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

  1. An action of $G$ on $P$, analogous to $(x,g)h = (x, gh)$ for a product space.
  2. A projection onto $X$. For a product space, this is just the projection onto the first factor, $(x,g) \to x$.

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of $(x,e)$. Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, $X \times G \to G$ 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 (Wikipedia).

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