Trivial fiber bundle vs product space

[cianfa72]

TL;DR

Is the trivial fiber bundle of topological spaces the same as the product space ?

I've a doubt about the following: take the trivial fiber bundle of the base space $B$ and the fiber $F$ vs the product $B \times F$ of topological spaces $B$ and $F$.

Are they really the same ?

As far as I can tell, in the trivial fiber bundle each fiber in the bundle in a distinct/separated copy of the topological space called "the fiber space" $F$, whereas in the product $B \times F$ it is the same copy of $F$ that, let me say, enters multiple times in the product (i.e. all the copies of $F$ are actually identified).

Then, it's true that they are canonically identified, but that's another story....

 

[martinbn]

What is the definition of a trivial bundle that you have seen?

 

[cianfa72]

What is the definition of a trivial bundle that you have seen?

https://en.wikipedia.org/wiki/Fiber_bundle#Trivial_bundle

I'd add that my doubt comes from this picture of $\mathbb E^1 \times \mathbb E^3$ in Penrose book:

 

[martinbn]

https://en.wikipedia.org/wiki/Fiber_bundle#Trivial_bundle

Seems quite clear and unambiguous.

I'd add that my doubt comes from this picture of $\mathbb E^1 \times \mathbb E^3$ in Penrose book:

What is your doubt?