This is in marked contrast with Cartesian products two points <a, b> and <c, d> may be related by the fact that a = c or alternately by b = d. Every point in a fiber bundle is on a definite fiber. In math notation there is a function f: fb -> b, from the bundle to what is called the base space. Two points, x & y, are on the same fiber when f(x) = f(y). A point in the base space corresponds to a fiber. The bundle is not like cooked celery, however, the fibers are glued together. There is another topological space called the fiber space F to which all fibers are homeomorphic.

The inverse image of some open sets in the base space is homeomorphic to the Cartesian product of the fiber space and the open set. This is the glue. If you collect an open covering of the base space then these inverse images cannot in general be merged together to form a homeomorphism of the Cartesian product of the fiber space and the whole base space.

Consider a möbeus strip. It is sort of like the Cartesian product of a circle and line segment, but not quite. It is, however a fiber bundle with the fibers running the short way. They can’t run the long way because they don’t meet up once around the circle. The line segment is the fiber space and the circle is the base space. There is the group of two elements that flips the fiber space end for end. In general there is a topological group on the fiber space which can be used to join together these inverse images.

The example that originally motivated fiber bundles was the vector space on a manifold. Consider the sphere as the manifold. We will consider the vectors tangent to the sphere at each point on the sphere. These are the fiber bundle. Two vectors at the same point are clearly related so the sphere, itself, is the base space. The two dimensional real vector space is the fiber space.