I'm currently studying Vector Bundles from Husemoller and I have two questions concerning Proposition $2.5$ (p.$142$) on Clutching Maps over $X \times S^2$.
$1.$ Husemoller afirms that there is an extension of Fejer's theorem for Vector Bundles without referencing, any one knows what's the theorem is thinking about or have a reference to it?
$2.$ The more important one: Husemoller afirms that homotopy classes of clutching maps are open sets in the uniform topology. I don't understand if he restrict to the case where the vector bundle is over $X \times S^2$, so basically a clutching map lives in Aut($E \times S^1$) or if this is a general fact, if so, open in which space? In addition is not clear to me how the vector bundle Hom($\xi,\eta$) should be defined (where $\xi,\eta$ are two vector bundle over the same space $X$). In particular the topology on it.
The question have a similar question here, but Hatcher seems less general, although providing "understandable" proofs.
Any help in order to clarify this details would be appreciated.