I am studying Algebraic Topology and struggling with isomorphism of fiber bundles, which is not explicitly defined by the professor. So what I am looking for is an explicit defintion of this.
The definition of fiber bundles he uses is as follows:
A continuoius map $p:E\rightarrow B$ between topological spaces is a fiber bundle if for every $x$ in $B$ there exist
i) an open neigbhourhood $U$ of $x$ in $B$
ii) A topological space $F$
iii) a homeomorphism $h:p^{-1}(U)\rightarrow U\times F$
such that $p=\pi_U\circ h$ where $\pi_U$ is the projection onto $U$.
Note that the prof is vehement about not using the term "fiber bundle" for a "total space".
Now, I know there is a category theory approach to this definition which some people have told me is simpler. But as someone who has little knowledge of category theory, I would appreciate a topological definition of isomorphism of fiber bundles and perhaps some correponding intuition.