## Fiber Bundle

A fiber bundle (also called simply a Bundle) with Fiber is a Map where is called the Total Space of the fiber bundle and the Base Space of the fiber bundle. The main condition for the Map to be a fiber bundle is that every point in the Base Space has a Neighborhood such that is Homeomorphic to in a special way. Namely, if

is the Homeomorphism, then

where the Map means projection onto the component. The homeomorphisms which commute with projection'' are called local Trivializations for the fiber bundle . In other words, looks like the product (at least locally), except that the fibers for may be a bit twisted.''

Examples of fiber bundles include any product (which is a bundle over with Fiber ), the Möbius Strip (which is a fiber bundle over the Circle with Fiber given by the unit interval [0,1]; i.e, the Base Space is the Circle), and (which is a bundle over with fiber ). A special class of fiber bundle is the Vector Bundle, in which the Fiber is a Vector Space.