Corollary

Let B be a locally connected topological space. Let $b$ be a point of B and let $(E, x)$ be a universal covering of $(B, b)$. For a subspace A of B, the following two properties are equivalent:

(i) The covering E is trivializable over A;

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

Property (ii) obviously implies property (i). The converse follows from the fact that every covering of B can be associated with the covering E (I, p. 105, Proposition 7).