SO(3): The Principal Bundle Structure

In this article, the special orthogonal group SO(3) is considered as a topological group. We show that SO(3) has the structure of a principal SO(2)-bundle over the sphere S2. As a consequence, we prove that every orbit of an SO(3)-action on a topological space is either trivial or homeomorphic to S2. We also introduce a topological atlas on SO(3), by means of its principal bundle structure, and prove that this atlas is smooth.