(a) $h:Y_0\to Y$ is a homeomorphism, it can be extended to a continuous $f:Z\to Y$ , then $h^{-1}\circ f$ is a retraction.(b) So we assume $K=[0,1]^J$ is compact and Hausdorff, i.e. normal. If we show that $Y$ is homeomorphic to a closed subspace $S$ of $K$ then, since $Y$ is an absolute retract, $S$ is a retract of $K$ , and, similar to 5(a)(b), $Y$ has the universal extension property. The Imbedding theorem provides a way to construct $F$ which is an imbedding. For each $(x,U)$ we define $f_{(x,U)}$ that equals 1 at $x$ and vanishes outside of $U$ . If $t\in[0,1]^J$ is not in the image, then for every $y\in Y$ there is some $(x,U)$ such that $f_{(x,U)}(y)\neq t_{(x,U)}$ . Choose a disjoint neighborhoods $V_y$ for $f(y)$ and $U_y$ for $t$ . The preimages of $V_y$ cover $Y$ which is compact, there is a finite subcovering. The corresponding finite intersection of $U_y$ is a neighborhood of $t$ disjoint from the image. (In other words, the continuous image of a compact is compact and closed in a Hausdorff space.)