Exercise 29, § 5

Let A be a closed subset of $\mathbf{R}^2$ homeomorphic to a closed subset B of $\mathbf{R}$.

(a) Prove that the subspaces $A\times \{0\}$ and $\{(0,0)\} \times B$ of $\mathbf{R}^3$ are isotopic.

(b) Suppose that B $=\not\mathbf{R}$; prove that $\mathbf{R}^2-$ A is arcwise connected. (c) Suppose that $B =\mathbf{R}$; prove that $\mathbf{R}^2-$ A has exactly two arcwise connected components and that A is the boundary of each of them. (d) Let A be a subset of $\mathbf{S}_2$ homeomorphic to $\mathbf{S}_1$; prove that $\mathbf{S}_2-$ A has exactly two arcwise connected components and that A is the boundary of each of them.

(e) Let A be a subset of $\mathbf{R}^2$ homeomorphic to $\mathbf{S}_1$; prove that $\mathbf{R}^2-$ A has exactly two arcwise connected components and that A is the boundary of each of them.