Proposition 9

Let B be a topological space, let E be a principal covering of B of group G, connected and nonempty. Let F be a nonempty G-set endowed with the discrete topology. In order that the space $E\times^GF$ be connected, it is necessary and sufficient that G operate transitively in F.

Let U be a subset open and closed of $E\times^GF$. If $\pi : E\times F\rightarrow$ $E\times^GF$ denotes the canonical surjection, $\overset{-1}{\pi}(U)$ is a subset open and closed of $E\times F$ which is stable under G. Since E is connected, there exists a subset $F'\subset F$, stable under G, such that $\overset{-1}{\pi}(U) = E\times F'$, whence $U =\pi (E\times F')$. Let $F'$ and $F''$ be subsets of F stable under G such that $\pi (E\times F') =\pi (E\times F'')$; since E is not empty, one has $F'= F''$. The mapping $F'\mapsto \pi (E\times F')$ is a bijection of the set of subsets of F stable under G onto the set of subsets open and closed of $E\times^GF$. The proposition follows.