Corollary 4

Let B be a connected and locally connected topological space, let $(E, p)$ be a covering of B, and let $b$ be a point of B. In order that E be a trivializable covering, it is necessary and sufficient that every point of the fibre $E_b$ belong to the image of a continuous section of $p$.

The condition is necessary. Let $E'$ be the union of the images of the continuous sections of $p$, and let $E''$ be its complement in E. The set $E'$ is open and closed in E (cf. I, p. 79). Hence $(E', p|E')$ is a covering of B (Corollary 1), and this covering is trivializable by virtue of Corollary 2 (I, p. 70). Suppose that $E'$ contains $E_b$ and let us prove that $E''$ is empty. Since $E''$ is open and closed in E, $p(E'')$ is open and closed in B (Corollary 1). Since B is connected and $b$ does not belong to $p(E''),p(E'') =\emptyset$, hence $E''=\emptyset$.