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 $\varphi: U \times F \to p^{-1}(U)$ (where $U \times F$ is endowed with the product topology) with the property that $p \circ \varphi$ 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.