Exercise 22, § 10

(a) Let X be an accessible space, $f : X \to Y$ a continuous closed mapping, such that for every $y \in f(X)$, the boundary of $f^{-1}(y)$ in X is quasi-compact. For every $y \in f(X)$, let $L(y)$ be the interior of $f^{-1}(y)$ in X if the boundary of $f^{-1}(y)$ is not empty, and the complement in $f^{-1}(y)$ of a point of this set if the boundary of $f^{-1}(y)$ is empty. Now prove that if F is the complement of the union of the $L(y)$, the restriction of $f$ to F is a proper mapping of F into Y.

(b) Let $X_i$ ($i = 1, 2$) be two accessible spaces, $f_i : X_i \to Y_i$ two continuous, closed and surjective mappings ($i = 1, 2$). Suppose in addition that for every $y_i \in Y_i$, the boundary of $f_i^{-1}(y_i)$ in $X_i$ is quasi-compact ($i = 1, 2$). Now prove that if

the bijection $(X_1 \times X_2)/R \to Y_1 \times Y_2$ associated with $f$ (E, II, p. 44) is a homeomorphism (use (a)). Give an example where $f_1 \times f_2$ is not closed.