- is surjective
- For any point , the fiber , given the subspace topology from , is homeomorphic to .
- For any point , there exists an open subset of such that , and such that there exists a homeomorphism (where is endowed with the product topology) with the property that coincides with the projection map from to . In other words, the bundle is locally trivial, i.e., locally like a product space.
We use the following terminology:
- The space is termed the fiber space or fiber or fiber type.
- The space is termed the total space.
- The space is termed the base space.
The letters are typical in this context.
A special case of a fiber bundle is the trivial fiber bundle, where with being the projection onto the first coordinate. We can also think of this as being globally trivial.