# Definition:Fiber Bundle

## Definition

Let $M, E, F$ be topological spaces.

Let $\pi: E \to M$ be a continuous surjection.

Let $\UU := \set {U_\alpha \subseteq M: \alpha \in I}$ be an open cover of $M$ with index set $I$.

Let $\pr_{1, \alpha}: U_\alpha \times F \to U_\alpha$ be the first projection on $U_\alpha \times F$.

Let there exist homeomorphisms:

- $\chi_\alpha: \map {\pi^{-1} } {U_\alpha} \to U_\alpha \times F$

such that for all $\alpha \in I$:

- $\pi {\restriction}_{U_\alpha} = \pr_{1, \alpha} \circ \chi_\alpha$

where $\pi {\restriction}_{U_\alpha}$ is the restriction of $\pi$ to $U_\alpha \in \UU$.

Then the ordered tuple $\struct {E, M, \pi, F}$ is called a **fiber bundle over $M$**.

### Total Space

The topological space $E$ is called the **total space of $B$**.

### Base Space

The topological space $M$ is called the **base space of $B$**.

### Bundle Projection

The continuous surjection $\pi: E \to M$ is called the **bundle projection of $B$**.

### Model Fiber

The topological space $F$ is called the **model fiber of $B$**.

### System of Local Trivializations

The set $\set {\struct {U_\alpha, \chi_\alpha}: \alpha \in I}$ is called a **system of local trivializations** of $E$ on $M$.

### Base Point

A point $m \in M$ is called a **base point of $B$**.

## Also known as

Some sources refer to a **fiber bundle** just as a **bundle**.

Some sources use the term **twisted product**.

By an abuse of language, it is common to say that **$E$ is a fiber bundle over $M$**.

One also finds the formulation **Let $E \overset \pi \to M$ be a fiber bundle** in the literature.

## Also see

- Definition:Local Trivialization
- Definition:Transition Mapping
- Definition:Fiber (Relation)
- Definition:Section (Topology)
- Definition:Smooth Fiber Bundle
- Definition:Vector Bundle

- Results about
**fiber bundles**can be found**here**.

## Linguistic Note

In British English, the word **fibre** is used instead of **fiber**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**bundle** - 2003: John M. Lee:
*Introduction to Smooth Manifolds*: $\S 10$: Fiber Bundles - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**bundle**