Let $\pi: E \rightarrow B$ be a surjective continuous map of topological spaces. Suppose we have an open cover of $B$ such that each set $U$ in the cover admits a continuous section $s: U \rightarrow E$. Are there some (well-known/common) general conditions under which these sections guarantee that $E$ is a fiber bundle? What about a principal $G$-bundle with $G$ a topological group acting continuously on $E$? What about the smooth case?
What motivated me to ask is that I encountered the statement "we can show directly that $S^{2n+1} \rightarrow \mathbb{C}P^n$ is locally trivial by producing sections over an open cover", which in general doesn't seem at all obvious to me, and in this case seems to depend on seeing $\mathbb{C}P^n$ as the orbit space of $S^{2n+1}$ acted on by $S^1$ and on the fact that this action is free, which is already begging the question.