Proposition 3

Let B be a topological space and let $p: E\rightarrow B$ be a covering. Let us denote by $\widetilde{p}:\mathscr{C}_c(\mathbf{I}; E)\rightarrow \mathscr{C}_c(\mathbf{I}; B)$ the mapping $c\mapsto p\circ c$. Let us denote by $o_E:\mathscr{C}_c(\mathbf{I}; E)\rightarrow E$ ( resp. $o_B:\mathscr{C}_c(\mathbf{I}; B)\rightarrow B$) the mapping defined by $c\mapsto c(0)$. Then, the diagram

$\mathscr{C}_c(\mathbf{I}; E)^{o_E}$ E

$\widetilde{p}p$

$\mathscr{C}_c(\mathbf{I}; B)^{o_B}$ B is a cartesian square.

The topological space $\mathbf{I}$ is locally connected, locally compact and simply connected. The proposition thus follows from prop. 9 of I, p. 131, applied with $T =\mathbf{I}$ and $t= 0$.