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.

## Best Answer

I found the definition.

Two fiber bundles $p_1:E_1\rightarrow B$ and $p_2:E_2\rightarrow B$ with the same base space $B$ are isomorphic if ther exists a homeomorphism $g:E_1\rightarrow E_2$ such that $$p_2\circ g=p_1$$ Then, $g$ is called an isomorphism of fiber bundles $p_1$ and $p_2$.

(This is basically the definition given by Vercassivelaunos in the comments written explicitly.)