Bundle map

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.

Stronger properties

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

Incomparable properties

 * Etale map
 * Local homeomorphism