Exercise 15, § 9

It is said that a topological space X is a Lindelöf space if every open covering of X contains a countable covering of X. Every space admitting a countable basis is a Lindelöf space; every quasi-compact space is a Lindelöf space.

a) Every closed subspace of a Lindelöf space is a Lindelöf space. In order that every subspace of a Lindelöf space be a Lindelöf space, it suffices that every open subspace be a Lindelöf space (cf. exercise.