---
tag: CX4M8K61AB
kind: concept
name: Fiber bundle
msc: [55R10, 57R22]
sources:
  - url: https://en.wikipedia.org/wiki/Fiber_bundle
same_as:
  wikidata: Q1154996
  wikipedia: Fiber bundle
---

## Summary

A fiber bundle is a space that looks locally like a product of a base space and a fixed fiber, but may be twisted globally.
It comes with a projection to the base whose preimages, the fibers, are all copies of the same space.
Vector bundles, covering spaces, principal bundles and the Hopf fibration are all fiber bundles.
They describe families of spaces parametrized by a base, and they are central in topology, geometry and physics.

## Explanations

### L1 human

Take a strip of paper: it is a product of a long segment and a short one.
Give it a half twist before gluing the ends and you get a Möbius band.
Over each small piece of the central circle the band still looks like a product of an arc and a segment, but as a whole it is twisted.
The Möbius band is a fiber bundle over the circle with fiber a segment, and the plain cylinder is the untwisted one.