Fibered Spaces

Abstract

Given a map p from a space E onto a space B, we can partition the domain into disjoint sets F _b = p ⁻¹ (b), for b ∈ B. (Recall that p ⁻¹(b) is the inverse image of b, that is, the set of points of E that are mapped to b.) We say that p defines a fibration if all the sets F _b are homeomorphic to one another. In this case F _b is called the fiber over b; the space B is called the base space of the fibration, E is the total space, and p is the projection. If the fibers of a fibration are homeomorphic to a space F, we say that F is the model fiber, or simply the fiber. A fibration with total space E, base space B, fiber F and projection p is denoted by (E,B,F,p), or simply by (E,B,F).