Exercise 3, § 6

[^1] Let $A$ be a closed subset of the numerical plane $\mathbf{R}^2$ which is homeomorphic to $\mathbf{R}$. (a) Prove that there exists a homeomorphism $g:\mathbf{R}^3\rightarrow \mathbf{R}^3$ such that, for every $t\in \mathbf{R}$, the set $g(A\times \{0\})\cap (\mathbf{R}^2\times \{t\})$ is reduced to an element.

(b) Prove that there exists a homeomorphism $h:\mathbf{R}^3\rightarrow \mathbf{R}^3$ which maps $A\times \{0\}$ onto the right $D =\{(0,0)\} \times \mathbf{R}$.

(c) Prove that $\mathbf{R}^3-$ D is not simply connected.

(d) Prove that $\mathbf{R}^2-$ A is not connected.

[^1]: This exercise and the following one are due to P. H. Doyle, « Plane separation », Proc. Camb. Phil. Soc. 64 (1968), p. 291.