Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.
Let represent the entire space, base space and the fiber respectively where are connected. If is a continuous surjection onto the base space, then the structure is said to be a fiber bundle if for every , there exists a neighborhood of such that there exists a homeomorphism such that .
What this basically means is that locally, the fiber bundle looks like the product but globally, it may have different topological properties.
A trivial fiber bundle is a fiber bundle which in which the total space is . In fact, any fiber bundle over a contractible CW Complex is trivial.