Chapter 4.35, Problem 4E

Let $Z$ be a topological space. If $Y$ is a subspace of $Z$, we say that $Y$ is a retract of $Z$ if there is a continuous map $r:Z\to Y$ such that $r\left(y\right)=y$ for each $y\in Y$.

(a) Show that if $Z$ is Hausdorff and $Y$ is a retract of $Z$, then $Y$ is closed in $Z$.

(b) Let $A$ be a two-point set in ${ℝ}^2$. Show that $A$ is not a retract of ${ℝ}^2$.

(c) Let $S^1$ be the unit circle in ${ℝ}^2$; show that $S^1$ is a retract of ${ℝ}^2-\left\{0\right\}$, where $0$ is the origin. Can you conjecture whether or not $S^1$ is a retract of ${ℝ}^2$?