TITLE: Difference between a function and a section of a fibre bundle
QUESTION [6 upvotes]: Suppose $E \rightarrow B$ with projection map $\pi$ and fibre F is a fibre bundle, with a section $\sigma$. How is $\sigma$ different from a function $f:B \rightarrow F$? The standard answer I find everywhere is that $\sigma$ is only a function if E is the trivial bundle $B \times F$, or that in general no such function exists with 'global structure', but I do not see why the former is true, and I do not understand what the latter means. A section seems like it is associating an element of the fibre F to every point in B. How is this not a function, even if E has some sort of non trivial topology? Why should 'twists' or other aberrations from a product in the space E matter, since the fibre over any point in B is the same?

REPLY [3 votes]: Long story short and summarizing the answers, since I think the OP had my same doubt, a function $f : B \rightarrow F$ takes any base point $m$ and maps it to an element of the same, fixed, topological space $F$. The intuition behind a section $\sigma : B \rightarrow E, \pi(\sigma(m))=m$, is that it maps any point $m$ to a different space, namely the fiber $\pi^{-1}(m)$, which are different copies of F, one for each base point $m$.
Only in particular cases ( e.g. when the bundle is trivial ) the two operations are equivalent in a well defined way.