Example 3

Let X be a separated differentiable manifold locally of finite dimension over $\mathbf{R}$, and let $\widetilde{X}$ be the manifold of orientations of the tangent bundle of X (VAR, R, 10.2.4). The space $\widetilde{X}$, endowed with its canonical projection onto X, is a principal covering of X of group $\{1,-1\}$.