Exercise 16, § 9

b) For every continuous mapping $f$ of a Lindelöf space X into a topological space X’, the subspace $f(X)$ of X’ is a Lindelöf space.

c) Every space which is a countable union of Lindelöf subspaces is a Lindelöf space. In particular, every countable space is a Lindelöf space.

d) In order that a Lindelöf space X be quasi-compact, it suffices that every sequence of points of X admit an adherence value (now prove that the axiom (C'') is then verified).

e) In an accessible Lindelof space X, a closed discrete subspace is necessarily countable, and, a fortiori, X verifies the condition (D_v) of I, p. 90, exerc. 7.

¶ 16) a) Let X be a space in which every open subspace is a Lindelöf space. Prove that X verifies the condition (D_{III}) of exerc. 7 of I, p. 90; if X is regular, every closed subset of X is a countable intersection of open sets (cf. I, p. 109, exerc. 24 c)).

b) Let X be a space in which every open subspace is paracompact. In order that every open subspace of X be a Lindelöf space, it is necessary and sufficient that X verify the condition (D_{III}) of exerc. 7 of I, p. 90.

c) In order that a compact space X be such that every open subspace of X be a Lindelöf space, it is necessary and sufficient that it verify the condition (D_{III}) of exerc. 7 of I, p. 90, and that every closed subset of X be a countable intersection of open sets (use b) and Theorem 5 of I, p. 70).