Theorem
Every closed subspace of a Lindelof space is a Lindelof space.
Proof
Supposeis a closed subspace of a Lindelof space
and
is an open cover of
Each
for some
open in
Sinceis closed,
is open in
hence
is an open cover of
There is a countable subcoverso
is a countable cover of
Henceis a countable open subcover of
and
is a Lindelof space.