Corollary 1

Let B be a connected topological space locally connected by paths, and let $b$ be a point of B. In order that a covering $(E, p)$ of B be trivializable, it is necessary and sufficient that, for every point $x$ of the fibre $E_b$, the homomorphism $\pi_1(p, x)$ be bijective.

Recall that the homomorphism $\pi_1(p, x)$ is injective (III, p. 303, cor. 1 of prop. 4). The statement therefore follows from prop. 1 and I, p. 81, cor. 4 of prop. 6.