Vector bundle definition

Question

Is the condition

$$ \pi \circ \varphi (x,v) = x $$

in the definition of a vector bundle needed? In Milnor/Stasheff "Characteristic classes" the definition is given without it.

Answer 1

It's not strictly necessary, because it is a consequence of the following

the map $v \mapsto \varphi (x, v)$ is an isomorphism between the vector spaces $\mathbb{R}^k$ and $\pi^{−1}(\{x\})$.

However, for a site like wikipedia, it is preferable to state the fact plainly instead of relying on all readers to deduce it.

Answer 2

You do need it. Adapted to Wikipedia's notation, for each $x \in U$ Milnor-Stasheff require $v \mapsto \varphi(x, v)$ to induce an isomorphism between $\mathbb{R}^k$ and $\pi^{-1}(x)$, so in particular $\varphi(x, v)$ really has to lie in the fibre.

Answer 3

Of course it's in Milnor-Stasheff's definition:

For each $b\in U$, the correspondence $x\mapsto h(b,x)$ defines an isomorphism between the vector space $\mathbb R^n$ and the vector space $\pi^{-1}(b)$.