# Context/history on fibers, bundles and trivilizations in topology/differential geometry

This is a soft-question in the sense that I have no confusion over the definitions of a bundle, fiber (see: Wikipedia), trivializing open sets and/or a locally trivial rank of $$r$$ (below are definitions for the latter two)

A surjective smooth map $$\pi:E\to M$$ of manifoldsis said to be locally trivial of rank $$r$$ if i.) each fiber $$\pi^{-1}(p)$$ has a structure of a vector space of dimension $$r$$, ii.) for each $$p \in M$$ there are open sets $$U$$ of $$p$$ and a fiber-preserving diffeomorphism $$\phi:\pi^{-1}(U)\to U\times \mathbb{R}^r$$ s.t. $$\forall q \in U$$ the restriction $$\phi\bigg|_{\pi^{-1}(q)}:\pi^{-1}(q)\to \{q\}\times\mathbb{R}^r$$ is a vector space isomorphism. Such an open set $$U$$ is called a trivializing open set for $$E$$, and $$\phi$$ the trivialization of $$E$$ over $$U$$

But what I am interested in hearing is that what was/is the motivation i.) for the fibers/bundles/trivializations, ii.) to name the constructs the way they were? As a novice to both topology and differential geometry, the fibers/bundles etc. seem to come out of nowhere, so thinking the bigger picture, and the why are we doing this is rather difficult.

• Are you only looking for the history? I would say that the naming of these objects are quite intuitive: A fiber bundle bundles fibers (a fiber is the preimage of a map at one point) in a continuous way and a trivialization makes the this bundling trivial in a small region. Feb 6, 2022 at 14:02