Proposition 3

Let B be an unrollable topological space and let A be a connected and locally arcwise connected subspace of B (for example an open and connected subset). Let $a$ be a point of A. The following properties are equivalent:

(i) Every covering of B is trivializable over A.

(ii) The image of the homomorphism from $\pi_1(A, a)$ into $\pi_1(B, a)$ induced by the canonical injection is reduced to the identity element.

The implication (ii)$\Rightarrow$(i) follows from corollary 3 (III, p. 309).

Conversely, suppose that every covering of B is trivializable over A. This is in particular the case for the simply arcwise connected covering $(\lambda_a(B), \varepsilon_B)$ (IV, p. 342, th. 1), so that the homomorphism $\pi_1(A, a)\rightarrow \pi_1(B, a)$ is trivial (III, p. 309, cor. 3 of prop. 1).