Exercise 4, § 6

a) Let $A$ be a closed subset of the sphere $\mathbf{S}_2$ which is homeomorphic to a circle $\mathbf{S}_1$. Prove that $\mathbf{S}_2-$ A is not connected.

b) Let $A$ be a closed subset of the numerical plane $\mathbf{R}^2$ which is homeomorphic to a circle $\mathbf{S}_1$. Prove that $\mathbf{R}^2-$ A is not connected.