In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product  of a space with a group . In the same way as with the Cartesian product, a principal bundle is equipped with

1. An action of on , analogous to for a product space.
2. A projection onto . For a product space, this is just the projection onto the first factor, .

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of . Likewise, there is not generally a projection onto generalizing the projection onto the second factor, which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.