Exercise 14, § 9

A topological space $X$ is said to be a Lindelöf space if every open covering of $X$ contains a countable covering of $X$. Every space with a countable base is a Lindelöf space, and so is every quasi-compact space.

a) Every closed subspace of a Lindelöf space is a Lindelöf space. For every subspace of a Lindelöf space to be a Lindelöf space it is necessary and sufficient that every open subspace is Lindelöf (cf. Exercise 15).

b) If $f$ is any continuous mapping of a Lindelöf space $X$ into a topological space $X'$, then the subspace $f(X)$ of $X'$ is Lindelöf.

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

d) A Lindelöf space $X$ is quasi-compact if every sequence in $X$ has a cluster point [show that axiom $(C''')$ is then verified].