Bundle map

This article defines a property of continuous maps between topological spaces

Definition

A surjective continuous map $p:E\to B$ is termed a bundle map or fiber bundle with fiber $F$ (where $F$ is an abstract topological space) if the following is true:

• The fiber at any point is homeomorphic to $F$
• Every point in $B$ has an open neighbourhood $U$ such that the map $p^{-1}(U)\to U$ looks like the projection $U\times F\to U$ (this is called a local triviality condition)

If there is a homeomorphism from $E$ to $B\times F$ under which $p$ gets sent to the projection map, then we say that the bundle map is trivial.

Relation with other properties

Stronger properties

• Covering map: This is a bundle map with discrete fibers

Incomparable properties

• Etale map
• Local homeomorphism