Proposition 5

Let B be a topological space, let E and $E'$ be B-spaces, and let $f: E'\rightarrow E$ be a B-morphism. We assume that E is a locally finite covering and that the space $E'$, endowed with the mapping $f$, is a locally trivial fibred E-space ( resp. a covering). Then, $E'$ is a locally trivial fibred B-space ( resp. a covering).

Let us denote by $p$ and $p'$ the respective projections of the B-spaces E and $E'$. Let $b$ be a point of B. There exists an open neighbourhood U of $b$ in B and a finite partition $(V_i)_{i\in I}$ of $\overset{-1}{p}(U)$ formed of open sets of E such that, for every $i\in I$, the mapping $p$ induces a homeomorphism of

$V_i$ onto U. Put $V'_i=\overset{-1}{f}(V_i)$. The sets $V'_i$ are open in $E'$,

form a partition of $(^-{p'}^1)(U)$ and the mapping from $V'_i$ to U deduced from $p'$ by passing to the subspaces makes $V'_i$ into a locally trivial fibred U-space. It follows that the space $(^-{p'}^1)(U)$, endowed with the mapping

$p'_U:(^-{p'}^1)(U)\rightarrow U$, is a locally trivial fibred U-space (I, p. 69, remark 2). If $(E', f)$ is a covering of E, the fibres of $p'_U$ are discrete because their intersections with each of the open sets $V'_i$ are so. This completes the proof.