Corollary 1

Let $(E, p)$ be a principal covering of B of group G; suppose that E is connected and nonempty. A covering X of B is associable to E if and only if the covering $p^*(X)$ is trivializable.

If the covering $p^*(X)$ is trivializable, it follows from proposition 8, b), that the covering X is associable to E. In this case, the mapping $\beta_X$ identifies the covering X with the covering of fibre-type $\mathscr{C}_B(E; X)$ associated to E. Conversely, let F be a G-set endowed with the discrete topology and suppose that one has $X = E\times^GF$. Then $\alpha_F$ is an isomorphism (loc. cit., a)) and formula (4) implies that $\beta_X$ is an isomorphism. Consequently, the covering $p^*(X)$ is trivializable (prop. 8, b)), whence the corollary.