Exercise 1, § 5

Let $A$ be a closed and discrete subset of $\mathbf{R}^2$.

(a) For $R>0$, let $D_R$ be the set of $z\in \mathbf{C}$ such that $|z|<R$; prove that the fundamental group of $D_R-(A\cap D_R)$ is isomorphic to the free group on $A\cap D_R$.

(b) Prove that there exists a homeomorphism $f$ of $\mathbf{R}^2$ onto itself such that $f(A)\subset \mathbf{N}\times \{0\}$.

(c) Prove that the fundamental group of $\mathbf{R}^2-$ A is isomorphic to F(A).