# I Product space vs fiber bundle

#### cianfa72

Summary
product space as example of trivial fiber bundle
Hi,

I'm not a really mathematician...I've a doubt about the difference between a trivial example of fiber bundle and the cartesian product space. Consider the product space $B \times F$ : from sources I read it is an example of trivial fiber bundle with $B$ as base space and $F$ the fiber.

As far as I understood Fiber bundle requires fibers "attached" on base space to be actually disjoint. With that in mind should we understand (conceive) the cartesian product $B \times F$ itself as a disjoint union where there exist for instance multiple copies of F space over B ?

hoping I was able to explain the point...

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#### fresh_42

Mentor
2018 Award
Summary: product space as example of trivial fiber bundle

With that in mind should we understand (conceive) the cartesian product $B \times F$ itself as a disjoint union where there exist for instance multiple copies of $F$ space over $B$?
Yes.
$B\times F = \{\,(b,f)\,|\,b\in B,f\in F\,\} = \{\,(b,F)\,|\,b\in B\,\} = \{\,b\,|\,b\in B\,\} \times F$
The fibers are $p^{-1}(b) =\{\,b\,\} \times F \cong F$ with the projection $p(b,f)=b$.

#### cianfa72

$\{\,(b,F)\,|\,b\in B\,\}$
That is just a notation where the set $F$ itself appears instead of its elements, I guess
$\{\,b\,|\,b\in B\,\} \times F$
That is basically the union of sets of type $\{\,b\} \times F$ with $b \in B$
$\{\,b\,\} \times F \cong F$
that means an identification (isomorphism ?) with $F$ right ?

Last edited:

#### fresh_42

Mentor
2018 Award
Yes, as soon as $b$ is fixed we could as well drop it. It is only a bijection. But as we talk about sets, no other structures can be expected, so a bijection is already all we can expect.

And yes, the others are just notational differences of the same set.

The crucial point is, that fiber bundles are local direct sums (in a neighborhood of $b$), but not necessarily global.

A simple example of a non-trivial bundle is the Möbius strip. The base $B$ is here $S^{1}$ (the circle line), the fiber $F$ is a closed interval. The corresponding trivial bundle would be a cylinder from which the Möbius strip differs by twisting the fiber.

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