In A Concise Course in Algebraic Topology, P. May,
It is written that
a vector bundle morphism $(f, g)$ is an isomorphism if and only if $g$ is a homeomorphism and $f$ restricts to linear isomorphisms on each fiber.
It seems to me that if $(f, g)$ should be an isomorphism, then $f$ must also be a homeomorphism. I was only able to show that it is bijective (and it is continuous by definition of vector bundle morphisms). The only way I know how to prove that such maps have continuous inverses is to show that they are local homeomorphisms, but it doesn't seem to be a very practical approach here, since open sets in the total space will intersect many fibers.
I would greatly appreciate hints or online accessible references.
Thank you.
Actually that seems the way to go. To prove it's a homeomorphism you'll have to show that the morphism $f:E_1\to E_2$ is a local homeomorphism.
First choose an open cover, $\{U_\alpha\}$, of your base space in such a way that it has a local trivialization for both bundles ($E_1$ and $E_2$), let's say $\varphi_\alpha:U_\alpha\times K^n\to K^n$ and $\psi_\alpha: U_\alpha\times K^n\to K^n$, respectively.
It's easily shown that this induces a continuous function $\gamma: U_\alpha\times K^n \to U_\alpha\times K^n$ such that $\gamma=\psi_\alpha\circ f|_{U_\alpha}\circ \varphi_\alpha^{-1}$. An immediate consequence is that $\gamma(x,v)=(x,\delta(x,v))$ where $\delta(x,v) = A_x(v)$ and $A_x\in GL_n(K)$.
So we must show that $\gamma^{-1}=\varphi_\alpha\circ f|_{U_\alpha}^{-1}\circ \psi_\alpha^{-1}$ is also continuous. This is equivalent to showing that $\delta$ is a homeomorphism. I actually don't know how to prove this so I started a thread to get some help: Topology of $GL_n(K)$