Some words about the context of the proposition. Hatcher is defining operations on vector bundles over the same base space $B$. We are speaking about the Whitney sum here, and it is proving that if $B$ is compact Hausdorff then exists the inverse under Whitney sum of a given vector bundles.
[Hatcher vol. 2 Vector Bundles & K-theory pag. 13]
My question is: I don't see the necessity of using Urysohn's lemma (U-L) in this proof. Am I wrong?
for what I've understood, Hatcher uses U-L to have an open cover of $B$, but by definition exists already a cover (the same Hatcher consider!) of $B$ with the trivialization property (and I can assume it is open). Moreover I can't appreciate the need of define $g_i : E \to \mathbb{R}^n$ by $g_i(v)=\varphi_i(p(v))[\pi_i(h_i(v))]$ instead of $g_i(v)=\pi_i(h_i(v))$ for $v \in U_x$ or $0$ elsewhere. The properties required holds thanks to $h_i$ not thanks to the continuous function $\varphi_i$. So why passing through U-L?
Maybe I can't appreciate some delicate passage in this proof. Thanks in advance!
Hatcher uses the $\varphi_i$ so that he can obtain a maps $g_i:E\rightarrow \mathbb{R}^n$, instead of just having maps $p^{-1}(U_i)\rightarrow \mathbb{R}^n$.