Exercise 13, § 5

Let $D$ and $D'$ be two distinct lines of $\mathbf{R}^3$, and let $U =\mathbf{R}^3-(D\cup D')$. a) Prove that U is arcwise connected.

b) Suppose that D and $D'$ are disjoint; prove that, for every point $a\in U$, the group $\pi_1(U, a)$ is a free group with two generators.

c) Suppose that D and $D'$ have a common point; prove that, for every point $a\in U$, the group $\pi_1(U, a)$ is a free group with three generators.