Tu's - Introduction to Smooth Manifolds. Definition of Section vs Inverse map.

Question

A section of a vector bundle $\pi : E \to M$ is defined as map $s : M \to E$ such that $\pi \circ s = \text{Id}_M$.

I cannot figure the difference between this definition and the definition of inverse map. Can anyone clarify?

Answer 1

$\pi$ in general will not be invertible. For $s$ to be $\pi^{-1}$, we would also need $s\circ \pi=\mathrm{Id}_E$. I think the definition of $s$ should also say that $s$ is continuous.