Problem
Problem 3.9 Prove that if a $C^{\infty}$ manifold $M$ admits an atlas formed by two charts $(U, \varphi),(V, \psi)$, and $U \cap V$ is connected, then $M$ is orientable. Apply this result to the sphere $S^{n}, n>1$, with the atlas formed by the stereographic projections from the poles (see Problem 1.28).