Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my doubt is slightly different
Every definition I've encountered of Vector bundle $(E,\pi, B)$ ($E$ is the total space, $B$ the base one, and $\pi \colon E \to B$ the projection) required that $E$ is a topological space. But, as stated in the linked question, in fact we care only about the topology induced by the trivializations and the local isomorphisms.
So the question is why don't we define a vector bundle as a space whose topology is generated by the image of open sets (contained in $B\times \mathbb{F}^n$) through the local isomorphisms?
Why mathematicians defined $E$ as "only" a Topological space (with a topology that may be finer than the one constructed above)? Is there the risk of the existence of an open that is not the union of open induced as above (and maybe this open can cause trouble in some proofs)?