(a) The quotient space of $A=\mathbb{R}^2-\{(0,0)\}$ obtained by identifying points with the same distance to the origin is homeomorphic to $\mathbb{R}$ and to $C'=C-\{(0,0)\}$ . This gives us a continuous function $f'$ from $A$ to $C'$ . The explicit expression for the function will be something like this: $f'(x,y)=(e^t\cos t,e^t\sin t)$ where $t=\sqrt{x^2+y^2}$ . Every sequence of points converging to the origin maps to a sequence converging to the origin as well, therefore, we can extend $f'$ to a continuous function $f:\mathbb{R}^2\to C$ .(b) $K$ is homeomorphic to $\mathbb{R}$ , which is normal. Therefore, using the two previous exercises, it has the universal extension property and is an absolute retract. Being closed in $\mathbb{R}^3$ , it is a retract of $\mathbb{R}^3$ .