Corollary 2

Let B be a topological space, let E be a B-space and let $p$ be its projection. Suppose that $p$ is étale and separated and that the space B is connected. In order that E be a trivializable covering of B, it is necessary and sufficient that, for every point $x$ of E, there exist a continuous section $s: B\rightarrow E$ of $p$ such that $s\circ p(x) =x$.

This is obviously necessary. Conversely, for every continuous section $s$ of $p$, the set $s(B)$ is open and the mapping $p$ induces a homeomorphism of $s(B)$ onto B (I, p. 30, Cor. 3 of Prop. 6). Moreover, when $s$ ranges over the set $\mathscr{S}(B; E)$ of sections of $p$, the sets $s(B)$ are pairwise disjoint, since B is connected (I, p. 34, Cor. 1 of Prop. 11). It then follows from proposition 1 that if the sets $s(B)$ cover E, the space E is a trivializable covering of B.