Exercise 15, § 9

a) Let $X$ be a topological space in which every open subspace is Lindelöf. Show that $X$ satisfies condition $(D_{III})$ of § 1, Exercise 7, and that if $X$ is regular then every closed subset of $X$ is a countable intersection of open sets [cf. Exercise 23 c)].

b) Let $X$ be a topological space in which every open subspace is paracompact. Show that every open subspace of $X$ is Lindelöf if and only if $X$ satisfies condition $(D_{III})$ of § 1, Exercise 7.

c) Let $X$ be a compact space. Show that every open subspace of $X$ is Lindelöf if and only if (i) $X$ satisfies condition $(D_{III})$ of § 1, Exercise 7, and (ii) every closed subset of $X$ is a countable intersection of open sets (use b) and Theorem 5 of no. 10).