^*_0(\sigma^*(E))$ and $i^*_1(\sigma^*(E))$ of $B'$ are isomorphic (I, p. 130, cor. 1 of prop. 8).