We recall that a complex line bundle consists of a triple $(\pi,E,B)$ where $E,B$ are topological spaces, $\pi : E \to B$ a continuous map satisfying the following local triviality condition:
Local Triviality: There is a cover $\{U_\alpha\}$ of $B$ such that for every $\alpha$, we have homeomorphisms $\varphi_\alpha: \pi^{-1}(U_\alpha) \stackrel{{\cong}}{\longrightarrow} U_\alpha \times \Bbb{C}$ such that $\pi = \operatorname{proj} \circ \varphi_\alpha$, where $\operatorname{proj} : U_\alpha \times \Bbb{C} \to U_\alpha$ is just projection onto the first factor.
This local triviality condition also gives that $ \varphi_{\beta}\varphi_{\alpha}^{-1}(x,u) = (x, t_{\beta\alpha})$ for some $t_{\beta\alpha} : U_\alpha \cap U_{\beta} \to \Bbb{C}$; the $t_{\beta \alpha}$ are called transition functions.
My question is: Say $B = \Bbb{P}^n$. Can we replace the condition "there is a cover such that..." with the condition "for every open affine cover..." ?