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.

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