Exercise 5, § 6

(a) Let $X$ be a paracompact topological space. Prove that every principal fibre space with base $X$ and group $\mathbf{R}$ is trivializable.

(b) Let $X$ be the Alexandroff half-line (TG, IV, p. 49, exerc. 12). Construct a principal fibre space with base $X$ and group $\mathbf{R}$ which is not trivializable.