Chapter 4.35, Problem 8E

Prove the following:

Theorem. Let $Y$ be a normal space. Then $Y$ is an absolute retract if and only if $Y$ has the universal extension property.

[Hint: If $X$ and $Y$ are disjoint normal spaces, $A$ is closed in $X$, and $f:A\to Y$ is a continuous map, define the adjunction space $Z_f$ to be quotient space obtained from $X\cup Y$ by identifying each point $a$ of $A$ with the point $f\left(a\right)$ and with all the points of $f^{-1}\left(\left\{f\left(a\right)\right\}\right)$. Using the Tietze theorem, show that $Z_f$ is normal. If $p:X\cup Y\to Z_f$ is the quotient map, show that $p|Y$ is a homeomorphism of $Y$ with a closed subspace of $Z_f$.]