Corollary 5

Let

${E'}^{f'}$ E

$p'p$

${B'}^f$ B

be a cartesian square. Suppose that the B-space $(E, p)$ is a covering and that the space $B'$ is connected and locally connected. Let $b'$ be a point of $B'$. In order that the $B'$-space $(E', p')$ be a trivializable covering, it is necessary and sufficient that for every point $x$ of E such that $p(x) =f(b')$, there exist a continuous lifting $g: B'\rightarrow E$ of $f$ such that $g(b') =x$.

Let us recall that $(E', p')$ is a covering of $B'($I, p. 71, Cor. 2) and that the mapping $f'$ induces a bijection $E'_{b'}\rightarrow E_{f(b')}($I, p. 10, Cor.). By Prop. 3 of I, p. 9, the mapping $s\mapsto f'\circ s$ defines a bijection between the set of continuous sections of $p'$ and the set of continuous liftings of $f$ to E. The corollary therefore follows from Corollary 4.