I'm learning about fiber bundles and I have the following question.
Let $p:E\rightarrow B$ be a fiber bundle (let's assume it is trivial) and let $U$ be an open connected set of $E$. Let $V=p(U)$. Is it always true that there exists a section $s$ such that $s(V)\subseteq U$?
Thank you (any reference about this kind of problems is welcome).