Exercise 29, § 9

(a) Let K be the discrete space $\{0, 1\}$. For every compact space X, let F be the set of mappings $f$ from K into the set $\mathfrak{F}(X)$ of closed subsets of X, such that $X = f(0) \cup f(1)$. Consider the compact space $Y = K^F$; now prove that for every $y = (y_f)_{f \in F}$ in Y, the intersection of the closed sets $f(y_f)$ of X is empty or reduced to one point. The subset Z of the $y \in Y$ such that this intersection is not empty is a closed subset of Y; if $\varphi(y)$ is the unique common point of the $f(y_f)$ for $y \in Z$, now prove that $\varphi$ is a continuous surjective mapping from Z onto X.

(b) Give an example of a compact space X such that there exists no continuous surjective mapping from a compact space of the form $K^A$ onto X (use exerc. 4 b) of I, p. 95 and exerc. 27 of I, p. 110).