# An isomorphism of fiber bundles

algebraic-topologyfiber-bundles

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.

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 $$p_2$$.