Exercise 16, § 5

Let A be a subset of $\mathbf{S}_2$ which is homeomorphic to $\mathbf{S}_1$. Let $a$ and $b$ be distinct points of A, denote by $A'$ and $A''$ the connected components of A$-\{a, b\}$ and put $X =\mathbf{S}_2-\{a, b\},U = X-A'$ and $V = X-A''$.

(a) Prove that U and V are connected. (Use Exercise 15.)

(b) Prove that, for every point $x\in U$, the homomorphism of $\pi_1(U, x)$ into $\pi_1(X, x)$ induced by the inclusion of U in X is trivial.

(c) Prove that $\mathbf{S}_2-$ A has exactly two connected components and that their boundary is equal to A. (d) Let S be a subset of $\mathbf{R}^2$ which is homeomorphic to $\mathbf{S}_1$; prove that $\mathbf{R}^2-$ S has exactly two connected components and that their boundary is equal to S (Jordan's theorem).