Problem
Problem 2.19 Prove that the infinite Möbius strip $M$ (see Problem 1.31) can be considered as the total space of a vector bundle over $S^{1}$. Specifically:
(i) Determine the base space, the fibre and the projection map $\pi$.
(ii) Prove that the vector bundle $\left(M, \pi, S^{1}\right)$ is locally trivial but not trivial.