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.

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.