# Fiber bundle

From Topospaces

## Definition

Suppose are topological spaces. A continuous map is termed a **fiber bundle** with fiber if:

- is surjective
- For any point , the fiber , given the subspace topology from , is homeomorphic to .
- For any point , there exists an open subset of such that , and such that there exists a homeomorphism (where is endowed with the product topology) with the property that coincides with the projection map from to . In other words, the bundle is
*locally trivial*, i.e.,*locally like a product space*.

We use the following terminology:

- The space is termed the
*fiber space*or*fiber*or*fiber type*. - The space is termed the
*total space*. - The space is termed the
*base space*.

The letters are typical in this context.

## Particular cases

### Product bundle

A special case of a fiber bundle is the *trivial* fiber bundle, where with being the projection onto the first coordinate. We can also think of this as being *globally trivial.*

### Covering map

When is a discrete space, then a fiber bundle with fiber is a covering map. In fact, a covering map can be defined as a fiber bundle with discrete fiber.