Exercise 9, § 10

a) Let $X, Y$ be two topological spaces and $f : X \to Y$ a surjective closed (but not necessarily continuous) mapping. Show that if $\overline{f}^{-1}(y)$ is quasi-compact for each $y \in Y$, then $\overline{f}^{-1}(K)$ is quasi-compact for each quasi-compact subset $K$ of $Y$ (either prove the result directly or else use Exercise 7 of § 5 and Exercise 12 of § 9).

b) Suppose in addition that $X$ is regular. Show that if $\overline{f}^{-1}(y)$ is closed in $X$ for each $y \in Y$, then $\overline{f}^{-1}(K)$ is closed in $X$ for each quasi-compact subset $K$ of $Y$ (same methods). If also $Y$ is locally compact then $f$ is continuous.