universal example of a line bundle

Question

I've read the definition of the universal bundle, but still don't see. What does it mean that $L\rightarrow \mathbb{C}P^\infty$ is the universal example of line bundles?

Thanks in advance.

Answer 1

The statement is that if you have a (sufficiently nice - say paracompact) topological space $M$ and a continuous complex line bundle $E\to M$, then there is a continuous map $f:M\to \mathbb CP^\infty$ such that $f^*L\cong E$. Otherwise, put, $f$ lifts to a continuous map $\hat f:E\to L$ which is a complex linear isomorphism on each fiber.

This in particular implies that a local trivialization (open sets and transition functions) of $L$ can be pulled back to a local trivialization of $E$. In that sense, $L$ is less trivial than $E$, so $L$ is the "most nontrivial" line bundle.