Fiber bundle

Definition

Suppose $E, B, F$ are topological spaces. A continuous map $p : E \to B$ is termed a fiber bundle with fiber $F$ if:

$p$ is surjective
For any point $b \in B$ , the fiber $p^{- 1} ({b})$ , given the subspace topology from $E$ , is homeomorphic to $F$ .
For any point $b \in B$ , there exists an open subset $U$ of $B$ such that $b \in U$ , and such that there exists a homeomorphism $φ : U \times F \to p^{- 1} (U)$ (where $U \times F$ is endowed with the product topology) with the property that $p \circ φ$ coincides with the projection map from $U \times F$ to $U$ . In other words, the bundle is locally trivial, i.e., locally like a product space.

We use the following terminology:

The space $F$ is termed the fiber space or fiber or fiber type.
The space $E$ is termed the total space.
The space $B$ is termed the base space.

The letters $F, E, B$ are typical in this context.

Particular cases

Product bundle

A special case of a fiber bundle is the trivial fiber bundle, where $E = B \times F$ with $p : E \to B$ being the projection onto the first coordinate. We can also think of this as being globally trivial.

Covering map

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