Line Bundle with Everywhere Nonzero Section is Trivial

Question

I'd like to show that a topological line bundle $(\pi,E,B)$ is trivial if there exists a section $\sigma : B \to E$ such that $\sigma(b) \neq 0$ for all $b$. I've been considering the map $B \times \mathbb{R} \to E$ given by $(b,\lambda) \mapsto \lambda \sigma(b)$, but I'm having trouble verifying that this map is continuous. Any ideas?

Answer 1

You have defined $\Psi:B\times \mathbb R\to E$ by $(b,\lambda)\mapsto\lambda \sigma(b).$ Take a $b\in U\subseteq B$ and a local trivialization $\Phi:\pi^{-1}(U)\to U\times \mathbb R.$ Then, $\Phi\circ \sigma|_U:U\to U\times \mathbb R$ is continuous so $\Phi\circ \sigma|_U(b)=(b,f(b))$ for some continuous $f:U\to \mathbb R$ and so $\Phi\circ \Psi(b,\lambda)=\Phi(\lambda\sigma (b))=(\lambda\sigma(b),f(\lambda\sigma (b)).$ This means that $\Phi\circ \Psi$ is continuous, and therefore, since $\Phi$ is a homeomorphism, that $\Psi$ is continuous, too.