More than once now I’ve read where a specific bundle, over a given smooth manifold base space $M$, has been defined by ‘stating’ it’s sections. Can someone please explain why this is justified? Or if it is not justified, then perhaps explain why?

Moreover, can someone please briefly describe (with loose justification) the ‘extent’ to which sections define a bundle? For instance, if we fix the base space, does specifying the sections *always* specify the bundle?

I think every time I’ve seen this done, it’s been for a line bundle. Is this important, or coincidence? What about a general $k$-vector bundle? Intuitively (and I could be very wrong here…) it seems like specifying all the sections over any bundle (or smooth functions over a manifold) *would* be enough to define it. For example, my intuition with a smooth manifold is that we could choose some coordinate function. Then we could consider the smooth function which projects onto each each coordinate, use these to ‘walk’ over the manifold, and reconstruct it that way. However as a differential geometry novice, I’m not sure this works for manifolds, let alone bundles over manifolds in general.

Please keep in mind I’m new to dealing with bundles. If my question is not well formed then please be patient, comment, and I will fix it.