Exercise 28, § 9

a) Let $K$ be the discrete space $\{0, 1\}$. If $X$ is any compact space, let $F$ be the set of mappings $f$ of $K$ into the set $\mathfrak{F}(X)$ of closed subsets of $X$ such that $X = f(0) \cup f(1)$. Consider the compact space $Y = K^F$; show that for each $y = (y_f)_{f \in F}$ in $Y$, the intersection of the closed sets $f(y_f)$ of $X$ consists of at most one point. The set $Z$ of $y \in Y$ for which this intersection is not empty is a closed subset of $Y$. If $\varphi(y)$ is the unique point common to the $f(y_f)$ for $y \in Z$, show that $\varphi$ is a continuous surjection of $Z$ onto $X$.

b) Give an example of a compact space $X$ such that there is no surjective continuous mapping of a compact space of the form $K^A$ onto $X$ [use Exercise 4 b) of § 4 and Exercise 26 of § 9].