From Wikipedia:
A fiber bundle consists of the data $(E, B, π, F)$, where $E, B, $and $F$ are topological spaces and $π : E → B$ is a continuous surjection satisfying a local triviality condition outlined below:
for every $x$ in $E$, there is an open neighborhood $U ⊂ B$ of $π(x)$ (which will be called a trivializing neighborhood) such that $π^{-1}(U)$ is homeomorphic to the product space U × F, in such a way that $π$ carries over to the projection onto the first factor.
I was wondering why the local triviality condition (the second paragraph) is initiated from "every $x$ in $E$"? In other words, can it be instead initiated from $B$ as follows:
there is an open cover of $B$ such that each open subset in the cover is homeomorphic to the product space U × F, in such a way that $π$ carries over to the projection onto the first factor.
- What does "the first factor" mean?
- Generally, what does "a mapping carries over to another mapping onto another thing" mean?
Thanks and regards!