Remark

Let B be a topological space locally connected by paths, let $b$ be a point of B, and let V be an open connected neighbourhood of $b$ such that the image of the homomorphism from $\pi_1(V, b)$ into $\pi_1(B, b)$ is the subgroup reduced to the identity element. By cor. 1, every covering of B is trivializable over V, and fortiori over every subset of B contained in V.