Some questions about Bott & Tu - Differential Forms in Algebraic Topology, chapter 11. I am reading chapter 11 of Bott & Tu - Differential forms in algebraic topology. And I have some questions about this section.
1: Let $\pi:E\to M$ be a sphere bundle with fiber $S^n$. For each $x\in M$ the fiber of $x$ is denoted by $F_x$. This bundle is said to be orientable if it is possible to choose a generator $[\sigma_x]\in H^n(F_x)=\Bbb R$ for each $x\in M$, satisfying the local compatibility condition: each $x$ has a neighborhood $U\subset M$ and a generator $[\sigma_U]\in H^n(E|_U)$ such that $[\sigma|_U]|_{F_x}=[\sigma_x]$.
The book says that orientability is equivalent to the following: there is an open cover $\{U_\alpha\}$ of $M$ and generators $[\sigma_\alpha]$ of $H^n(E|_{U_\alpha})$ so that $[\sigma_\alpha]=[\sigma_\beta]$ in $H^n(E|_{U_\alpha\cap U_\beta})$.
I can't see why these two conditions are equivalent. I think a "generator" of $H^n(E|_U)$ should be interpreted as it restricts to a generator of each fiber. Then the second condition obviously implies the first condition. But how does the first condition imply the second one? Is the following true? For $[\sigma_1], [\sigma_2] \in H^n(E|_U)$, if $[\sigma_1]|_{F_x}=[\sigma_2]|_{F_x}$ for each $x\in U$, then $[\sigma_1]=[\sigma_2]$. If this is not true, then I think there is no need that two conditions to be equivalent.
2: The book uses a good cover of a manifold $M$, which is by definition an open cover $\mathfrak{U}$ of $M$ such that for every finitely many open sets in $\mathfrak{U}$, their intersection is either empty or diffeomorphic to $\Bbb R^n$. Is the following true? For any open cover of $M$, there is a good cover which refines the open cover. It seems that throughout the chapter, the book assumes this as true, but I can't see why it is true.
 A: Question 1:
Speaking about a generator of $H^n(E \mid_ U)$ should in my opinion be interpreted in the literal sense which implictly assumes that $H^n(E \mid_ U) \approx \mathbb Z$. You cannot expect that for arbitrary $U$, but certainly we can take a neigborhood $U$ of the point $x_0$ which is contractible (e.g. diffeomorphic to $\mathbb R^m$) and admits a homeomorphism $E \mid_U \to U \times S^n$. Then $H^n(E \mid_ U) \approx H^n(S^n) \approx \mathbb Z$. Call such $U$ simple. It has the benefit that all inclusions $i_x = i_{x,U} : F_x \to E \mid_ U$ induce isomorphisms $i_x^* : H^n(E \mid_ U) \to H^n(F_x)$. If in addition there is a generator $[\sigma_U]$ of $H^n(E \mid_ U)$ such that $[\sigma|_U]|_{F_x} =i_x^*([\sigma_U]) =[\sigma_x]$, we call $(U,[\sigma_U])$ a nice pair.
Alternatively you can interpret it for an arbitrary $U$ as you do. This means that there exists an element $[\sigma_U] \in H^n(E \mid_ U)$ such that $[\sigma|_U]|_{F_x}=[\sigma_x]$. Call such $(U,[\sigma_U])$ a quasi-nice pair. If you shrink this $U$ to a simple $U'$ as above, then certainly $[\sigma_U] \mid_{E \mid_{U'}}$ is generator of $H^n(E \mid_{U'})$, i.e. $(U', [\sigma_U] \mid_{E \mid_{U'}})$ is a nice pair.
Thus it is equivalent to require the existence of nice pairs or the existence of quasi-nice pairs.
Now let the bundle be orientable. Take a good cover $\{U_\alpha\}$ consisting of simple $U_\alpha$ and let $[\sigma_\alpha])$ be such that all $(U_\alpha,[\sigma_\alpha])$ are nice pairs. Then $U_\alpha \cap U_\beta$ is simple and we have for $x \in U_\alpha \cap U_\beta$ (with $i_\alpha : U_\alpha \cap U_\beta \to U_\alpha, i_\beta : U_\alpha \cap U_\beta \to U_\beta$)
$$i_{x,U_\alpha \cap U_\beta}^*(i_\alpha^*([\sigma_\alpha])) = i_{x,U_\alpha}^*[\sigma_\alpha]) = [\sigma_x] = i_{x,U_\beta}^*[\sigma_\beta]) =  i_{x,U_\alpha \cap U_\beta}^*(i_\beta^*([\sigma_\beta])) ,$$
thus since $i_{x,U_\alpha \cap U_\beta}^*$ is an isomorphism
$$i_\alpha^*([\sigma_\alpha]) = i_\beta^*([\sigma_\beta]) .$$
Conversely, if we have an open cover $\{U_\alpha\}$ of $M$ and generators $[\sigma_\alpha]$ of $H^n(E|_{U_\alpha})$ so that $[\sigma_\alpha]=[\sigma_\beta]$ in $H^n(E|_{U_\alpha\cap U_\beta})$, then define $[\sigma_x] = [\sigma_\alpha] \mid_{F_x}$ for any $\alpha$ such that $x \in U_\alpha$. This is well-defined. Clearly the $[\sigma_x]$ are an orientation.
Question 2:
This is answered by Theorem 5.1 and Corollary 5.2.
